Theorem ltexprlemrl 6574

Description: Lemma for ltexpri 6577. 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 6480	. . . . . . . 8 ⊢ <P ⊆ (P × P)
2	1	brel 4334	. . . . . . 7 ⊢ (A<P B → (A ∈ P ∧ B ∈ P))
3	2	simprd 107	. . . . . 6 ⊢ (A<P B → B ∈ P)
4		prop 6450	. . . . . 6 ⊢ (B ∈ P → ⟨(1st ‘B), (2nd ‘B)⟩ ∈ P)
5	3, 4	syl 14	. . . . 5 ⊢ (A<P B → ⟨(1st ‘B), (2nd ‘B)⟩ ∈ P)
6		prnmaddl 6465	. . . . 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 6450	. . . . . . . 8 ⊢ (A ∈ P → ⟨(1st ‘A), (2nd ‘A)⟩ ∈ P)
10	8, 9	syl 14	. . . . . . 7 ⊢ (A<P B → ⟨(1st ‘A), (2nd ‘A)⟩ ∈ P)
11		prarloc 6478	. . . . . . 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 478	. . . . 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 485	. . . . . . . . . . 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 487	. . . . . . . . . 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 6456	. . . . . . . . . . 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 391	. . . . . . . . 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 6456	. . . . . . . . . . 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 6373	. . . . . . . . 9 ⊢ ((z ∈ Q ∧ w ∈ Q) → (z <Q w ∨ z = w ∨ w <Q z))
24	19, 22, 23	syl2anc 391	. . . . . . . 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 6384	. . . . . . . . . . . . 13 ⊢ ((z ∈ Q ∧ w ∈ Q) → (z <Q w ↔ ∃𝑠 ∈ Q (z +Q 𝑠) = w))
26	19, 22, 25	syl2anc 391	. . . . . . . . . . . 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 484	. . . . . . . . . . . 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 457	. . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . 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 6460	. . . . . . . . . . . . . . . . . . 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 391	. . . . . . . . . . . . . . . . 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 457	. . . . . . . . . . . . . . 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 457	. . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . 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 6358	. . . . . . . . . . . . . . . . . 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 6359	. . . . . . . . . . . . . . . . . 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 5620	. . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . 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 5459	. . . . . . . . . . . . . . . . . . 19 ⊢ ((z +Q 𝑠) = w → ((z +Q 𝑠) +Q v) = (w +Q v))
49	48	eleq1d 2103	. . . . . . . . . . . . . . . . . 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 2111	. . . . . . . . . . . . . . 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 2097	. . . . . . . . . . . . . . . . . 18 ⊢ (y = (z +Q v) → (y ∈ (2nd ‘A) ↔ (z +Q v) ∈ (2nd ‘A)))
54		oveq1 5459	. . . . . . . . . . . . . . . . . . 19 ⊢ (y = (z +Q v) → (y +Q 𝑠) = ((z +Q v) +Q 𝑠))
55	54	eleq1d 2103	. . . . . . . . . . . . . . . . . 18 ⊢ (y = (z +Q v) → ((y +Q 𝑠) ∈ (1st ‘B) ↔ ((z +Q v) +Q 𝑠) ∈ (1st ‘B)))
56	53, 55	anbi12d 442	. . . . . . . . . . . . . . . . 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 2635	. . . . . . . . . . . . . . . 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 513	. . . . . . . . . . . . . . 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 391	. . . . . . . . . . . . . 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 6562	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (1st ‘𝐶) ↔ (𝑠 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑠) ∈ (1st ‘B))))
62	30, 59, 61	sylanbrc 394	. . . . . . . . . . . . 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 457	. . . . . . . . . . . . . 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 6572	. . . . . . . . . . . . . . . 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 457	. . . . . . . . . . . . . 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 6443	. . . . . . . . . . . . . . 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 6352	. . . . . . . . . . . . . . 15 ⊢ ((f ∈ Q ∧ v ∈ Q) → (f +Q v) ∈ Q)
70	68, 69	genpprecll 6489	. . . . . . . . . . . . . 14 ⊢ ((A ∈ P ∧ 𝐶 ∈ P) → ((z ∈ (1st ‘A) ∧ 𝑠 ∈ (1st ‘𝐶)) → (z +Q 𝑠) ∈ (1st ‘(A +P 𝐶))))
71	64, 67, 70	syl2anc 391	. . . . . . . . . . . . 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 409	. . . . . . . . . . . 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 2112	. . . . . . . . . . 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 2431	. . . . . . . . . 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 457	. . . . . . . . . . 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 2112	. . . . . . . . . . 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 6561	. . . . . . . . . . . . 13 ⊢ ((A ∈ P ∧ 𝐶 ∈ P) → A<P (A +P 𝐶))
81	8, 65, 80	syl2anc 391	. . . . . . . . . . . 12 ⊢ (A<P B → A<P (A +P 𝐶))
82		ltprordil 6555	. . . . . . . . . . . . 13 ⊢ (A<P (A +P 𝐶) → (1st ‘A) ⊆ (1st ‘(A +P 𝐶)))
83	82	sseld 2938	. . . . . . . . . . . 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 6459	. . . . . . . . . . . 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 391	. . . . . . . . . 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 1198	. . . . . . . 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 2434	. . . . 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 2431	. . 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 2945	1 ⊢ (A<P B → (1st ‘B) ⊆ (1st ‘(A +P 𝐶)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ w3o 883   ∧ w3a 884   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  {crab 2304   ⊆ wss 2911  ⟨cop 3369   class class class wbr 3754  ‘cfv 4844  (class class class)co 5452  1^st c1st 5704  2^nd c2nd 5705  Qcnq 6257   +_Q cplq 6259   <_Q cltq 6262  Pcnp 6268   +_P cpp 6270  <_P cltp 6272

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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3862  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219  ax-iinf 4253

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-int 3606  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-eprel 4016  df-id 4020  df-po 4023  df-iso 4024  df-iord 4068  df-on 4070  df-suc 4073  df-iom 4256  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-ov 5455  df-oprab 5456  df-mpt2 5457  df-1st 5706  df-2nd 5707  df-recs 5858  df-irdg 5894  df-1o 5933  df-2o 5934  df-oadd 5937  df-omul 5938  df-er 6035  df-ec 6037  df-qs 6041  df-ni 6281  df-pli 6282  df-mi 6283  df-lti 6284  df-plpq 6321  df-mpq 6322  df-enq 6324  df-nqqs 6325  df-plqqs 6326  df-mqqs 6327  df-1nqqs 6328  df-rq 6329  df-ltnqqs 6330  df-enq0 6399  df-nq0 6400  df-0nq0 6401  df-plq0 6402  df-mq0 6403  df-inp 6441  df-iplp 6443  df-iltp 6445

This theorem is referenced by:  ltexpri  6577