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Theorem cantnflt 8452

Description: An upper bound on the partial sums of the CNF function. Since each term dominates all previous terms, by induction we can bound the whole sum with any exponent 𝐴 ↑_𝑜 𝐶 where 𝐶 is larger than any exponent (𝐺‘𝑥), 𝑥 ∈ 𝐾 which has been summed so far. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.)

**Hypotheses**
Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
cantnfcl.g	⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
cantnfcl.f	⊢ (𝜑 → 𝐹 ∈ 𝑆)
cantnfval.h	⊢ 𝐻 = seq_𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_𝑜 (𝐺‘𝑘)) ·_𝑜 (𝐹‘(𝐺‘𝑘))) +_𝑜 𝑧)), ∅)
cantnflt.a	⊢ (𝜑 → ∅ ∈ 𝐴)
cantnflt.k	⊢ (𝜑 → 𝐾 ∈ suc dom 𝐺)
cantnflt.c	⊢ (𝜑 → 𝐶 ∈ On)
cantnflt.s	⊢ (𝜑 → (𝐺 “ 𝐾) ⊆ 𝐶)

**Assertion**
Ref	Expression
cantnflt	⊢ (𝜑 → (𝐻‘𝐾) ∈ (𝐴 ↑_𝑜 𝐶))

Distinct variable groups: 𝑧,𝑘,𝐵 𝑧,𝐶 𝐴,𝑘,𝑧 𝑘,𝐹,𝑧 𝑆,𝑘,𝑧 𝑘,𝐺,𝑧 𝑘,𝐾,𝑧 𝜑,𝑘,𝑧 Allowed substitution hints: 𝐶(𝑘) 𝐻(𝑧,𝑘)

Proof of Theorem cantnflt Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfs.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ On)
2		cantnflt.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ On)
3		cantnflt.a	. . . 4 ⊢ (𝜑 → ∅ ∈ 𝐴)
4		oen0 7553	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑_𝑜 𝐶))
5	1, 2, 3, 4	syl21anc 1317	. . 3 ⊢ (𝜑 → ∅ ∈ (𝐴 ↑_𝑜 𝐶))
6		fveq2 6103	. . . . 5 ⊢ (𝐾 = ∅ → (𝐻‘𝐾) = (𝐻‘∅))
7		0ex 4718	. . . . . 6 ⊢ ∅ ∈ V
8		cantnfval.h	. . . . . . 7 ⊢ 𝐻 = seq_𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_𝑜 (𝐺‘𝑘)) ·_𝑜 (𝐹‘(𝐺‘𝑘))) +_𝑜 𝑧)), ∅)
9	8	seqom0g 7438	. . . . . 6 ⊢ (∅ ∈ V → (𝐻‘∅) = ∅)
10	7, 9	ax-mp 5	. . . . 5 ⊢ (𝐻‘∅) = ∅
11	6, 10	syl6eq 2660	. . . 4 ⊢ (𝐾 = ∅ → (𝐻‘𝐾) = ∅)
12	11	eleq1d 2672	. . 3 ⊢ (𝐾 = ∅ → ((𝐻‘𝐾) ∈ (𝐴 ↑_𝑜 𝐶) ↔ ∅ ∈ (𝐴 ↑_𝑜 𝐶)))
13	5, 12	syl5ibrcom 236	. 2 ⊢ (𝜑 → (𝐾 = ∅ → (𝐻‘𝐾) ∈ (𝐴 ↑_𝑜 𝐶)))
14	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → 𝐶 ∈ On)
15		eloni 5650	. . . . . . 7 ⊢ (𝐶 ∈ On → Ord 𝐶)
16	14, 15	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → Ord 𝐶)
17		cantnflt.s	. . . . . . . 8 ⊢ (𝜑 → (𝐺 “ 𝐾) ⊆ 𝐶)
18	17	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → (𝐺 “ 𝐾) ⊆ 𝐶)
19		cantnfcl.g	. . . . . . . . . 10 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
20	19	oif 8318	. . . . . . . . 9 ⊢ 𝐺:dom 𝐺⟶(𝐹 supp ∅)
21		ffn 5958	. . . . . . . . 9 ⊢ (𝐺:dom 𝐺⟶(𝐹 supp ∅) → 𝐺 Fn dom 𝐺)
22	20, 21	mp1i 13	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → 𝐺 Fn dom 𝐺)
23		cantnflt.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ suc dom 𝐺)
24	19	oicl 8317	. . . . . . . . . . . . 13 ⊢ Ord dom 𝐺
25		ordsuc 6906	. . . . . . . . . . . . 13 ⊢ (Ord dom 𝐺 ↔ Ord suc dom 𝐺)
26	24, 25	mpbi 219	. . . . . . . . . . . 12 ⊢ Ord suc dom 𝐺
27		ordelon 5664	. . . . . . . . . . . 12 ⊢ ((Ord suc dom 𝐺 ∧ 𝐾 ∈ suc dom 𝐺) → 𝐾 ∈ On)
28	26, 23, 27	sylancr 694	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ On)
29		ordsssuc 5729	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ On ∧ Ord dom 𝐺) → (𝐾 ⊆ dom 𝐺 ↔ 𝐾 ∈ suc dom 𝐺))
30	28, 24, 29	sylancl 693	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾 ⊆ dom 𝐺 ↔ 𝐾 ∈ suc dom 𝐺))
31	23, 30	mpbird 246	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ⊆ dom 𝐺)
32	31	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → 𝐾 ⊆ dom 𝐺)
33		vex 3176	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
34	33	sucid 5721	. . . . . . . . 9 ⊢ 𝑥 ∈ suc 𝑥
35		simprr 792	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → 𝐾 = suc 𝑥)
36	34, 35	syl5eleqr 2695	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → 𝑥 ∈ 𝐾)
37		fnfvima 6400	. . . . . . . 8 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝐾 ⊆ dom 𝐺 ∧ 𝑥 ∈ 𝐾) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐾))
38	22, 32, 36, 37	syl3anc 1318	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐾))
39	18, 38	sseldd 3569	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → (𝐺‘𝑥) ∈ 𝐶)
40		ordsucss 6910	. . . . . 6 ⊢ (Ord 𝐶 → ((𝐺‘𝑥) ∈ 𝐶 → suc (𝐺‘𝑥) ⊆ 𝐶))
41	16, 39, 40	sylc 63	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → suc (𝐺‘𝑥) ⊆ 𝐶)
42		suppssdm 7195	. . . . . . . . . . 11 ⊢ (𝐹 supp ∅) ⊆ dom 𝐹
43		cantnfcl.f	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
44		cantnfs.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
45		cantnfs.b	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ On)
46	44, 1, 45	cantnfs 8446	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
47	43, 46	mpbid 221	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))
48	47	simpld 474	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴)
49		fdm 5964	. . . . . . . . . . . 12 ⊢ (𝐹:𝐵⟶𝐴 → dom 𝐹 = 𝐵)
50	48, 49	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = 𝐵)
51	42, 50	syl5sseq 3616	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐵)
52		onss 6882	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
53	45, 52	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ On)
54	51, 53	sstrd 3578	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ On)
55	54	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → (𝐹 supp ∅) ⊆ On)
56	23	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → 𝐾 ∈ suc dom 𝐺)
57	35, 56	eqeltrrd 2689	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → suc 𝑥 ∈ suc dom 𝐺)
58		ordsucelsuc 6914	. . . . . . . . . . 11 ⊢ (Ord dom 𝐺 → (𝑥 ∈ dom 𝐺 ↔ suc 𝑥 ∈ suc dom 𝐺))
59	24, 58	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝐺 ↔ suc 𝑥 ∈ suc dom 𝐺)
60	57, 59	sylibr 223	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → 𝑥 ∈ dom 𝐺)
61	20	ffvelrni 6266	. . . . . . . . 9 ⊢ (𝑥 ∈ dom 𝐺 → (𝐺‘𝑥) ∈ (𝐹 supp ∅))
62	60, 61	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → (𝐺‘𝑥) ∈ (𝐹 supp ∅))
63	55, 62	sseldd 3569	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → (𝐺‘𝑥) ∈ On)
64		suceloni 6905	. . . . . . 7 ⊢ ((𝐺‘𝑥) ∈ On → suc (𝐺‘𝑥) ∈ On)
65	63, 64	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → suc (𝐺‘𝑥) ∈ On)
66	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → 𝐴 ∈ On)
67	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → ∅ ∈ 𝐴)
68		oewordi 7558	. . . . . 6 ⊢ (((suc (𝐺‘𝑥) ∈ On ∧ 𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (suc (𝐺‘𝑥) ⊆ 𝐶 → (𝐴 ↑_𝑜 suc (𝐺‘𝑥)) ⊆ (𝐴 ↑_𝑜 𝐶)))
69	65, 14, 66, 67, 68	syl31anc 1321	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → (suc (𝐺‘𝑥) ⊆ 𝐶 → (𝐴 ↑_𝑜 suc (𝐺‘𝑥)) ⊆ (𝐴 ↑_𝑜 𝐶)))
70	41, 69	mpd 15	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → (𝐴 ↑_𝑜 suc (𝐺‘𝑥)) ⊆ (𝐴 ↑_𝑜 𝐶))
71	35	fveq2d 6107	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → (𝐻‘𝐾) = (𝐻‘suc 𝑥))
72		simprl 790	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → 𝑥 ∈ ω)
73		simpl 472	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → 𝜑)
74		eleq1 2676	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ∈ dom 𝐺 ↔ ∅ ∈ dom 𝐺))
75		suceq 5707	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
76	75	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐻‘suc 𝑥) = (𝐻‘suc ∅))
77		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝐺‘𝑥) = (𝐺‘∅))
78		suceq 5707	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑥) = (𝐺‘∅) → suc (𝐺‘𝑥) = suc (𝐺‘∅))
79	77, 78	syl 17	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → suc (𝐺‘𝑥) = suc (𝐺‘∅))
80	79	oveq2d 6565	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐴 ↑_𝑜 suc (𝐺‘𝑥)) = (𝐴 ↑_𝑜 suc (𝐺‘∅)))
81	76, 80	eleq12d 2682	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝐻‘suc 𝑥) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑥)) ↔ (𝐻‘suc ∅) ∈ (𝐴 ↑_𝑜 suc (𝐺‘∅))))
82	74, 81	imbi12d 333	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥 ∈ dom 𝐺 → (𝐻‘suc 𝑥) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑥))) ↔ (∅ ∈ dom 𝐺 → (𝐻‘suc ∅) ∈ (𝐴 ↑_𝑜 suc (𝐺‘∅)))))
83		eleq1 2676	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ dom 𝐺 ↔ 𝑦 ∈ dom 𝐺))
84		suceq 5707	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
85	84	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐻‘suc 𝑥) = (𝐻‘suc 𝑦))
86		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐺‘𝑥) = (𝐺‘𝑦))
87		suceq 5707	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑥) = (𝐺‘𝑦) → suc (𝐺‘𝑥) = suc (𝐺‘𝑦))
88	86, 87	syl 17	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → suc (𝐺‘𝑥) = suc (𝐺‘𝑦))
89	88	oveq2d 6565	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 ↑_𝑜 suc (𝐺‘𝑥)) = (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))
90	85, 89	eleq12d 2682	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐻‘suc 𝑥) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑥)) ↔ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦))))
91	83, 90	imbi12d 333	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝐺 → (𝐻‘suc 𝑥) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑥))) ↔ (𝑦 ∈ dom 𝐺 → (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))))
92		eleq1 2676	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ dom 𝐺 ↔ suc 𝑦 ∈ dom 𝐺))
93		suceq 5707	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
94	93	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐻‘suc 𝑥) = (𝐻‘suc suc 𝑦))
95		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → (𝐺‘𝑥) = (𝐺‘suc 𝑦))
96		suceq 5707	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑥) = (𝐺‘suc 𝑦) → suc (𝐺‘𝑥) = suc (𝐺‘suc 𝑦))
97	95, 96	syl 17	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → suc (𝐺‘𝑥) = suc (𝐺‘suc 𝑦))
98	97	oveq2d 6565	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑_𝑜 suc (𝐺‘𝑥)) = (𝐴 ↑_𝑜 suc (𝐺‘suc 𝑦)))
99	94, 98	eleq12d 2682	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → ((𝐻‘suc 𝑥) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑥)) ↔ (𝐻‘suc suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘suc 𝑦))))
100	92, 99	imbi12d 333	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ∈ dom 𝐺 → (𝐻‘suc 𝑥) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑥))) ↔ (suc 𝑦 ∈ dom 𝐺 → (𝐻‘suc suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘suc 𝑦)))))
101	48	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → 𝐹:𝐵⟶𝐴)
102	20	ffvelrni 6266	. . . . . . . . . . . 12 ⊢ (∅ ∈ dom 𝐺 → (𝐺‘∅) ∈ (𝐹 supp ∅))
103	51	sselda 3568	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐺‘∅) ∈ (𝐹 supp ∅)) → (𝐺‘∅) ∈ 𝐵)
104	102, 103	sylan2 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝐺‘∅) ∈ 𝐵)
105	101, 104	ffvelrnd 6268	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝐹‘(𝐺‘∅)) ∈ 𝐴)
106	1	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → 𝐴 ∈ On)
107		onelon 5665	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝐹‘(𝐺‘∅)) ∈ 𝐴) → (𝐹‘(𝐺‘∅)) ∈ On)
108	106, 105, 107	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝐹‘(𝐺‘∅)) ∈ On)
109	54	sselda 3568	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐺‘∅) ∈ (𝐹 supp ∅)) → (𝐺‘∅) ∈ On)
110	102, 109	sylan2 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝐺‘∅) ∈ On)
111		oecl 7504	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝐺‘∅) ∈ On) → (𝐴 ↑_𝑜 (𝐺‘∅)) ∈ On)
112	106, 110, 111	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝐴 ↑_𝑜 (𝐺‘∅)) ∈ On)
113	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ 𝐴)
114		oen0 7553	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝐺‘∅) ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑_𝑜 (𝐺‘∅)))
115	106, 110, 113, 114	syl21anc 1317	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ (𝐴 ↑_𝑜 (𝐺‘∅)))
116		omord2 7534	. . . . . . . . . . 11 ⊢ ((((𝐹‘(𝐺‘∅)) ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ↑_𝑜 (𝐺‘∅)) ∈ On) ∧ ∅ ∈ (𝐴 ↑_𝑜 (𝐺‘∅))) → ((𝐹‘(𝐺‘∅)) ∈ 𝐴 ↔ ((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))) ∈ ((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 𝐴)))
117	108, 106, 112, 115, 116	syl31anc 1321	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → ((𝐹‘(𝐺‘∅)) ∈ 𝐴 ↔ ((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))) ∈ ((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 𝐴)))
118	105, 117	mpbid 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → ((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))) ∈ ((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 𝐴))
119		peano1 6977	. . . . . . . . . . . 12 ⊢ ∅ ∈ ω
120	119	a1i 11	. . . . . . . . . . 11 ⊢ (∅ ∈ dom 𝐺 → ∅ ∈ ω)
121	44, 1, 45, 19, 43, 8	cantnfsuc 8450	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∅ ∈ ω) → (𝐻‘suc ∅) = (((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))) +_𝑜 (𝐻‘∅)))
122	120, 121	sylan2 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝐻‘suc ∅) = (((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))) +_𝑜 (𝐻‘∅)))
123	10	oveq2i 6560	. . . . . . . . . . 11 ⊢ (((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))) +_𝑜 (𝐻‘∅)) = (((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))) +_𝑜 ∅)
124		omcl 7503	. . . . . . . . . . . . 13 ⊢ (((𝐴 ↑_𝑜 (𝐺‘∅)) ∈ On ∧ (𝐹‘(𝐺‘∅)) ∈ On) → ((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))) ∈ On)
125	112, 108, 124	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → ((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))) ∈ On)
126		oa0 7483	. . . . . . . . . . . 12 ⊢ (((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))) ∈ On → (((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))) +_𝑜 ∅) = ((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))))
127	125, 126	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))) +_𝑜 ∅) = ((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))))
128	123, 127	syl5eq 2656	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))) +_𝑜 (𝐻‘∅)) = ((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))))
129	122, 128	eqtrd 2644	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝐻‘suc ∅) = ((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))))
130		oesuc 7494	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝐺‘∅) ∈ On) → (𝐴 ↑_𝑜 suc (𝐺‘∅)) = ((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 𝐴))
131	106, 110, 130	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝐴 ↑_𝑜 suc (𝐺‘∅)) = ((𝐴 ↑_𝑜 (𝐺‘∅)) ·_𝑜 𝐴))
132	118, 129, 131	3eltr4d 2703	. . . . . . . 8 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝐻‘suc ∅) ∈ (𝐴 ↑_𝑜 suc (𝐺‘∅)))
133	132	ex 449	. . . . . . 7 ⊢ (𝜑 → (∅ ∈ dom 𝐺 → (𝐻‘suc ∅) ∈ (𝐴 ↑_𝑜 suc (𝐺‘∅))))
134		ordtr 5654	. . . . . . . . . . . 12 ⊢ (Ord dom 𝐺 → Tr dom 𝐺)
135	24, 134	ax-mp 5	. . . . . . . . . . 11 ⊢ Tr dom 𝐺
136		trsuc 5727	. . . . . . . . . . 11 ⊢ ((Tr dom 𝐺 ∧ suc 𝑦 ∈ dom 𝐺) → 𝑦 ∈ dom 𝐺)
137	135, 136	mpan 702	. . . . . . . . . 10 ⊢ (suc 𝑦 ∈ dom 𝐺 → 𝑦 ∈ dom 𝐺)
138	137	imim1i 61	. . . . . . . . 9 ⊢ ((𝑦 ∈ dom 𝐺 → (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦))) → (suc 𝑦 ∈ dom 𝐺 → (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦))))
139	1	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → 𝐴 ∈ On)
140		eloni 5650	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On → Ord 𝐴)
141	139, 140	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → Ord 𝐴)
142	48	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → 𝐹:𝐵⟶𝐴)
143	51	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐹 supp ∅) ⊆ 𝐵)
144	20	ffvelrni 6266	. . . . . . . . . . . . . . . . . 18 ⊢ (suc 𝑦 ∈ dom 𝐺 → (𝐺‘suc 𝑦) ∈ (𝐹 supp ∅))
145	144	ad2antrl 760	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐺‘suc 𝑦) ∈ (𝐹 supp ∅))
146	143, 145	sseldd 3569	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐺‘suc 𝑦) ∈ 𝐵)
147	142, 146	ffvelrnd 6268	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐹‘(𝐺‘suc 𝑦)) ∈ 𝐴)
148		ordsucss 6910	. . . . . . . . . . . . . . 15 ⊢ (Ord 𝐴 → ((𝐹‘(𝐺‘suc 𝑦)) ∈ 𝐴 → suc (𝐹‘(𝐺‘suc 𝑦)) ⊆ 𝐴))
149	141, 147, 148	sylc 63	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → suc (𝐹‘(𝐺‘suc 𝑦)) ⊆ 𝐴)
150		onelon 5665	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ (𝐹‘(𝐺‘suc 𝑦)) ∈ 𝐴) → (𝐹‘(𝐺‘suc 𝑦)) ∈ On)
151	139, 147, 150	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐹‘(𝐺‘suc 𝑦)) ∈ On)
152		suceloni 6905	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘(𝐺‘suc 𝑦)) ∈ On → suc (𝐹‘(𝐺‘suc 𝑦)) ∈ On)
153	151, 152	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → suc (𝐹‘(𝐺‘suc 𝑦)) ∈ On)
154	54	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐹 supp ∅) ⊆ On)
155	154, 145	sseldd 3569	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐺‘suc 𝑦) ∈ On)
156		oecl 7504	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ (𝐺‘suc 𝑦) ∈ On) → (𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ∈ On)
157	139, 155, 156	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ∈ On)
158		omwordi 7538	. . . . . . . . . . . . . . 15 ⊢ ((suc (𝐹‘(𝐺‘suc 𝑦)) ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ∈ On) → (suc (𝐹‘(𝐺‘suc 𝑦)) ⊆ 𝐴 → ((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 suc (𝐹‘(𝐺‘suc 𝑦))) ⊆ ((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 𝐴)))
159	153, 139, 157, 158	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (suc (𝐹‘(𝐺‘suc 𝑦)) ⊆ 𝐴 → ((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 suc (𝐹‘(𝐺‘suc 𝑦))) ⊆ ((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 𝐴)))
160	149, 159	mpd 15	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → ((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 suc (𝐹‘(𝐺‘suc 𝑦))) ⊆ ((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 𝐴))
161		oesuc 7494	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ (𝐺‘suc 𝑦) ∈ On) → (𝐴 ↑_𝑜 suc (𝐺‘suc 𝑦)) = ((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 𝐴))
162	139, 155, 161	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐴 ↑_𝑜 suc (𝐺‘suc 𝑦)) = ((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 𝐴))
163	160, 162	sseqtr4d 3605	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → ((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 suc (𝐹‘(𝐺‘suc 𝑦))) ⊆ (𝐴 ↑_𝑜 suc (𝐺‘suc 𝑦)))
164		eloni 5650	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺‘suc 𝑦) ∈ On → Ord (𝐺‘suc 𝑦))
165	155, 164	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → Ord (𝐺‘suc 𝑦))
166		vex 3176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑦 ∈ V
167	166	sucid 5721	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑦 ∈ suc 𝑦
168	166	sucex 6903	. . . . . . . . . . . . . . . . . . . . 21 ⊢ suc 𝑦 ∈ V
169	168	epelc 4951	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 E suc 𝑦 ↔ 𝑦 ∈ suc 𝑦)
170	167, 169	mpbir 220	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑦 E suc 𝑦
171	45, 51	ssexd 4733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V)
172	44, 1, 45, 19, 43	cantnfcl 8447	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
173	172	simpld 474	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → E We (𝐹 supp ∅))
174	19	oiiso 8325	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
175	171, 173, 174	syl2anc 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
176	175	ad2antrr 758	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
177	137	ad2antrl 760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → 𝑦 ∈ dom 𝐺)
178		simprl 790	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → suc 𝑦 ∈ dom 𝐺)
179		isorel 6476	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)) ∧ (𝑦 ∈ dom 𝐺 ∧ suc 𝑦 ∈ dom 𝐺)) → (𝑦 E suc 𝑦 ↔ (𝐺‘𝑦) E (𝐺‘suc 𝑦)))
180	176, 177, 178, 179	syl12anc 1316	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝑦 E suc 𝑦 ↔ (𝐺‘𝑦) E (𝐺‘suc 𝑦)))
181	170, 180	mpbii 222	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐺‘𝑦) E (𝐺‘suc 𝑦))
182		fvex 6113	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺‘suc 𝑦) ∈ V
183	182	epelc 4951	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺‘𝑦) E (𝐺‘suc 𝑦) ↔ (𝐺‘𝑦) ∈ (𝐺‘suc 𝑦))
184	181, 183	sylib 207	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐺‘𝑦) ∈ (𝐺‘suc 𝑦))
185		ordsucss 6910	. . . . . . . . . . . . . . . . 17 ⊢ (Ord (𝐺‘suc 𝑦) → ((𝐺‘𝑦) ∈ (𝐺‘suc 𝑦) → suc (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)))
186	165, 184, 185	sylc 63	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → suc (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦))
187	20	ffvelrni 6266	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ dom 𝐺 → (𝐺‘𝑦) ∈ (𝐹 supp ∅))
188	177, 187	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐺‘𝑦) ∈ (𝐹 supp ∅))
189	154, 188	sseldd 3569	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐺‘𝑦) ∈ On)
190		suceloni 6905	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺‘𝑦) ∈ On → suc (𝐺‘𝑦) ∈ On)
191	189, 190	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → suc (𝐺‘𝑦) ∈ On)
192	3	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → ∅ ∈ 𝐴)
193		oewordi 7558	. . . . . . . . . . . . . . . . 17 ⊢ (((suc (𝐺‘𝑦) ∈ On ∧ (𝐺‘suc 𝑦) ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (suc (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦) → (𝐴 ↑_𝑜 suc (𝐺‘𝑦)) ⊆ (𝐴 ↑_𝑜 (𝐺‘suc 𝑦))))
194	191, 155, 139, 192, 193	syl31anc 1321	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (suc (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦) → (𝐴 ↑_𝑜 suc (𝐺‘𝑦)) ⊆ (𝐴 ↑_𝑜 (𝐺‘suc 𝑦))))
195	186, 194	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐴 ↑_𝑜 suc (𝐺‘𝑦)) ⊆ (𝐴 ↑_𝑜 (𝐺‘suc 𝑦)))
196		simprr 792	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))
197	195, 196	sseldd 3569	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 (𝐺‘suc 𝑦)))
198		peano2 6978	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
199	198	ad2antlr 759	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → suc 𝑦 ∈ ω)
200	8	cantnfvalf 8445	. . . . . . . . . . . . . . . . 17 ⊢ 𝐻:ω⟶On
201	200	ffvelrni 6266	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑦 ∈ ω → (𝐻‘suc 𝑦) ∈ On)
202	199, 201	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐻‘suc 𝑦) ∈ On)
203		omcl 7503	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ∈ On ∧ (𝐹‘(𝐺‘suc 𝑦)) ∈ On) → ((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))) ∈ On)
204	157, 151, 203	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → ((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))) ∈ On)
205		oaord 7514	. . . . . . . . . . . . . . 15 ⊢ (((𝐻‘suc 𝑦) ∈ On ∧ (𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ∈ On ∧ ((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))) ∈ On) → ((𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ↔ (((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))) +_𝑜 (𝐻‘suc 𝑦)) ∈ (((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))) +_𝑜 (𝐴 ↑_𝑜 (𝐺‘suc 𝑦)))))
206	202, 157, 204, 205	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → ((𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ↔ (((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))) +_𝑜 (𝐻‘suc 𝑦)) ∈ (((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))) +_𝑜 (𝐴 ↑_𝑜 (𝐺‘suc 𝑦)))))
207	197, 206	mpbid 221	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))) +_𝑜 (𝐻‘suc 𝑦)) ∈ (((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))) +_𝑜 (𝐴 ↑_𝑜 (𝐺‘suc 𝑦))))
208	44, 1, 45, 19, 43, 8	cantnfsuc 8450	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ suc 𝑦 ∈ ω) → (𝐻‘suc suc 𝑦) = (((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))) +_𝑜 (𝐻‘suc 𝑦)))
209	198, 208	sylan2 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (𝐻‘suc suc 𝑦) = (((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))) +_𝑜 (𝐻‘suc 𝑦)))
210	209	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐻‘suc suc 𝑦) = (((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))) +_𝑜 (𝐻‘suc 𝑦)))
211		omsuc 7493	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ∈ On ∧ (𝐹‘(𝐺‘suc 𝑦)) ∈ On) → ((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 suc (𝐹‘(𝐺‘suc 𝑦))) = (((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))) +_𝑜 (𝐴 ↑_𝑜 (𝐺‘suc 𝑦))))
212	157, 151, 211	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → ((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 suc (𝐹‘(𝐺‘suc 𝑦))) = (((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))) +_𝑜 (𝐴 ↑_𝑜 (𝐺‘suc 𝑦))))
213	207, 210, 212	3eltr4d 2703	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐻‘suc suc 𝑦) ∈ ((𝐴 ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 suc (𝐹‘(𝐺‘suc 𝑦))))
214	163, 213	sseldd 3569	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)))) → (𝐻‘suc suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘suc 𝑦)))
215	214	exp32 629	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (suc 𝑦 ∈ dom 𝐺 → ((𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦)) → (𝐻‘suc suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘suc 𝑦)))))
216	215	a2d 29	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((suc 𝑦 ∈ dom 𝐺 → (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦))) → (suc 𝑦 ∈ dom 𝐺 → (𝐻‘suc suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘suc 𝑦)))))
217	138, 216	syl5 33	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((𝑦 ∈ dom 𝐺 → (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦))) → (suc 𝑦 ∈ dom 𝐺 → (𝐻‘suc suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘suc 𝑦)))))
218	217	expcom 450	. . . . . . 7 ⊢ (𝑦 ∈ ω → (𝜑 → ((𝑦 ∈ dom 𝐺 → (𝐻‘suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑦))) → (suc 𝑦 ∈ dom 𝐺 → (𝐻‘suc suc 𝑦) ∈ (𝐴 ↑_𝑜 suc (𝐺‘suc 𝑦))))))
219	82, 91, 100, 133, 218	finds2 6986	. . . . . 6 ⊢ (𝑥 ∈ ω → (𝜑 → (𝑥 ∈ dom 𝐺 → (𝐻‘suc 𝑥) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑥)))))
220	72, 73, 60, 219	syl3c 64	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → (𝐻‘suc 𝑥) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑥)))
221	71, 220	eqeltrd 2688	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → (𝐻‘𝐾) ∈ (𝐴 ↑_𝑜 suc (𝐺‘𝑥)))
222	70, 221	sseldd 3569	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ω ∧ 𝐾 = suc 𝑥)) → (𝐻‘𝐾) ∈ (𝐴 ↑_𝑜 𝐶))
223	222	rexlimdvaa 3014	. 2 ⊢ (𝜑 → (∃𝑥 ∈ ω 𝐾 = suc 𝑥 → (𝐻‘𝐾) ∈ (𝐴 ↑_𝑜 𝐶)))
224	172	simprd 478	. . . . 5 ⊢ (𝜑 → dom 𝐺 ∈ ω)
225		peano2 6978	. . . . 5 ⊢ (dom 𝐺 ∈ ω → suc dom 𝐺 ∈ ω)
226	224, 225	syl 17	. . . 4 ⊢ (𝜑 → suc dom 𝐺 ∈ ω)
227		elnn 6967	. . . 4 ⊢ ((𝐾 ∈ suc dom 𝐺 ∧ suc dom 𝐺 ∈ ω) → 𝐾 ∈ ω)
228	23, 226, 227	syl2anc 691	. . 3 ⊢ (𝜑 → 𝐾 ∈ ω)
229		nn0suc 6982	. . 3 ⊢ (𝐾 ∈ ω → (𝐾 = ∅ ∨ ∃𝑥 ∈ ω 𝐾 = suc 𝑥))
230	228, 229	syl 17	. 2 ⊢ (𝜑 → (𝐾 = ∅ ∨ ∃𝑥 ∈ ω 𝐾 = suc 𝑥))
231	13, 223, 230	mpjaod 395	1 ⊢ (𝜑 → (𝐻‘𝐾) ∈ (𝐴 ↑_𝑜 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 Tr wtr 4680 E cep 4947 We wwe 4996 dom cdm 5038 “ cima 5041 Ord word 5639 Oncon0 5640 suc csuc 5642 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 Isom wiso 5805 (class class class)co 6549 ↦ cmpt2 6551 ωcom 6957 supp csupp 7182 seq_𝜔cseqom 7429 +_𝑜 coa 7444 ·_𝑜 comu 7445 ↑_𝑜 coe 7446 finSupp cfsupp 8158 OrdIsocoi 8297 CNF ccnf 8441

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seqom 7430 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-oexp 7453 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-cnf 8442

This theorem is referenced by: cantnflt2 8453 cnfcomlem 8479

W3C validator