Theorem ltexprlemrl 6441

Description: Lemma for ltexpri 6444. Reverse directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)

Hypothesis

Ref	Expression
ltexprlem.1	⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2nd ‘A) ∧ (y +Q x) ∈ (1st ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1st ‘A) ∧ (y +Q x) ∈ (2nd ‘B))}⟩

Assertion

Ref	Expression
ltexprlemrl	⊢ (A<P B → (1st ‘B) ⊆ (1st ‘(A +P 𝐶)))

Distinct variable groups:   x,y,A   x,B,y   x,𝐶,y

Proof of Theorem ltexprlemrl

Dummy variables z w u v f g ℎ 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelpr 6353	. . . . . . . 8 ⊢ <P ⊆ (P × P)
2	1	brel 4315	. . . . . . 7 ⊢ (A<P B → (A ∈ P ∧ B ∈ P))
3	2	simprd 107	. . . . . 6 ⊢ (A<P B → B ∈ P)
4		prop 6323	. . . . . 6 ⊢ (B ∈ P → ⟨(1st ‘B), (2nd ‘B)⟩ ∈ P)
5	3, 4	syl 14	. . . . 5 ⊢ (A<P B → ⟨(1st ‘B), (2nd ‘B)⟩ ∈ P)
6		prnmaddl 6338	. . . . 5 ⊢ ((⟨(1st ‘B), (2nd ‘B)⟩ ∈ P ∧ w ∈ (1st ‘B)) → ∃v ∈ Q (w +Q v) ∈ (1st ‘B))
7	5, 6	sylan 267	. . . 4 ⊢ ((A<P B ∧ w ∈ (1st ‘B)) → ∃v ∈ Q (w +Q v) ∈ (1st ‘B))
8	2	simpld 105	. . . . . . . 8 ⊢ (A<P B → A ∈ P)
9		prop 6323	. . . . . . . 8 ⊢ (A ∈ P → ⟨(1st ‘A), (2nd ‘A)⟩ ∈ P)
10	8, 9	syl 14	. . . . . . 7 ⊢ (A<P B → ⟨(1st ‘A), (2nd ‘A)⟩ ∈ P)
11		prarloc 6351	. . . . . . 7 ⊢ ((⟨(1st ‘A), (2nd ‘A)⟩ ∈ P ∧ v ∈ Q) → ∃z ∈ (1st ‘A)∃u ∈ (2nd ‘A)u <Q (z +Q v))
12	10, 11	sylan 267	. . . . . 6 ⊢ ((A<P B ∧ v ∈ Q) → ∃z ∈ (1st ‘A)∃u ∈ (2nd ‘A)u <Q (z +Q v))
13	12	ad2ant2r 466	. . . . 5 ⊢ (((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) → ∃z ∈ (1st ‘A)∃u ∈ (2nd ‘A)u <Q (z +Q v))
14		simplll 473	. . . . . . . . . . 11 ⊢ ((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) → A<P B)
15	14	adantr 261	. . . . . . . . . 10 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → A<P B)
16		simplrl 475	. . . . . . . . . 10 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → z ∈ (1st ‘A))
17		elprnql 6329	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘A), (2nd ‘A)⟩ ∈ P ∧ z ∈ (1st ‘A)) → z ∈ Q)
18	10, 17	sylan 267	. . . . . . . . . 10 ⊢ ((A<P B ∧ z ∈ (1st ‘A)) → z ∈ Q)
19	15, 16, 18	syl2anc 393	. . . . . . . . 9 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → z ∈ Q)
20		elprnql 6329	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘B), (2nd ‘B)⟩ ∈ P ∧ w ∈ (1st ‘B)) → w ∈ Q)
21	5, 20	sylan 267	. . . . . . . . . 10 ⊢ ((A<P B ∧ w ∈ (1st ‘B)) → w ∈ Q)
22	21	ad3antrrr 461	. . . . . . . . 9 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → w ∈ Q)
23		nqtri3or 6249	. . . . . . . . 9 ⊢ ((z ∈ Q ∧ w ∈ Q) → (z <Q w ∨ z = w ∨ w <Q z))
24	19, 22, 23	syl2anc 393	. . . . . . . 8 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → (z <Q w ∨ z = w ∨ w <Q z))
25		ltexnqq 6260	. . . . . . . . . . . . 13 ⊢ ((z ∈ Q ∧ w ∈ Q) → (z <Q w ↔ ∃𝑠 ∈ Q (z +Q 𝑠) = w))
26	19, 22, 25	syl2anc 393	. . . . . . . . . . . 12 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → (z <Q w ↔ ∃𝑠 ∈ Q (z +Q 𝑠) = w))
27	26	biimpa 280	. . . . . . . . . . 11 ⊢ ((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) → ∃𝑠 ∈ Q (z +Q 𝑠) = w)
28		simprr 472	. . . . . . . . . . . 12 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → (z +Q 𝑠) = w)
29	16	ad2antrr 460	. . . . . . . . . . . . 13 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → z ∈ (1st ‘A))
30		simprl 471	. . . . . . . . . . . . . 14 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → 𝑠 ∈ Q)
31		simpr 103	. . . . . . . . . . . . . . . . 17 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → u <Q (z +Q v))
32		simplrr 476	. . . . . . . . . . . . . . . . . 18 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → u ∈ (2nd ‘A))
33		prcunqu 6333	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨(1st ‘A), (2nd ‘A)⟩ ∈ P ∧ u ∈ (2nd ‘A)) → (u <Q (z +Q v) → (z +Q v) ∈ (2nd ‘A)))
34	10, 33	sylan 267	. . . . . . . . . . . . . . . . . 18 ⊢ ((A<P B ∧ u ∈ (2nd ‘A)) → (u <Q (z +Q v) → (z +Q v) ∈ (2nd ‘A)))
35	15, 32, 34	syl2anc 393	. . . . . . . . . . . . . . . . 17 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → (u <Q (z +Q v) → (z +Q v) ∈ (2nd ‘A)))
36	31, 35	mpd 13	. . . . . . . . . . . . . . . 16 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → (z +Q v) ∈ (2nd ‘A))
37	36	ad2antrr 460	. . . . . . . . . . . . . . 15 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → (z +Q v) ∈ (2nd ‘A))
38	19	ad2antrr 460	. . . . . . . . . . . . . . . . 17 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → z ∈ Q)
39		simplrl 475	. . . . . . . . . . . . . . . . . 18 ⊢ ((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) → v ∈ Q)
40	39	ad3antrrr 461	. . . . . . . . . . . . . . . . 17 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → v ∈ Q)
41		addcomnqg 6234	. . . . . . . . . . . . . . . . . 18 ⊢ ((f ∈ Q ∧ g ∈ Q) → (f +Q g) = (g +Q f))
42	41	adantl 262	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) ∧ (f ∈ Q ∧ g ∈ Q)) → (f +Q g) = (g +Q f))
43		addassnqg 6235	. . . . . . . . . . . . . . . . . 18 ⊢ ((f ∈ Q ∧ g ∈ Q ∧ ℎ ∈ Q) → ((f +Q g) +Q ℎ) = (f +Q (g +Q ℎ)))
44	43	adantl 262	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) ∧ (f ∈ Q ∧ g ∈ Q ∧ ℎ ∈ Q)) → ((f +Q g) +Q ℎ) = (f +Q (g +Q ℎ)))
45	38, 40, 30, 42, 44	caov32d 5600	. . . . . . . . . . . . . . . 16 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → ((z +Q v) +Q 𝑠) = ((z +Q 𝑠) +Q v))
46		simplrr 476	. . . . . . . . . . . . . . . . . 18 ⊢ ((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) → (w +Q v) ∈ (1st ‘B))
47	46	ad3antrrr 461	. . . . . . . . . . . . . . . . 17 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → (w +Q v) ∈ (1st ‘B))
48		oveq1 5439	. . . . . . . . . . . . . . . . . . 19 ⊢ ((z +Q 𝑠) = w → ((z +Q 𝑠) +Q v) = (w +Q v))
49	48	eleq1d 2084	. . . . . . . . . . . . . . . . . 18 ⊢ ((z +Q 𝑠) = w → (((z +Q 𝑠) +Q v) ∈ (1st ‘B) ↔ (w +Q v) ∈ (1st ‘B)))
50	28, 49	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → (((z +Q 𝑠) +Q v) ∈ (1st ‘B) ↔ (w +Q v) ∈ (1st ‘B)))
51	47, 50	mpbird 156	. . . . . . . . . . . . . . . 16 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → ((z +Q 𝑠) +Q v) ∈ (1st ‘B))
52	45, 51	eqeltrd 2092	. . . . . . . . . . . . . . 15 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → ((z +Q v) +Q 𝑠) ∈ (1st ‘B))
53		eleq1 2078	. . . . . . . . . . . . . . . . . 18 ⊢ (y = (z +Q v) → (y ∈ (2nd ‘A) ↔ (z +Q v) ∈ (2nd ‘A)))
54		oveq1 5439	. . . . . . . . . . . . . . . . . . 19 ⊢ (y = (z +Q v) → (y +Q 𝑠) = ((z +Q v) +Q 𝑠))
55	54	eleq1d 2084	. . . . . . . . . . . . . . . . . 18 ⊢ (y = (z +Q v) → ((y +Q 𝑠) ∈ (1st ‘B) ↔ ((z +Q v) +Q 𝑠) ∈ (1st ‘B)))
56	53, 55	anbi12d 445	. . . . . . . . . . . . . . . . 17 ⊢ (y = (z +Q v) → ((y ∈ (2nd ‘A) ∧ (y +Q 𝑠) ∈ (1st ‘B)) ↔ ((z +Q v) ∈ (2nd ‘A) ∧ ((z +Q v) +Q 𝑠) ∈ (1st ‘B))))
57	56	spcegv 2614	. . . . . . . . . . . . . . . 16 ⊢ ((z +Q v) ∈ (2nd ‘A) → (((z +Q v) ∈ (2nd ‘A) ∧ ((z +Q v) +Q 𝑠) ∈ (1st ‘B)) → ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑠) ∈ (1st ‘B))))
58	57	anabsi5 500	. . . . . . . . . . . . . . 15 ⊢ (((z +Q v) ∈ (2nd ‘A) ∧ ((z +Q v) +Q 𝑠) ∈ (1st ‘B)) → ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑠) ∈ (1st ‘B)))
59	37, 52, 58	syl2anc 393	. . . . . . . . . . . . . 14 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑠) ∈ (1st ‘B)))
60		ltexprlem.1	. . . . . . . . . . . . . . 15 ⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2nd ‘A) ∧ (y +Q x) ∈ (1st ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1st ‘A) ∧ (y +Q x) ∈ (2nd ‘B))}⟩
61	60	ltexprlemell 6429	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (1st ‘𝐶) ↔ (𝑠 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑠) ∈ (1st ‘B))))
62	30, 59, 61	sylanbrc 396	. . . . . . . . . . . . 13 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → 𝑠 ∈ (1st ‘𝐶))
63	15, 8	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → A ∈ P)
64	63	ad2antrr 460	. . . . . . . . . . . . . 14 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → A ∈ P)
65	60	ltexprlempr 6439	. . . . . . . . . . . . . . . 16 ⊢ (A<P B → 𝐶 ∈ P)
66	15, 65	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → 𝐶 ∈ P)
67	66	ad2antrr 460	. . . . . . . . . . . . . 14 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → 𝐶 ∈ P)
68		df-iplp 6316	. . . . . . . . . . . . . . 15 ⊢ +P = (x ∈ P, w ∈ P ↦ ⟨{z ∈ Q ∣ ∃f ∈ Q ∃v ∈ Q (f ∈ (1st ‘x) ∧ v ∈ (1st ‘w) ∧ z = (f +Q v))}, {z ∈ Q ∣ ∃f ∈ Q ∃v ∈ Q (f ∈ (2nd ‘x) ∧ v ∈ (2nd ‘w) ∧ z = (f +Q v))}⟩)
69		addclnq 6228	. . . . . . . . . . . . . . 15 ⊢ ((f ∈ Q ∧ v ∈ Q) → (f +Q v) ∈ Q)
70	68, 69	genpprecll 6362	. . . . . . . . . . . . . 14 ⊢ ((A ∈ P ∧ 𝐶 ∈ P) → ((z ∈ (1st ‘A) ∧ 𝑠 ∈ (1st ‘𝐶)) → (z +Q 𝑠) ∈ (1st ‘(A +P 𝐶))))
71	64, 67, 70	syl2anc 393	. . . . . . . . . . . . 13 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → ((z ∈ (1st ‘A) ∧ 𝑠 ∈ (1st ‘𝐶)) → (z +Q 𝑠) ∈ (1st ‘(A +P 𝐶))))
72	29, 62, 71	mp2and 411	. . . . . . . . . . . 12 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → (z +Q 𝑠) ∈ (1st ‘(A +P 𝐶)))
73	28, 72	eqeltrrd 2093	. . . . . . . . . . 11 ⊢ (((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) ∧ (𝑠 ∈ Q ∧ (z +Q 𝑠) = w)) → w ∈ (1st ‘(A +P 𝐶)))
74	27, 73	rexlimddv 2411	. . . . . . . . . 10 ⊢ ((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z <Q w) → w ∈ (1st ‘(A +P 𝐶)))
75	74	ex 108	. . . . . . . . 9 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → (z <Q w → w ∈ (1st ‘(A +P 𝐶))))
76	14	ad2antrr 460	. . . . . . . . . . 11 ⊢ ((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z = w) → A<P B)
77		simpr 103	. . . . . . . . . . . 12 ⊢ ((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z = w) → z = w)
78	16	adantr 261	. . . . . . . . . . . 12 ⊢ ((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z = w) → z ∈ (1st ‘A))
79	77, 78	eqeltrrd 2093	. . . . . . . . . . 11 ⊢ ((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z = w) → w ∈ (1st ‘A))
80		ltaddpr 6428	. . . . . . . . . . . . 13 ⊢ ((A ∈ P ∧ 𝐶 ∈ P) → A<P (A +P 𝐶))
81	8, 65, 80	syl2anc 393	. . . . . . . . . . . 12 ⊢ (A<P B → A<P (A +P 𝐶))
82		ltprordil 6422	. . . . . . . . . . . . 13 ⊢ (A<P (A +P 𝐶) → (1st ‘A) ⊆ (1st ‘(A +P 𝐶)))
83	82	sseld 2917	. . . . . . . . . . . 12 ⊢ (A<P (A +P 𝐶) → (w ∈ (1st ‘A) → w ∈ (1st ‘(A +P 𝐶))))
84	81, 83	syl 14	. . . . . . . . . . 11 ⊢ (A<P B → (w ∈ (1st ‘A) → w ∈ (1st ‘(A +P 𝐶))))
85	76, 79, 84	sylc 56	. . . . . . . . . 10 ⊢ ((((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) ∧ z = w) → w ∈ (1st ‘(A +P 𝐶)))
86	85	ex 108	. . . . . . . . 9 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → (z = w → w ∈ (1st ‘(A +P 𝐶))))
87		prcdnql 6332	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘A), (2nd ‘A)⟩ ∈ P ∧ z ∈ (1st ‘A)) → (w <Q z → w ∈ (1st ‘A)))
88	10, 87	sylan 267	. . . . . . . . . . 11 ⊢ ((A<P B ∧ z ∈ (1st ‘A)) → (w <Q z → w ∈ (1st ‘A)))
89	15, 16, 88	syl2anc 393	. . . . . . . . . 10 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → (w <Q z → w ∈ (1st ‘A)))
90	15, 89, 84	sylsyld 52	. . . . . . . . 9 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → (w <Q z → w ∈ (1st ‘(A +P 𝐶))))
91	75, 86, 90	3jaod 1183	. . . . . . . 8 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → ((z <Q w ∨ z = w ∨ w <Q z) → w ∈ (1st ‘(A +P 𝐶))))
92	24, 91	mpd 13	. . . . . . 7 ⊢ (((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) ∧ u <Q (z +Q v)) → w ∈ (1st ‘(A +P 𝐶)))
93	92	ex 108	. . . . . 6 ⊢ ((((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) ∧ (z ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) → (u <Q (z +Q v) → w ∈ (1st ‘(A +P 𝐶))))
94	93	rexlimdvva 2414	. . . . 5 ⊢ (((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) → (∃z ∈ (1st ‘A)∃u ∈ (2nd ‘A)u <Q (z +Q v) → w ∈ (1st ‘(A +P 𝐶))))
95	13, 94	mpd 13	. . . 4 ⊢ (((A<P B ∧ w ∈ (1st ‘B)) ∧ (v ∈ Q ∧ (w +Q v) ∈ (1st ‘B))) → w ∈ (1st ‘(A +P 𝐶)))
96	7, 95	rexlimddv 2411	. . 3 ⊢ ((A<P B ∧ w ∈ (1st ‘B)) → w ∈ (1st ‘(A +P 𝐶)))
97	96	ex 108	. 2 ⊢ (A<P B → (w ∈ (1st ‘B) → w ∈ (1st ‘(A +P 𝐶))))
98	97	ssrdv 2924	1 ⊢ (A<P B → (1st ‘B) ⊆ (1st ‘(A +P 𝐶)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ w3o 870   ∧ w3a 871   = wceq 1226  ∃wex 1358   ∈ wcel 1370  ∃wrex 2281  {crab 2284   ⊆ wss 2890  ⟨cop 3349   class class class wbr 3734  ‘cfv 4825  (class class class)co 5432  1^st c1st 5684  2^nd c2nd 5685  Qcnq 6134   +_Q cplq 6136   <_Q cltq 6139  Pcnp 6145   +_P cpp 6147  <_P cltp 6149

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200  ax-iinf 4234

This theorem depends on definitions:  df-bi 110  df-dc 731  df-3or 872  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-eprel 3996  df-id 4000  df-po 4003  df-iso 4004  df-iord 4048  df-on 4050  df-suc 4053  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-1st 5686  df-2nd 5687  df-recs 5838  df-irdg 5874  df-1o 5912  df-2o 5913  df-oadd 5916  df-omul 5917  df-er 6013  df-ec 6015  df-qs 6019  df-ni 6158  df-pli 6159  df-mi 6160  df-lti 6161  df-plpq 6197  df-mpq 6198  df-enq 6200  df-nqqs 6201  df-plqqs 6202  df-mqqs 6203  df-1nqqs 6204  df-rq 6205  df-ltnqqs 6206  df-enq0 6273  df-nq0 6274  df-0nq0 6275  df-plq0 6276  df-mq0 6277  df-inp 6314  df-iplp 6316  df-iltp 6318

This theorem is referenced by:  ltexpri  6444