Theorem ltexprlemm 6572

Description: Our constructed difference is inhabited. Lemma for ltexpri 6585. (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 6487	. . . . . . . . 9 ⊢ <P ⊆ (P × P)
2	1	brel 4335	. . . . . . . 8 ⊢ (A<P B → (A ∈ P ∧ B ∈ P))
3		ltdfpr 6488	. . . . . . . . 9 ⊢ ((A ∈ P ∧ B ∈ P) → (A<P B ↔ ∃y ∈ Q (y ∈ (2nd ‘A) ∧ y ∈ (1st ‘B))))
4	3	biimpd 132	. . . . . . . 8 ⊢ ((A ∈ P ∧ B ∈ P) → (A<P B → ∃y ∈ Q (y ∈ (2nd ‘A) ∧ y ∈ (1st ‘B))))
5	2, 4	mpcom 32	. . . . . . 7 ⊢ (A<P B → ∃y ∈ Q (y ∈ (2nd ‘A) ∧ y ∈ (1st ‘B)))
6		simprrl 491	. . . . . . . . . 10 ⊢ ((A<P B ∧ (y ∈ Q ∧ (y ∈ (2nd ‘A) ∧ y ∈ (1st ‘B)))) → y ∈ (2nd ‘A))
7	2	simprd 107	. . . . . . . . . . . . 13 ⊢ (A<P B → B ∈ P)
8		prop 6457	. . . . . . . . . . . . . . . . . 18 ⊢ (B ∈ P → ⟨(1st ‘B), (2nd ‘B)⟩ ∈ P)
9		prnmaxl 6470	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘B), (2nd ‘B)⟩ ∈ P ∧ y ∈ (1st ‘B)) → ∃w ∈ (1st ‘B)y <Q w)
10	8, 9	sylan 267	. . . . . . . . . . . . . . . . 17 ⊢ ((B ∈ P ∧ y ∈ (1st ‘B)) → ∃w ∈ (1st ‘B)y <Q w)
11		ltexnqi 6392	. . . . . . . . . . . . . . . . . 18 ⊢ (y <Q w → ∃𝑞 ∈ Q (y +Q 𝑞) = w)
12	11	reximi 2410	. . . . . . . . . . . . . . . . 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 2306	. . . . . . . . . . . . . . . 16 ⊢ (∃w ∈ (1st ‘B)∃𝑞 ∈ Q (y +Q 𝑞) = w ↔ ∃w(w ∈ (1st ‘B) ∧ ∃𝑞 ∈ Q (y +Q 𝑞) = w))
15	13, 14	sylib 127	. . . . . . . . . . . . . . 15 ⊢ ((B ∈ P ∧ y ∈ (1st ‘B)) → ∃w(w ∈ (1st ‘B) ∧ ∃𝑞 ∈ Q (y +Q 𝑞) = w))
16		r19.42v 2461	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑞 ∈ Q (w ∈ (1st ‘B) ∧ (y +Q 𝑞) = w) ↔ (w ∈ (1st ‘B) ∧ ∃𝑞 ∈ Q (y +Q 𝑞) = w))
17	16	exbii 1493	. . . . . . . . . . . . . . 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 137	. . . . . . . . . . . . . 14 ⊢ ((B ∈ P ∧ y ∈ (1st ‘B)) → ∃w∃𝑞 ∈ Q (w ∈ (1st ‘B) ∧ (y +Q 𝑞) = w))
19		eleq1 2097	. . . . . . . . . . . . . . . . 17 ⊢ ((y +Q 𝑞) = w → ((y +Q 𝑞) ∈ (1st ‘B) ↔ w ∈ (1st ‘B)))
20	19	biimparc 283	. . . . . . . . . . . . . . . 16 ⊢ ((w ∈ (1st ‘B) ∧ (y +Q 𝑞) = w) → (y +Q 𝑞) ∈ (1st ‘B))
21	20	reximi 2410	. . . . . . . . . . . . . . 15 ⊢ (∃𝑞 ∈ Q (w ∈ (1st ‘B) ∧ (y +Q 𝑞) = w) → ∃𝑞 ∈ Q (y +Q 𝑞) ∈ (1st ‘B))
22	21	exlimiv 1486	. . . . . . . . . . . . . 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 267	. . . . . . . . . . . 12 ⊢ ((A<P B ∧ y ∈ (1st ‘B)) → ∃𝑞 ∈ Q (y +Q 𝑞) ∈ (1st ‘B))
25	24	adantrl 447	. . . . . . . . . . 11 ⊢ ((A<P B ∧ (y ∈ (2nd ‘A) ∧ y ∈ (1st ‘B))) → ∃𝑞 ∈ Q (y +Q 𝑞) ∈ (1st ‘B))
26	25	adantrl 447	. . . . . . . . . 10 ⊢ ((A<P B ∧ (y ∈ Q ∧ (y ∈ (2nd ‘A) ∧ y ∈ (1st ‘B)))) → ∃𝑞 ∈ Q (y +Q 𝑞) ∈ (1st ‘B))
27	6, 26	jca 290	. . . . . . . . 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 357	. . . . . . . 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 2415	. . . . . . 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 2461	. . . . . . 7 ⊢ (∃𝑞 ∈ Q (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈ (1st ‘B)) ↔ (y ∈ (2nd ‘A) ∧ ∃𝑞 ∈ Q (y +Q 𝑞) ∈ (1st ‘B)))
32	31	rexbii 2325	. . . . . 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 137	. . . . 5 ⊢ (A<P B → ∃y ∈ Q ∃𝑞 ∈ Q (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈ (1st ‘B)))
34		rexcom 2468	. . . . 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 127	. . . 4 ⊢ (A<P B → ∃𝑞 ∈ Q ∃y ∈ Q (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈ (1st ‘B)))
36	2	simpld 105	. . . . . . . . . . . 12 ⊢ (A<P B → A ∈ P)
37		prop 6457	. . . . . . . . . . . . 13 ⊢ (A ∈ P → ⟨(1st ‘A), (2nd ‘A)⟩ ∈ P)
38		elprnqu 6464	. . . . . . . . . . . . 13 ⊢ ((⟨(1st ‘A), (2nd ‘A)⟩ ∈ P ∧ y ∈ (2nd ‘A)) → y ∈ Q)
39	37, 38	sylan 267	. . . . . . . . . . . 12 ⊢ ((A ∈ P ∧ y ∈ (2nd ‘A)) → y ∈ Q)
40	36, 39	sylan 267	. . . . . . . . . . 11 ⊢ ((A<P B ∧ y ∈ (2nd ‘A)) → y ∈ Q)
41	40	ex 108	. . . . . . . . . 10 ⊢ (A<P B → (y ∈ (2nd ‘A) → y ∈ Q))
42	41	pm4.71rd 374	. . . . . . . . 9 ⊢ (A<P B → (y ∈ (2nd ‘A) ↔ (y ∈ Q ∧ y ∈ (2nd ‘A))))
43	42	anbi1d 438	. . . . . . . 8 ⊢ (A<P B → ((y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈ (1st ‘B)) ↔ ((y ∈ Q ∧ y ∈ (2nd ‘A)) ∧ (y +Q 𝑞) ∈ (1st ‘B))))
44		anass 381	. . . . . . . 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 185	. . . . . . 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 1703	. . . . . 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 2321	. . . . 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 2306	. . . . . 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 2325	. . . . 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 187	. . . 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 156	. . 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 6570	. . . . 5 ⊢ (𝑞 ∈ (1st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈ (1st ‘B))))
54	53	rexbii 2325	. . . 4 ⊢ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈ (1st ‘B))))
55		ssid 2958	. . . . 5 ⊢ Q ⊆ Q
56		rexss 3001	. . . . 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 176	. . 3 ⊢ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ↔ ∃𝑞 ∈ Q ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈ (1st ‘B)))
59	51, 58	sylibr 137	. 2 ⊢ (A<P B → ∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶))
60		nfv 1418	. . 3 ⊢ Ⅎ𝑟 A<P B
61		nfre1 2359	. . 3 ⊢ Ⅎ𝑟∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶)
62		prmu 6460	. . . . 5 ⊢ (⟨(1st ‘B), (2nd ‘B)⟩ ∈ P → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘B))
63		rexex 2362	. . . . 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 6464	. . . . . . 7 ⊢ ((⟨(1st ‘B), (2nd ‘B)⟩ ∈ P ∧ 𝑟 ∈ (2nd ‘B)) → 𝑟 ∈ Q)
67	8, 66	sylan 267	. . . . . 6 ⊢ ((B ∈ P ∧ 𝑟 ∈ (2nd ‘B)) → 𝑟 ∈ Q)
68	7, 67	sylan 267	. . . . 5 ⊢ ((A<P B ∧ 𝑟 ∈ (2nd ‘B)) → 𝑟 ∈ Q)
69		prml 6459	. . . . . . . . 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 2362	. . . . . . . 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 261	. . . . . 6 ⊢ ((A<P B ∧ 𝑟 ∈ (2nd ‘B)) → ∃y y ∈ (1st ‘A))
74	68	3adant3 923	. . . . . . . . 9 ⊢ ((A<P B ∧ 𝑟 ∈ (2nd ‘B) ∧ y ∈ (1st ‘A)) → 𝑟 ∈ Q)
75		simp3 905	. . . . . . . . . 10 ⊢ ((A<P B ∧ 𝑟 ∈ (2nd ‘B) ∧ y ∈ (1st ‘A)) → y ∈ (1st ‘A))
76		elprnql 6463	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘A), (2nd ‘A)⟩ ∈ P ∧ y ∈ (1st ‘A)) → y ∈ Q)
77	37, 76	sylan 267	. . . . . . . . . . . . . 14 ⊢ ((A ∈ P ∧ y ∈ (1st ‘A)) → y ∈ Q)
78	36, 77	sylan 267	. . . . . . . . . . . . 13 ⊢ ((A<P B ∧ y ∈ (1st ‘A)) → y ∈ Q)
79	78	3adant2 922	. . . . . . . . . . . 12 ⊢ ((A<P B ∧ 𝑟 ∈ (2nd ‘B) ∧ y ∈ (1st ‘A)) → y ∈ Q)
80		addcomnqg 6365	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ Q ∧ y ∈ Q) → (𝑟 +Q y) = (y +Q 𝑟))
81	74, 79, 80	syl2anc 391	. . . . . . . . . . 11 ⊢ ((A<P B ∧ 𝑟 ∈ (2nd ‘B) ∧ y ∈ (1st ‘A)) → (𝑟 +Q y) = (y +Q 𝑟))
82		ltaddnq 6390	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ Q ∧ y ∈ Q) → 𝑟 <Q (𝑟 +Q y))
83	74, 79, 82	syl2anc 391	. . . . . . . . . . . 12 ⊢ ((A<P B ∧ 𝑟 ∈ (2nd ‘B) ∧ y ∈ (1st ‘A)) → 𝑟 <Q (𝑟 +Q y))
84		prcunqu 6467	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘B), (2nd ‘B)⟩ ∈ P ∧ 𝑟 ∈ (2nd ‘B)) → (𝑟 <Q (𝑟 +Q y) → (𝑟 +Q y) ∈ (2nd ‘B)))
85	8, 84	sylan 267	. . . . . . . . . . . . . 14 ⊢ ((B ∈ P ∧ 𝑟 ∈ (2nd ‘B)) → (𝑟 <Q (𝑟 +Q y) → (𝑟 +Q y) ∈ (2nd ‘B)))
86	7, 85	sylan 267	. . . . . . . . . . . . 13 ⊢ ((A<P B ∧ 𝑟 ∈ (2nd ‘B)) → (𝑟 <Q (𝑟 +Q y) → (𝑟 +Q y) ∈ (2nd ‘B)))
87	86	3adant3 923	. . . . . . . . . . . 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 2112	. . . . . . . . . 10 ⊢ ((A<P B ∧ 𝑟 ∈ (2nd ‘B) ∧ y ∈ (1st ‘A)) → (y +Q 𝑟) ∈ (2nd ‘B))
90		19.8a 1479	. . . . . . . . . 10 ⊢ ((y ∈ (1st ‘A) ∧ (y +Q 𝑟) ∈ (2nd ‘B)) → ∃y(y ∈ (1st ‘A) ∧ (y +Q 𝑟) ∈ (2nd ‘B)))
91	75, 89, 90	syl2anc 391	. . . . . . . . 9 ⊢ ((A<P B ∧ 𝑟 ∈ (2nd ‘B) ∧ y ∈ (1st ‘A)) → ∃y(y ∈ (1st ‘A) ∧ (y +Q 𝑟) ∈ (2nd ‘B)))
92	74, 91	jca 290	. . . . . . . 8 ⊢ ((A<P B ∧ 𝑟 ∈ (2nd ‘B) ∧ y ∈ (1st ‘A)) → (𝑟 ∈ Q ∧ ∃y(y ∈ (1st ‘A) ∧ (y +Q 𝑟) ∈ (2nd ‘B))))
93	52	ltexprlemelu 6571	. . . . . . . 8 ⊢ (𝑟 ∈ (2nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃y(y ∈ (1st ‘A) ∧ (y +Q 𝑟) ∈ (2nd ‘B))))
94	92, 93	sylibr 137	. . . . . . 7 ⊢ ((A<P B ∧ 𝑟 ∈ (2nd ‘B) ∧ y ∈ (1st ‘A)) → 𝑟 ∈ (2nd ‘𝐶))
95	94	3expa 1103	. . . . . 6 ⊢ (((A<P B ∧ 𝑟 ∈ (2nd ‘B)) ∧ y ∈ (1st ‘A)) → 𝑟 ∈ (2nd ‘𝐶))
96	73, 95	exlimddv 1775	. . . . 5 ⊢ ((A<P B ∧ 𝑟 ∈ (2nd ‘B)) → 𝑟 ∈ (2nd ‘𝐶))
97		19.8a 1479	. . . . 5 ⊢ ((𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)) → ∃𝑟(𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)))
98	68, 96, 97	syl2anc 391	. . . 4 ⊢ ((A<P B ∧ 𝑟 ∈ (2nd ‘B)) → ∃𝑟(𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)))
99		df-rex 2306	. . . 4 ⊢ (∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶) ↔ ∃𝑟(𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)))
100	98, 99	sylibr 137	. . 3 ⊢ ((A<P B ∧ 𝑟 ∈ (2nd ‘B)) → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶))
101	60, 61, 65, 100	exlimdd 1749	. 2 ⊢ (A<P B → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶))
102	59, 101	jca 290	1 ⊢ (A<P B → (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  {crab 2304   ⊆ wss 2911  ⟨cop 3370   class class class wbr 3755  ‘cfv 4845  (class class class)co 5455  1^st c1st 5707  2^nd c2nd 5708  Qcnq 6264   +_Q cplq 6266   <_Q cltq 6269  Pcnp 6275  <_P cltp 6279

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 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

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 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-ltnqqs 6337  df-inp 6448  df-iltp 6452

This theorem is referenced by:  ltexprlempr  6580