Theorem ltexprlemm 7820

Description: Our constructed difference is inhabited. Lemma for ltexpri 7833. (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 7725	. . . . . . . . 9 ⊢ <P ⊆ (P × P)
2	1	brel 4778	. . . . . . . 8 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3		ltdfpr 7726	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ∃𝑦 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵))))
4	3	biimpd 144	. . . . . . . 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 541	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 ∈ Q ∧ (𝑦 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵)))) → 𝑦 ∈ (2nd ‘𝐴))
7	2	simprd 114	. . . . . . . . . . . . 13 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
8		prop 7695	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
9		prnmaxl 7708	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐵)) → ∃𝑤 ∈ (1st ‘𝐵)𝑦 <Q 𝑤)
10	8, 9	sylan 283	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (1st ‘𝐵)) → ∃𝑤 ∈ (1st ‘𝐵)𝑦 <Q 𝑤)
11		ltexnqi 7629	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 <Q 𝑤 → ∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑤)
12	11	reximi 2629	. . . . . . . . . . . . . . . . 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 2516	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤 ∈ (1st ‘𝐵)∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑤 ↔ ∃𝑤(𝑤 ∈ (1st ‘𝐵) ∧ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑤))
15	13, 14	sylib 122	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (1st ‘𝐵)) → ∃𝑤(𝑤 ∈ (1st ‘𝐵) ∧ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑤))
16		r19.42v 2690	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑞 ∈ Q (𝑤 ∈ (1st ‘𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) ↔ (𝑤 ∈ (1st ‘𝐵) ∧ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑤))
17	16	exbii 1653	. . . . . . . . . . . . . . 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 134	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (1st ‘𝐵)) → ∃𝑤∃𝑞 ∈ Q (𝑤 ∈ (1st ‘𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤))
19		eleq1 2294	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 +Q 𝑞) = 𝑤 → ((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ↔ 𝑤 ∈ (1st ‘𝐵)))
20	19	biimparc 299	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ (1st ‘𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) → (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))
21	20	reximi 2629	. . . . . . . . . . . . . . 15 ⊢ (∃𝑞 ∈ Q (𝑤 ∈ (1st ‘𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) → ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))
22	21	exlimiv 1646	. . . . . . . . . . . . . 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 283	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ (1st ‘𝐵)) → ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))
25	24	adantrl 478	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵))) → ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))
26	25	adantrl 478	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 ∈ Q ∧ (𝑦 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵)))) → ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))
27	6, 26	jca 306	. . . . . . . . 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 375	. . . . . . . 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 2634	. . . . . . 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 2690	. . . . . . 7 ⊢ (∃𝑞 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ (𝑦 ∈ (2nd ‘𝐴) ∧ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
32	31	rexbii 2539	. . . . . 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 134	. . . . 5 ⊢ (𝐴<P 𝐵 → ∃𝑦 ∈ Q ∃𝑞 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
34		rexcom 2697	. . . . 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 122	. . . 4 ⊢ (𝐴<P 𝐵 → ∃𝑞 ∈ Q ∃𝑦 ∈ Q (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
36	2	simpld 112	. . . . . . . . . . . 12 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P)
37		prop 7695	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
38		elprnqu 7702	. . . . . . . . . . . . 13 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
39	37, 38	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
40	36, 39	sylan 283	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
41	40	ex 115	. . . . . . . . . 10 ⊢ (𝐴<P 𝐵 → (𝑦 ∈ (2nd ‘𝐴) → 𝑦 ∈ Q))
42	41	pm4.71rd 394	. . . . . . . . 9 ⊢ (𝐴<P 𝐵 → (𝑦 ∈ (2nd ‘𝐴) ↔ (𝑦 ∈ Q ∧ 𝑦 ∈ (2nd ‘𝐴))))
43	42	anbi1d 465	. . . . . . . 8 ⊢ (𝐴<P 𝐵 → ((𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ ((𝑦 ∈ Q ∧ 𝑦 ∈ (2nd ‘𝐴)) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
44		anass 401	. . . . . . . 8 ⊢ (((𝑦 ∈ Q ∧ 𝑦 ∈ (2nd ‘𝐴)) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ (𝑦 ∈ Q ∧ (𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
45	43, 44	bitrdi 196	. . . . . . 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 1873	. . . . . 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 2533	. . . . 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 2516	. . . . . 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 2539	. . . . 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	bitr4di 198	. . . 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 167	. . 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 7818	. . . . 5 ⊢ (𝑞 ∈ (1st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
54	53	rexbii 2539	. . . 4 ⊢ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
55		ssid 3247	. . . . 5 ⊢ Q ⊆ Q
56		rexss 3294	. . . . 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 5	. . . 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 187	. . 3 ⊢ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ↔ ∃𝑞 ∈ Q ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
59	51, 58	sylibr 134	. 2 ⊢ (𝐴<P 𝐵 → ∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶))
60		nfv 1576	. . 3 ⊢ Ⅎ𝑟 𝐴<P 𝐵
61		nfre1 2575	. . 3 ⊢ Ⅎ𝑟∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶)
62		prmu 7698	. . . . 5 ⊢ (⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐵))
63		rexex 2578	. . . . 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 7702	. . . . . . 7 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑟 ∈ (2nd ‘𝐵)) → 𝑟 ∈ Q)
67	8, 66	sylan 283	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝑟 ∈ (2nd ‘𝐵)) → 𝑟 ∈ Q)
68	7, 67	sylan 283	. . . . 5 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵)) → 𝑟 ∈ Q)
69		prml 7697	. . . . . . . . 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 2578	. . . . . . . 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 276	. . . . . 6 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵)) → ∃𝑦 𝑦 ∈ (1st ‘𝐴))
74	68	3adant3 1043	. . . . . . . . 9 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑟 ∈ Q)
75		simp3 1025	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ (1st ‘𝐴))
76		elprnql 7701	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
77	37, 76	sylan 283	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
78	36, 77	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
79	78	3adant2 1042	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
80		addcomnqg 7601	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ Q ∧ 𝑦 ∈ Q) → (𝑟 +Q 𝑦) = (𝑦 +Q 𝑟))
81	74, 79, 80	syl2anc 411	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → (𝑟 +Q 𝑦) = (𝑦 +Q 𝑟))
82		ltaddnq 7627	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ Q ∧ 𝑦 ∈ Q) → 𝑟 <Q (𝑟 +Q 𝑦))
83	74, 79, 82	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑟 <Q (𝑟 +Q 𝑦))
84		prcunqu 7705	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑟 ∈ (2nd ‘𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd ‘𝐵)))
85	8, 84	sylan 283	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ P ∧ 𝑟 ∈ (2nd ‘𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd ‘𝐵)))
86	7, 85	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd ‘𝐵)))
87	86	3adant3 1043	. . . . . . . . . . . 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 2309	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))
90		19.8a 1638	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))
91	75, 89, 90	syl2anc 411	. . . . . . . . 9 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))
92	74, 91	jca 306	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))))
93	52	ltexprlemelu 7819	. . . . . . . 8 ⊢ (𝑟 ∈ (2nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))))
94	92, 93	sylibr 134	. . . . . . 7 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑟 ∈ (2nd ‘𝐶))
95	94	3expa 1229	. . . . . 6 ⊢ (((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑟 ∈ (2nd ‘𝐶))
96	73, 95	exlimddv 1947	. . . . 5 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵)) → 𝑟 ∈ (2nd ‘𝐶))
97		19.8a 1638	. . . . 5 ⊢ ((𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)) → ∃𝑟(𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)))
98	68, 96, 97	syl2anc 411	. . . 4 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵)) → ∃𝑟(𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)))
99		df-rex 2516	. . . 4 ⊢ (∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶) ↔ ∃𝑟(𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)))
100	98, 99	sylibr 134	. . 3 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ (2nd ‘𝐵)) → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶))
101	60, 61, 65, 100	exlimdd 1920	. 2 ⊢ (𝐴<P 𝐵 → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶))
102	59, 101	jca 306	1 ⊢ (𝐴<P 𝐵 → (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∃wrex 2511  {crab 2514   ⊆ wss 3200  ⟨cop 3672   class class class wbr 4088  ‘cfv 5326  (class class class)co 6018  1^st c1st 6301  2^nd c2nd 6302  Qcnq 7500   +_Q cplq 7502   <_Q cltq 7505  Pcnp 7511  <_P cltp 7515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-ltnqqs 7573  df-inp 7686  df-iltp 7690

This theorem is referenced by:  ltexprlempr  7828