Theorem ltexprlemm 6790

Description: Our constructed difference is inhabited. Lemma for ltexpri 6803. (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
ltexprlemm	⊢ (𝐴<P 𝐵 → (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶)))

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

Proof of Theorem ltexprlemm

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelpr 6695	. . . . . . . . 9 ⊢ <P ⊆ (P × P)
2	1	brel 4410	. . . . . . . 8 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3		ltdfpr 6696	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ∃𝑦 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵))))
4	3	biimpd 142	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 → ∃𝑦 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵))))
5	2, 4	mpcom 36	. . . . . . 7 ⊢ (𝐴<P 𝐵 → ∃𝑦 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵)))
6		simprrl 505	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 ∈ Q ∧ (𝑦 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵)))) → 𝑦 ∈ (2nd ‘𝐴))
7	2	simprd 112	. . . . . . . . . . . . 13 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
8		prop 6665	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
9		prnmaxl 6678	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐵)) → ∃𝑤 ∈ (1st ‘𝐵)𝑦 <Q 𝑤)
10	8, 9	sylan 277	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (1st ‘𝐵)) → ∃𝑤 ∈ (1st ‘𝐵)𝑦 <Q 𝑤)
11		ltexnqi 6599	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 <Q 𝑤 → ∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑤)
12	11	reximi 2458	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑤 ∈ (1st ‘𝐵)𝑦 <Q 𝑤 → ∃𝑤 ∈ (1st ‘𝐵)∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑤)
13	10, 12	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (1st ‘𝐵)) → ∃𝑤 ∈ (1st ‘𝐵)∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑤)
14		df-rex 2354	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤 ∈ (1st ‘𝐵)∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑤 ↔ ∃𝑤(𝑤 ∈ (1st ‘𝐵) ∧ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑤))
15	13, 14	sylib 120	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (1st ‘𝐵)) → ∃𝑤(𝑤 ∈ (1st ‘𝐵) ∧ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑤))
16		r19.42v 2511	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑞 ∈ Q (𝑤 ∈ (1st ‘𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) ↔ (𝑤 ∈ (1st ‘𝐵) ∧ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑤))
17	16	exbii 1536	. . . . . . . . . . . . . . 15 ⊢ (∃𝑤∃𝑞 ∈ Q (𝑤 ∈ (1st ‘𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) ↔ ∃𝑤(𝑤 ∈ (1st ‘𝐵) ∧ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑤))
18	15, 17	sylibr 132	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (1st ‘𝐵)) → ∃𝑤∃𝑞 ∈ Q (𝑤 ∈ (1st ‘𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤))
19		eleq1 2141	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 +Q 𝑞) = 𝑤 → ((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ↔ 𝑤 ∈ (1st ‘𝐵)))
20	19	biimparc 293	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ (1st ‘𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) → (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))
21	20	reximi 2458	. . . . . . . . . . . . . . 15 ⊢ (∃𝑞 ∈ Q (𝑤 ∈ (1st ‘𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) → ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))
22	21	exlimiv 1529	. . . . . . . . . . . . . 14 ⊢ (∃𝑤∃𝑞 ∈ Q (𝑤 ∈ (1st ‘𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) → ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))
23	18, 22	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (1st ‘𝐵)) → ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))
24	7, 23	sylan 277	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ (1st ‘𝐵)) → ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))
25	24	adantrl 461	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵))) → ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))
26	25	adantrl 461	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 ∈ Q ∧ (𝑦 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵)))) → ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))
27	6, 26	jca 300	. . . . . . . . 9 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 ∈ Q ∧ (𝑦 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵)))) → (𝑦 ∈ (2nd ‘𝐴) ∧ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
28	27	expr 367	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ Q) → ((𝑦 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵)) → (𝑦 ∈ (2nd ‘𝐴) ∧ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
29	28	reximdva 2463	. . . . . . 7 ⊢ (𝐴<P 𝐵 → (∃𝑦 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵)) → ∃𝑦 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
30	5, 29	mpd 13	. . . . . 6 ⊢ (𝐴<P 𝐵 → ∃𝑦 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
31		r19.42v 2511	. . . . . . 7 ⊢ (∃𝑞 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ (𝑦 ∈ (2nd ‘𝐴) ∧ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
32	31	rexbii 2373	. . . . . 6 ⊢ (∃𝑦 ∈ Q ∃𝑞 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ ∃𝑦 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
33	30, 32	sylibr 132	. . . . 5 ⊢ (𝐴<P 𝐵 → ∃𝑦 ∈ Q ∃𝑞 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
34		rexcom 2518	. . . . 5 ⊢ (∃𝑦 ∈ Q ∃𝑞 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ ∃𝑞 ∈ Q ∃𝑦 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
35	33, 34	sylib 120	. . . 4 ⊢ (𝐴<P 𝐵 → ∃𝑞 ∈ Q ∃𝑦 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
36	2	simpld 110	. . . . . . . . . . . 12 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P)
37		prop 6665	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
38		elprnqu 6672	. . . . . . . . . . . . 13 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
39	37, 38	sylan 277	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
40	36, 39	sylan 277	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
41	40	ex 113	. . . . . . . . . 10 ⊢ (𝐴<P 𝐵 → (𝑦 ∈ (2nd ‘𝐴) → 𝑦 ∈ Q))
42	41	pm4.71rd 386	. . . . . . . . 9 ⊢ (𝐴<P 𝐵 → (𝑦 ∈ (2nd ‘𝐴) ↔ (𝑦 ∈ Q ∧ 𝑦 ∈ (2nd ‘𝐴))))
43	42	anbi1d 452	. . . . . . . 8 ⊢ (𝐴<P 𝐵 → ((𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ ((𝑦 ∈ Q ∧ 𝑦 ∈ (2nd ‘𝐴)) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
44		anass 393	. . . . . . . 8 ⊢ (((𝑦 ∈ Q ∧ 𝑦 ∈ (2nd ‘𝐴)) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ (𝑦 ∈ Q ∧ (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
45	43, 44	syl6bb 194	. . . . . . 7 ⊢ (𝐴<P 𝐵 → ((𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ (𝑦 ∈ Q ∧ (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))))
46	45	exbidv 1746	. . . . . 6 ⊢ (𝐴<P 𝐵 → (∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ ∃𝑦(𝑦 ∈ Q ∧ (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))))
47	46	rexbidv 2369	. . . . 5 ⊢ (𝐴<P 𝐵 → (∃𝑞 ∈ Q ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ ∃𝑞 ∈ Q ∃𝑦(𝑦 ∈ Q ∧ (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))))
48		df-rex 2354	. . . . . 6 ⊢ (∃𝑦 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ ∃𝑦(𝑦 ∈ Q ∧ (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
49	48	rexbii 2373	. . . . 5 ⊢ (∃𝑞 ∈ Q ∃𝑦 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ ∃𝑞 ∈ Q ∃𝑦(𝑦 ∈ Q ∧ (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
50	47, 49	syl6bbr 196	. . . 4 ⊢ (𝐴<P 𝐵 → (∃𝑞 ∈ Q ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ ∃𝑞 ∈ Q ∃𝑦 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
51	35, 50	mpbird 165	. . 3 ⊢ (𝐴<P 𝐵 → ∃𝑞 ∈ Q ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
52		ltexprlem.1	. . . . . 6 ⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩
53	52	ltexprlemell 6788	. . . . 5 ⊢ (𝑞 ∈ (1st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
54	53	rexbii 2373	. . . 4 ⊢ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
55		ssid 3018	. . . . 5 ⊢ Q ⊆ Q
56		rexss 3061	. . . . 5 ⊢ (Q ⊆ Q → (∃𝑞 ∈ Q ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))))
57	55, 56	ax-mp 7	. . . 4 ⊢ (∃𝑞 ∈ Q ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
58	54, 57	bitr4i 185	. . 3 ⊢ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ↔ ∃𝑞 ∈ Q ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
59	51, 58	sylibr 132	. 2 ⊢ (𝐴<P 𝐵 → ∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶))
60		nfv 1461	. . 3 ⊢ Ⅎ𝑟 𝐴<P 𝐵
61		nfre1 2407	. . 3 ⊢ Ⅎ𝑟∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶)
62		prmu 6668	. . . . 5 ⊢ (⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐵))
63		rexex 2410	. . . . 5 ⊢ (∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐵) → ∃𝑟 𝑟 ∈ (2nd ‘𝐵))
64	62, 63	syl 14	. . . 4 ⊢ (⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P → ∃𝑟 𝑟 ∈ (2nd ‘𝐵))
65	7, 8, 64	3syl 17	. . 3 ⊢ (𝐴<P 𝐵 → ∃𝑟 𝑟 ∈ (2nd ‘𝐵))
66		elprnqu 6672	. . . . . . 7 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑟 ∈ (2nd ‘𝐵)) → 𝑟 ∈ Q)
67	8, 66	sylan 277	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝑟 ∈ (2nd ‘𝐵)) → 𝑟 ∈ Q)
68	7, 67	sylan 277	. . . . 5 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵)) → 𝑟 ∈ Q)
69		prml 6667	. . . . . . . . 9 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P → ∃𝑦 ∈ Q 𝑦 ∈ (1st ‘𝐴))
70	37, 69	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ P → ∃𝑦 ∈ Q 𝑦 ∈ (1st ‘𝐴))
71		rexex 2410	. . . . . . . 8 ⊢ (∃𝑦 ∈ Q 𝑦 ∈ (1st ‘𝐴) → ∃𝑦 𝑦 ∈ (1st ‘𝐴))
72	36, 70, 71	3syl 17	. . . . . . 7 ⊢ (𝐴<P 𝐵 → ∃𝑦 𝑦 ∈ (1st ‘𝐴))
73	72	adantr 270	. . . . . 6 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵)) → ∃𝑦 𝑦 ∈ (1st ‘𝐴))
74	68	3adant3 958	. . . . . . . . 9 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑟 ∈ Q)
75		simp3 940	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ (1st ‘𝐴))
76		elprnql 6671	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
77	37, 76	sylan 277	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
78	36, 77	sylan 277	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
79	78	3adant2 957	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
80		addcomnqg 6571	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ Q ∧ 𝑦 ∈ Q) → (𝑟 +Q 𝑦) = (𝑦 +Q 𝑟))
81	74, 79, 80	syl2anc 403	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → (𝑟 +Q 𝑦) = (𝑦 +Q 𝑟))
82		ltaddnq 6597	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ Q ∧ 𝑦 ∈ Q) → 𝑟 <Q (𝑟 +Q 𝑦))
83	74, 79, 82	syl2anc 403	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑟 <Q (𝑟 +Q 𝑦))
84		prcunqu 6675	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑟 ∈ (2nd ‘𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd ‘𝐵)))
85	8, 84	sylan 277	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ P ∧ 𝑟 ∈ (2nd ‘𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd ‘𝐵)))
86	7, 85	sylan 277	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd ‘𝐵)))
87	86	3adant3 958	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd ‘𝐵)))
88	83, 87	mpd 13	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → (𝑟 +Q 𝑦) ∈ (2nd ‘𝐵))
89	81, 88	eqeltrrd 2156	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))
90		19.8a 1522	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))
91	75, 89, 90	syl2anc 403	. . . . . . . . 9 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))
92	74, 91	jca 300	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))))
93	52	ltexprlemelu 6789	. . . . . . . 8 ⊢ (𝑟 ∈ (2nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))))
94	92, 93	sylibr 132	. . . . . . 7 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑟 ∈ (2nd ‘𝐶))
95	94	3expa 1138	. . . . . 6 ⊢ (((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑟 ∈ (2nd ‘𝐶))
96	73, 95	exlimddv 1819	. . . . 5 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵)) → 𝑟 ∈ (2nd ‘𝐶))
97		19.8a 1522	. . . . 5 ⊢ ((𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)) → ∃𝑟(𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)))
98	68, 96, 97	syl2anc 403	. . . 4 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵)) → ∃𝑟(𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)))
99		df-rex 2354	. . . 4 ⊢ (∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶) ↔ ∃𝑟(𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)))
100	98, 99	sylibr 132	. . 3 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵)) → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶))
101	60, 61, 65, 100	exlimdd 1793	. 2 ⊢ (𝐴<P 𝐵 → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶))
102	59, 101	jca 300	1 ⊢ (𝐴<P 𝐵 → (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 919   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∃wrex 2349  {crab 2352   ⊆ wss 2973  ⟨cop 3401   class class class wbr 3785  ‘cfv 4922  (class class class)co 5532  1^st c1st 5785  2^nd c2nd 5786  Qcnq 6470   +_Q cplq 6472   <_Q cltq 6475  Pcnp 6481  <_P cltp 6485

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-ltnqqs 6543  df-inp 6656  df-iltp 6660

This theorem is referenced by:  ltexprlempr  6798