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Theorem prodmolem2a 14503

Description: Lemma for prodmo 14505. (Contributed by Scott Fenton, 4-Dec-2017.)

**Hypotheses**
Ref	Expression
prodmo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
prodmo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
prodmo.3	⊢ 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
prodmolem2.4	⊢ 𝐻 = (𝑗 ∈ ℕ ↦ ⦋(𝐾‘𝑗) / 𝑘⦌𝐵)
prodmolem2.5	⊢ (𝜑 → 𝑁 ∈ ℕ)
prodmolem2.6	⊢ (𝜑 → 𝑀 ∈ ℤ)
prodmolem2.7	⊢ (𝜑 → 𝐴 ⊆ (ℤ_≥‘𝑀))
prodmolem2.8	⊢ (𝜑 → 𝑓:(1...𝑁)–1-1-onto→𝐴)
prodmolem2.9	⊢ (𝜑 → 𝐾 Isom < , < ((1...(#‘𝐴)), 𝐴))

**Assertion**
Ref	Expression
prodmolem2a	⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑁))

Distinct variable groups: 𝐴,𝑘 𝑘,𝐹 𝜑,𝑘 𝐴,𝑗 𝐵,𝑗 𝑓,𝑗,𝑘 𝑗,𝐺 𝑗,𝑘,𝜑 𝑗,𝐾,𝑘 𝑗,𝑀,𝑘 𝑗,𝑁,𝑘 Allowed substitution hints: 𝜑(𝑓) 𝐴(𝑓) 𝐵(𝑓,𝑘) 𝐹(𝑓,𝑗) 𝐺(𝑓,𝑘) 𝐻(𝑓,𝑗,𝑘) 𝐾(𝑓) 𝑀(𝑓) 𝑁(𝑓)

Proof of Theorem prodmolem2a Dummy variables 𝑛 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prodmo.1	. . 3 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
2		prodmo.2	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
3		prodmolem2.7	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (ℤ_≥‘𝑀))
4		prodmolem2.9	. . . . . . 7 ⊢ (𝜑 → 𝐾 Isom < , < ((1...(#‘𝐴)), 𝐴))
5		prodmolem2.8	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑓:(1...𝑁)–1-1-onto→𝐴)
6		ovex 6577	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ∈ V
7	6	f1oen 7862	. . . . . . . . . . . 12 ⊢ (𝑓:(1...𝑁)–1-1-onto→𝐴 → (1...𝑁) ≈ 𝐴)
8	5, 7	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑁) ≈ 𝐴)
9		fzfid 12634	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
10	8	ensymd 7893	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ≈ (1...𝑁))
11		enfii 8062	. . . . . . . . . . . . 13 ⊢ (((1...𝑁) ∈ Fin ∧ 𝐴 ≈ (1...𝑁)) → 𝐴 ∈ Fin)
12	9, 10, 11	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ Fin)
13		hashen 12997	. . . . . . . . . . . 12 ⊢ (((1...𝑁) ∈ Fin ∧ 𝐴 ∈ Fin) → ((#‘(1...𝑁)) = (#‘𝐴) ↔ (1...𝑁) ≈ 𝐴))
14	9, 12, 13	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → ((#‘(1...𝑁)) = (#‘𝐴) ↔ (1...𝑁) ≈ 𝐴))
15	8, 14	mpbird 246	. . . . . . . . . 10 ⊢ (𝜑 → (#‘(1...𝑁)) = (#‘𝐴))
16		prodmolem2.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
17	16	nnnn0d 11228	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
18		hashfz1 12996	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (#‘(1...𝑁)) = 𝑁)
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (#‘(1...𝑁)) = 𝑁)
20	15, 19	eqtr3d 2646	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝐴) = 𝑁)
21	20	oveq2d 6565	. . . . . . . 8 ⊢ (𝜑 → (1...(#‘𝐴)) = (1...𝑁))
22		isoeq4 6470	. . . . . . . 8 ⊢ ((1...(#‘𝐴)) = (1...𝑁) → (𝐾 Isom < , < ((1...(#‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴)))
23	21, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐾 Isom < , < ((1...(#‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴)))
24	4, 23	mpbid 221	. . . . . 6 ⊢ (𝜑 → 𝐾 Isom < , < ((1...𝑁), 𝐴))
25		isof1o 6473	. . . . . 6 ⊢ (𝐾 Isom < , < ((1...𝑁), 𝐴) → 𝐾:(1...𝑁)–1-1-onto→𝐴)
26		f1of 6050	. . . . . 6 ⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → 𝐾:(1...𝑁)⟶𝐴)
27	24, 25, 26	3syl 18	. . . . 5 ⊢ (𝜑 → 𝐾:(1...𝑁)⟶𝐴)
28		nnuz 11599	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
29	16, 28	syl6eleq 2698	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
30		eluzfz2 12220	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
31	29, 30	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
32	27, 31	ffvelrnd 6268	. . . 4 ⊢ (𝜑 → (𝐾‘𝑁) ∈ 𝐴)
33	3, 32	sseldd 3569	. . 3 ⊢ (𝜑 → (𝐾‘𝑁) ∈ (ℤ_≥‘𝑀))
34	3	sselda 3568	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ (ℤ_≥‘𝑀))
35	24, 25	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)
36		f1ocnvfv2 6433	. . . . . . . . 9 ⊢ ((𝐾:(1...𝑁)–1-1-onto→𝐴 ∧ 𝑗 ∈ 𝐴) → (𝐾‘(^◡𝐾‘𝑗)) = 𝑗)
37	35, 36	sylan 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐾‘(^◡𝐾‘𝑗)) = 𝑗)
38		f1ocnv 6062	. . . . . . . . . . . 12 ⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → ^◡𝐾:𝐴–1-1-onto→(1...𝑁))
39		f1of 6050	. . . . . . . . . . . 12 ⊢ (^◡𝐾:𝐴–1-1-onto→(1...𝑁) → ^◡𝐾:𝐴⟶(1...𝑁))
40	35, 38, 39	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ^◡𝐾:𝐴⟶(1...𝑁))
41	40	ffvelrnda 6267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (^◡𝐾‘𝑗) ∈ (1...𝑁))
42		elfzle2 12216	. . . . . . . . . 10 ⊢ ((^◡𝐾‘𝑗) ∈ (1...𝑁) → (^◡𝐾‘𝑗) ≤ 𝑁)
43	41, 42	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (^◡𝐾‘𝑗) ≤ 𝑁)
44	24	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐾 Isom < , < ((1...𝑁), 𝐴))
45		fzssuz 12253	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ⊆ (ℤ_≥‘1)
46		uzssz 11583	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘1) ⊆ ℤ
47		zssre 11261	. . . . . . . . . . . . . 14 ⊢ ℤ ⊆ ℝ
48	46, 47	sstri 3577	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘1) ⊆ ℝ
49	45, 48	sstri 3577	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ ℝ
50		ressxr 9962	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℝ^*
51	49, 50	sstri 3577	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ ℝ^*
52	51	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (1...𝑁) ⊆ ℝ^*)
53		uzssz 11583	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
54	53, 47	sstri 3577	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘𝑀) ⊆ ℝ
55	54, 50	sstri 3577	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘𝑀) ⊆ ℝ^*
56	3, 55	syl6ss 3580	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
57	56	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐴 ⊆ ℝ^*)
58	31	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑁 ∈ (1...𝑁))
59		leisorel 13101	. . . . . . . . . 10 ⊢ ((𝐾 Isom < , < ((1...𝑁), 𝐴) ∧ ((1...𝑁) ⊆ ℝ^* ∧ 𝐴 ⊆ ℝ^*) ∧ ((^◡𝐾‘𝑗) ∈ (1...𝑁) ∧ 𝑁 ∈ (1...𝑁))) → ((^◡𝐾‘𝑗) ≤ 𝑁 ↔ (𝐾‘(^◡𝐾‘𝑗)) ≤ (𝐾‘𝑁)))
60	44, 52, 57, 41, 58, 59	syl122anc 1327	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((^◡𝐾‘𝑗) ≤ 𝑁 ↔ (𝐾‘(^◡𝐾‘𝑗)) ≤ (𝐾‘𝑁)))
61	43, 60	mpbid 221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐾‘(^◡𝐾‘𝑗)) ≤ (𝐾‘𝑁))
62	37, 61	eqbrtrrd 4607	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ≤ (𝐾‘𝑁))
63	3, 53	syl6ss 3580	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℤ)
64	63	sselda 3568	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ ℤ)
65		eluzelz 11573	. . . . . . . . . 10 ⊢ ((𝐾‘𝑁) ∈ (ℤ_≥‘𝑀) → (𝐾‘𝑁) ∈ ℤ)
66	33, 65	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘𝑁) ∈ ℤ)
67	66	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐾‘𝑁) ∈ ℤ)
68		eluz 11577	. . . . . . . 8 ⊢ ((𝑗 ∈ ℤ ∧ (𝐾‘𝑁) ∈ ℤ) → ((𝐾‘𝑁) ∈ (ℤ_≥‘𝑗) ↔ 𝑗 ≤ (𝐾‘𝑁)))
69	64, 67, 68	syl2anc 691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐾‘𝑁) ∈ (ℤ_≥‘𝑗) ↔ 𝑗 ≤ (𝐾‘𝑁)))
70	62, 69	mpbird 246	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐾‘𝑁) ∈ (ℤ_≥‘𝑗))
71		elfzuzb 12207	. . . . . 6 ⊢ (𝑗 ∈ (𝑀...(𝐾‘𝑁)) ↔ (𝑗 ∈ (ℤ_≥‘𝑀) ∧ (𝐾‘𝑁) ∈ (ℤ_≥‘𝑗)))
72	34, 70, 71	sylanbrc 695	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ (𝑀...(𝐾‘𝑁)))
73	72	ex 449	. . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝐴 → 𝑗 ∈ (𝑀...(𝐾‘𝑁))))
74	73	ssrdv 3574	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (𝑀...(𝐾‘𝑁)))
75	1, 2, 33, 74	fprodcvg 14499	. 2 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq𝑀( · , 𝐹)‘(𝐾‘𝑁)))
76		mulid2 9917	. . . . 5 ⊢ (𝑚 ∈ ℂ → (1 · 𝑚) = 𝑚)
77	76	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℂ) → (1 · 𝑚) = 𝑚)
78		mulid1 9916	. . . . 5 ⊢ (𝑚 ∈ ℂ → (𝑚 · 1) = 𝑚)
79	78	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℂ) → (𝑚 · 1) = 𝑚)
80		mulcl 9899	. . . . 5 ⊢ ((𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑚 · 𝑥) ∈ ℂ)
81	80	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑚 · 𝑥) ∈ ℂ)
82		1cnd 9935	. . . 4 ⊢ (𝜑 → 1 ∈ ℂ)
83	31, 21	eleqtrrd 2691	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (1...(#‘𝐴)))
84		iftrue 4042	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) = 𝐵)
85	84	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) = 𝐵)
86	85, 2	eqeltrd 2688	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
87	86	ex 449	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ))
88		iffalse 4045	. . . . . . . . 9 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) = 1)
89		ax-1cn 9873	. . . . . . . . 9 ⊢ 1 ∈ ℂ
90	88, 89	syl6eqel 2696	. . . . . . . 8 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
91	87, 90	pm2.61d1 170	. . . . . . 7 ⊢ (𝜑 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
92	91	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
93	92, 1	fmptd 6292	. . . . 5 ⊢ (𝜑 → 𝐹:ℤ⟶ℂ)
94		elfzelz 12213	. . . . 5 ⊢ (𝑚 ∈ (𝑀...(𝐾‘(#‘𝐴))) → 𝑚 ∈ ℤ)
95		ffvelrn 6265	. . . . 5 ⊢ ((𝐹:ℤ⟶ℂ ∧ 𝑚 ∈ ℤ) → (𝐹‘𝑚) ∈ ℂ)
96	93, 94, 95	syl2an 493	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...(𝐾‘(#‘𝐴)))) → (𝐹‘𝑚) ∈ ℂ)
97		fveq2 6103	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
98	97	eqeq1d 2612	. . . . . 6 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) = 1 ↔ (𝐹‘𝑚) = 1))
99		fzssuz 12253	. . . . . . . . . 10 ⊢ (𝑀...(𝐾‘(#‘𝐴))) ⊆ (ℤ_≥‘𝑀)
100	99, 53	sstri 3577	. . . . . . . . 9 ⊢ (𝑀...(𝐾‘(#‘𝐴))) ⊆ ℤ
101		eldifi 3694	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → 𝑘 ∈ (𝑀...(𝐾‘(#‘𝐴))))
102	100, 101	sseldi 3566	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → 𝑘 ∈ ℤ)
103		eldifn 3695	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → ¬ 𝑘 ∈ 𝐴)
104	103, 88	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) = 1)
105	104, 89	syl6eqel 2696	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
106	1	fvmpt2 6200	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
107	102, 105, 106	syl2anc 691	. . . . . . 7 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
108	107, 104	eqtrd 2644	. . . . . 6 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → (𝐹‘𝑘) = 1)
109	98, 108	vtoclga 3245	. . . . 5 ⊢ (𝑚 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → (𝐹‘𝑚) = 1)
110	109	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴)) → (𝐹‘𝑚) = 1)
111		isof1o 6473	. . . . . . . 8 ⊢ (𝐾 Isom < , < ((1...(#‘𝐴)), 𝐴) → 𝐾:(1...(#‘𝐴))–1-1-onto→𝐴)
112		f1of 6050	. . . . . . . 8 ⊢ (𝐾:(1...(#‘𝐴))–1-1-onto→𝐴 → 𝐾:(1...(#‘𝐴))⟶𝐴)
113	4, 111, 112	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐾:(1...(#‘𝐴))⟶𝐴)
114	113	ffvelrnda 6267	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(#‘𝐴))) → (𝐾‘𝑥) ∈ 𝐴)
115	114	iftrued 4044	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(#‘𝐴))) → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
116	63	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(#‘𝐴))) → 𝐴 ⊆ ℤ)
117	116, 114	sseldd 3569	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(#‘𝐴))) → (𝐾‘𝑥) ∈ ℤ)
118		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑘𝜑
119		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝐾‘𝑥) ∈ 𝐴
120		nfcsb1v 3515	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(𝐾‘𝑥) / 𝑘⦌𝐵
121		nfcv 2751	. . . . . . . . . . 11 ⊢ Ⅎ𝑘1
122	119, 120, 121	nfif 4065	. . . . . . . . . 10 ⊢ Ⅎ𝑘if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1)
123	122	nfel1 2765	. . . . . . . . 9 ⊢ Ⅎ𝑘if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ
124	118, 123	nfim 1813	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ)
125		fvex 6113	. . . . . . . 8 ⊢ (𝐾‘𝑥) ∈ V
126		eleq1 2676	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐾‘𝑥) → (𝑘 ∈ 𝐴 ↔ (𝐾‘𝑥) ∈ 𝐴))
127		csbeq1a 3508	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐾‘𝑥) → 𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
128	126, 127	ifbieq1d 4059	. . . . . . . . . 10 ⊢ (𝑘 = (𝐾‘𝑥) → if(𝑘 ∈ 𝐴, 𝐵, 1) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1))
129	128	eleq1d 2672	. . . . . . . . 9 ⊢ (𝑘 = (𝐾‘𝑥) → (if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ ↔ if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ))
130	129	imbi2d 329	. . . . . . . 8 ⊢ (𝑘 = (𝐾‘𝑥) → ((𝜑 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) ↔ (𝜑 → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ)))
131	124, 125, 130, 91	vtoclf 3231	. . . . . . 7 ⊢ (𝜑 → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ)
132	131	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(#‘𝐴))) → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ)
133		eleq1 2676	. . . . . . . 8 ⊢ (𝑛 = (𝐾‘𝑥) → (𝑛 ∈ 𝐴 ↔ (𝐾‘𝑥) ∈ 𝐴))
134		csbeq1 3502	. . . . . . . 8 ⊢ (𝑛 = (𝐾‘𝑥) → ⦋𝑛 / 𝑘⦌𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
135	133, 134	ifbieq1d 4059	. . . . . . 7 ⊢ (𝑛 = (𝐾‘𝑥) → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 1) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1))
136		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑛if(𝑘 ∈ 𝐴, 𝐵, 1)
137		nfv 1830	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ 𝐴
138		nfcsb1v 3515	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
139	137, 138, 121	nfif 4065	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 1)
140		eleq1 2676	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝑘 ∈ 𝐴 ↔ 𝑛 ∈ 𝐴))
141		csbeq1a 3508	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑛 / 𝑘⦌𝐵)
142	140, 141	ifbieq1d 4059	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 1))
143	136, 139, 142	cbvmpt 4677	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 1))
144	1, 143	eqtri 2632	. . . . . . 7 ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 1))
145	135, 144	fvmptg 6189	. . . . . 6 ⊢ (((𝐾‘𝑥) ∈ ℤ ∧ if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ) → (𝐹‘(𝐾‘𝑥)) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1))
146	117, 132, 145	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(#‘𝐴))) → (𝐹‘(𝐾‘𝑥)) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1))
147		elfznn 12241	. . . . . . 7 ⊢ (𝑥 ∈ (1...(#‘𝐴)) → 𝑥 ∈ ℕ)
148	147	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(#‘𝐴))) → 𝑥 ∈ ℕ)
149	115, 132	eqeltrrd 2689	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(#‘𝐴))) → ⦋(𝐾‘𝑥) / 𝑘⦌𝐵 ∈ ℂ)
150		fveq2 6103	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → (𝐾‘𝑗) = (𝐾‘𝑥))
151	150	csbeq1d 3506	. . . . . . 7 ⊢ (𝑗 = 𝑥 → ⦋(𝐾‘𝑗) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
152		prodmolem2.4	. . . . . . 7 ⊢ 𝐻 = (𝑗 ∈ ℕ ↦ ⦋(𝐾‘𝑗) / 𝑘⦌𝐵)
153	151, 152	fvmptg 6189	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ ⦋(𝐾‘𝑥) / 𝑘⦌𝐵 ∈ ℂ) → (𝐻‘𝑥) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
154	148, 149, 153	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(#‘𝐴))) → (𝐻‘𝑥) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
155	115, 146, 154	3eqtr4rd 2655	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(#‘𝐴))) → (𝐻‘𝑥) = (𝐹‘(𝐾‘𝑥)))
156	77, 79, 81, 82, 4, 83, 3, 96, 110, 155	seqcoll 13105	. . 3 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘(𝐾‘𝑁)) = (seq1( · , 𝐻)‘𝑁))
157		prodmo.3	. . . 4 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
158	16, 16	jca 553	. . . 4 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑁 ∈ ℕ))
159	1, 2, 157, 152, 158, 5, 35	prodmolem3 14502	. . 3 ⊢ (𝜑 → (seq1( · , 𝐺)‘𝑁) = (seq1( · , 𝐻)‘𝑁))
160	156, 159	eqtr4d 2647	. 2 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘(𝐾‘𝑁)) = (seq1( · , 𝐺)‘𝑁))
161	75, 160	breqtrd 4609	1 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⦋csb 3499 ∖ cdif 3537 ⊆ wss 3540 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 ^◡ccnv 5037 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 Isom wiso 5805 (class class class)co 6549 ≈ cen 7838 Fincfn 7841 ℂcc 9813 ℝcr 9814 1c1 9816 · cmul 9820 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 seqcseq 12663 #chash 12979 ⇝ cli 14063

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 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 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-nel 2783 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-int 4411 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-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-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067

This theorem is referenced by: prodmolem2 14504 zprod 14506

W3C validator