Theorem recexprlem1ssl 7623

Description: The lower cut of one is a subset of the lower cut of 𝐴 ·_P 𝐵. Lemma for recexpr 7628. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

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

Assertion

Ref	Expression
recexprlem1ssl	⊢ (𝐴 ∈ P → (1st ‘1P) ⊆ (1st ‘(𝐴 ·P 𝐵)))

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

Proof of Theorem recexprlem1ssl

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

Step	Hyp	Ref	Expression
1		1prl 7545	. . . 4 ⊢ (1st ‘1P) = {𝑤 ∣ 𝑤 <Q 1Q}
2	1	abeq2i 2288	. . 3 ⊢ (𝑤 ∈ (1st ‘1P) ↔ 𝑤 <Q 1Q)
3		rec1nq 7385	. . . . . . 7 ⊢ (*Q‘1Q) = 1Q
4		ltrnqi 7411	. . . . . . 7 ⊢ (𝑤 <Q 1Q → (*Q‘1Q) <Q (*Q‘𝑤))
5	3, 4	eqbrtrrid 4036	. . . . . 6 ⊢ (𝑤 <Q 1Q → 1Q <Q (*Q‘𝑤))
6		prop 7465	. . . . . . 7 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
7		prmuloc2 7557	. . . . . . 7 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 1Q <Q (*Q‘𝑤)) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))
8	6, 7	sylan 283	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 1Q <Q (*Q‘𝑤)) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))
9	5, 8	sylan2 286	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑤 <Q 1Q) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))
10		prnmaxl 7478	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → ∃𝑧 ∈ (1st ‘𝐴)𝑣 <Q 𝑧)
11	6, 10	sylan 283	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → ∃𝑧 ∈ (1st ‘𝐴)𝑣 <Q 𝑧)
12	11	ad2ant2r 509	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → ∃𝑧 ∈ (1st ‘𝐴)𝑣 <Q 𝑧)
13		elprnql 7471	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → 𝑣 ∈ Q)
14	6, 13	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → 𝑣 ∈ Q)
15	14	ad2ant2r 509	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → 𝑣 ∈ Q)
16	15	3adant3 1017	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑣 ∈ Q)
17		simp1r 1022	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑤 <Q 1Q)
18		ltrelnq 7355	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
19	18	brel 4675	. . . . . . . . . . . . 13 ⊢ (𝑤 <Q 1Q → (𝑤 ∈ Q ∧ 1Q ∈ Q))
20	19	simpld 112	. . . . . . . . . . . 12 ⊢ (𝑤 <Q 1Q → 𝑤 ∈ Q)
21	17, 20	syl 14	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑤 ∈ Q)
22		simp3 999	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑣 <Q 𝑧)
23		simp2r 1024	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))
24		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)))
25		ltrnqi 7411	. . . . . . . . . . . . . 14 ⊢ (𝑣 <Q 𝑧 → (*Q‘𝑧) <Q (*Q‘𝑣))
26		ltmnqg 7391	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ ·Q 𝑓) <Q (ℎ ·Q 𝑔)))
27	26	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <Q 𝑔 ↔ (ℎ ·Q 𝑓) <Q (ℎ ·Q 𝑔)))
28		simprl 529	. . . . . . . . . . . . . . . 16 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → 𝑣 <Q 𝑧)
29	18	brel 4675	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 <Q 𝑧 → (𝑣 ∈ Q ∧ 𝑧 ∈ Q))
30	29	simprd 114	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 <Q 𝑧 → 𝑧 ∈ Q)
31		recclnq 7382	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ Q → (*Q‘𝑧) ∈ Q)
32	28, 30, 31	3syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (*Q‘𝑧) ∈ Q)
33		recclnq 7382	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ Q → (*Q‘𝑣) ∈ Q)
34	33	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (*Q‘𝑣) ∈ Q)
35		simplr 528	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → 𝑤 ∈ Q)
36		mulcomnqg 7373	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
37	36	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
38	27, 32, 34, 35, 37	caovord2d 6038	. . . . . . . . . . . . . 14 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → ((*Q‘𝑧) <Q (*Q‘𝑣) ↔ ((*Q‘𝑧) ·Q 𝑤) <Q ((*Q‘𝑣) ·Q 𝑤)))
39	25, 38	imbitrid 154	. . . . . . . . . . . . 13 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (𝑣 <Q 𝑧 → ((*Q‘𝑧) ·Q 𝑤) <Q ((*Q‘𝑣) ·Q 𝑤)))
40		mulcomnqg 7373	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 ∈ Q ∧ (*Q‘𝑣) ∈ Q) → (𝑣 ·Q (*Q‘𝑣)) = ((*Q‘𝑣) ·Q 𝑣))
41	33, 40	mpdan 421	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ Q → (𝑣 ·Q (*Q‘𝑣)) = ((*Q‘𝑣) ·Q 𝑣))
42		recidnq 7383	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ Q → (𝑣 ·Q (*Q‘𝑣)) = 1Q)
43	41, 42	eqtr3d 2212	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ Q → ((*Q‘𝑣) ·Q 𝑣) = 1Q)
44		recidnq 7383	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ Q → (𝑤 ·Q (*Q‘𝑤)) = 1Q)
45	43, 44	oveqan12d 5888	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → (((*Q‘𝑣) ·Q 𝑣) ·Q (𝑤 ·Q (*Q‘𝑤))) = (1Q ·Q 1Q))
46	45	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (((*Q‘𝑣) ·Q 𝑣) ·Q (𝑤 ·Q (*Q‘𝑤))) = (1Q ·Q 1Q))
47		simpll 527	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → 𝑣 ∈ Q)
48		mulassnqg 7374	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 ·Q 𝑔) ·Q ℎ) = (𝑓 ·Q (𝑔 ·Q ℎ)))
49	48	adantl 277	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → ((𝑓 ·Q 𝑔) ·Q ℎ) = (𝑓 ·Q (𝑔 ·Q ℎ)))
50		recclnq 7382	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ Q → (*Q‘𝑤) ∈ Q)
51	35, 50	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (*Q‘𝑤) ∈ Q)
52		mulclnq 7366	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) ∈ Q)
53	52	adantl 277	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 ·Q 𝑔) ∈ Q)
54	34, 47, 35, 37, 49, 51, 53	caov4d 6053	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (((*Q‘𝑣) ·Q 𝑣) ·Q (𝑤 ·Q (*Q‘𝑤))) = (((*Q‘𝑣) ·Q 𝑤) ·Q (𝑣 ·Q (*Q‘𝑤))))
55	46, 54	eqtr3d 2212	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (1Q ·Q 1Q) = (((*Q‘𝑣) ·Q 𝑤) ·Q (𝑣 ·Q (*Q‘𝑤))))
56		1nq 7356	. . . . . . . . . . . . . . . . . 18 ⊢ 1Q ∈ Q
57		mulidnq 7379	. . . . . . . . . . . . . . . . . 18 ⊢ (1Q ∈ Q → (1Q ·Q 1Q) = 1Q)
58	56, 57	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (1Q ·Q 1Q) = 1Q
59	55, 58	eqtr3di 2225	. . . . . . . . . . . . . . . 16 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (((*Q‘𝑣) ·Q 𝑤) ·Q (𝑣 ·Q (*Q‘𝑤))) = 1Q)
60		mulclnq 7366	. . . . . . . . . . . . . . . . . . 19 ⊢ (((*Q‘𝑣) ∈ Q ∧ 𝑤 ∈ Q) → ((*Q‘𝑣) ·Q 𝑤) ∈ Q)
61	33, 60	sylan 283	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → ((*Q‘𝑣) ·Q 𝑤) ∈ Q)
62		mulclnq 7366	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ Q ∧ (*Q‘𝑤) ∈ Q) → (𝑣 ·Q (*Q‘𝑤)) ∈ Q)
63	50, 62	sylan2 286	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → (𝑣 ·Q (*Q‘𝑤)) ∈ Q)
64		recmulnqg 7381	. . . . . . . . . . . . . . . . . 18 ⊢ ((((*Q‘𝑣) ·Q 𝑤) ∈ Q ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ Q) → ((*Q‘((*Q‘𝑣) ·Q 𝑤)) = (𝑣 ·Q (*Q‘𝑤)) ↔ (((*Q‘𝑣) ·Q 𝑤) ·Q (𝑣 ·Q (*Q‘𝑤))) = 1Q))
65	61, 63, 64	syl2anc 411	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → ((*Q‘((*Q‘𝑣) ·Q 𝑤)) = (𝑣 ·Q (*Q‘𝑤)) ↔ (((*Q‘𝑣) ·Q 𝑤) ·Q (𝑣 ·Q (*Q‘𝑤))) = 1Q))
66	65	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → ((*Q‘((*Q‘𝑣) ·Q 𝑤)) = (𝑣 ·Q (*Q‘𝑤)) ↔ (((*Q‘𝑣) ·Q 𝑤) ·Q (𝑣 ·Q (*Q‘𝑤))) = 1Q))
67	59, 66	mpbird 167	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (*Q‘((*Q‘𝑣) ·Q 𝑤)) = (𝑣 ·Q (*Q‘𝑤)))
68	67	eleq1d 2246	. . . . . . . . . . . . . 14 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → ((*Q‘((*Q‘𝑣) ·Q 𝑤)) ∈ (2nd ‘𝐴) ↔ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)))
69	68	biimprd 158	. . . . . . . . . . . . 13 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → ((𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴) → (*Q‘((*Q‘𝑣) ·Q 𝑤)) ∈ (2nd ‘𝐴)))
70		breq2 4004	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ((*Q‘𝑣) ·Q 𝑤) → (((*Q‘𝑧) ·Q 𝑤) <Q 𝑦 ↔ ((*Q‘𝑧) ·Q 𝑤) <Q ((*Q‘𝑣) ·Q 𝑤)))
71		fveq2 5511	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ((*Q‘𝑣) ·Q 𝑤) → (*Q‘𝑦) = (*Q‘((*Q‘𝑣) ·Q 𝑤)))
72	71	eleq1d 2246	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ((*Q‘𝑣) ·Q 𝑤) → ((*Q‘𝑦) ∈ (2nd ‘𝐴) ↔ (*Q‘((*Q‘𝑣) ·Q 𝑤)) ∈ (2nd ‘𝐴)))
73	70, 72	anbi12d 473	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ((*Q‘𝑣) ·Q 𝑤) → ((((*Q‘𝑧) ·Q 𝑤) <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) ↔ (((*Q‘𝑧) ·Q 𝑤) <Q ((*Q‘𝑣) ·Q 𝑤) ∧ (*Q‘((*Q‘𝑣) ·Q 𝑤)) ∈ (2nd ‘𝐴))))
74	73	spcegv 2825	. . . . . . . . . . . . . . . 16 ⊢ (((*Q‘𝑣) ·Q 𝑤) ∈ Q → ((((*Q‘𝑧) ·Q 𝑤) <Q ((*Q‘𝑣) ·Q 𝑤) ∧ (*Q‘((*Q‘𝑣) ·Q 𝑤)) ∈ (2nd ‘𝐴)) → ∃𝑦(((*Q‘𝑧) ·Q 𝑤) <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
75	61, 74	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → ((((*Q‘𝑧) ·Q 𝑤) <Q ((*Q‘𝑣) ·Q 𝑤) ∧ (*Q‘((*Q‘𝑣) ·Q 𝑤)) ∈ (2nd ‘𝐴)) → ∃𝑦(((*Q‘𝑧) ·Q 𝑤) <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
76		recexpr.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩
77	76	recexprlemell 7612	. . . . . . . . . . . . . . 15 ⊢ (((*Q‘𝑧) ·Q 𝑤) ∈ (1st ‘𝐵) ↔ ∃𝑦(((*Q‘𝑧) ·Q 𝑤) <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))
78	75, 77	syl6ibr 162	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → ((((*Q‘𝑧) ·Q 𝑤) <Q ((*Q‘𝑣) ·Q 𝑤) ∧ (*Q‘((*Q‘𝑣) ·Q 𝑤)) ∈ (2nd ‘𝐴)) → ((*Q‘𝑧) ·Q 𝑤) ∈ (1st ‘𝐵)))
79	78	adantr 276	. . . . . . . . . . . . 13 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → ((((*Q‘𝑧) ·Q 𝑤) <Q ((*Q‘𝑣) ·Q 𝑤) ∧ (*Q‘((*Q‘𝑣) ·Q 𝑤)) ∈ (2nd ‘𝐴)) → ((*Q‘𝑧) ·Q 𝑤) ∈ (1st ‘𝐵)))
80	39, 69, 79	syl2and 295	. . . . . . . . . . . 12 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → ((𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) → ((*Q‘𝑧) ·Q 𝑤) ∈ (1st ‘𝐵)))
81	24, 80	mpd 13	. . . . . . . . . . 11 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → ((*Q‘𝑧) ·Q 𝑤) ∈ (1st ‘𝐵))
82	16, 21, 22, 23, 81	syl22anc 1239	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → ((*Q‘𝑧) ·Q 𝑤) ∈ (1st ‘𝐵))
83	30	3ad2ant3 1020	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑧 ∈ Q)
84		mulidnq 7379	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = 𝑤)
85		mulcomnqg 7373	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ Q ∧ 1Q ∈ Q) → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
86	56, 85	mpan2 425	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
87	84, 86	eqtr3d 2212	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ Q → 𝑤 = (1Q ·Q 𝑤))
88	87	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → 𝑤 = (1Q ·Q 𝑤))
89		recidnq 7383	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ Q → (𝑧 ·Q (*Q‘𝑧)) = 1Q)
90	89	oveq1d 5884	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ Q → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
91	90	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
92		mulassnqg 7374	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ Q ∧ (*Q‘𝑧) ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
93	31, 92	syl3an2 1272	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
94	93	3anidm12 1295	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
95	88, 91, 94	3eqtr2d 2216	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
96	83, 21, 95	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
97		oveq2 5877	. . . . . . . . . . . 12 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑧 ·Q 𝑥) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
98	97	eqeq2d 2189	. . . . . . . . . . 11 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑤 = (𝑧 ·Q 𝑥) ↔ 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤))))
99	98	rspcev 2841	. . . . . . . . . 10 ⊢ ((((*Q‘𝑧) ·Q 𝑤) ∈ (1st ‘𝐵) ∧ 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤))) → ∃𝑥 ∈ (1st ‘𝐵)𝑤 = (𝑧 ·Q 𝑥))
100	82, 96, 99	syl2anc 411	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → ∃𝑥 ∈ (1st ‘𝐵)𝑤 = (𝑧 ·Q 𝑥))
101	100	3expia 1205	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (𝑣 <Q 𝑧 → ∃𝑥 ∈ (1st ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
102	101	reximdv 2578	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (∃𝑧 ∈ (1st ‘𝐴)𝑣 <Q 𝑧 → ∃𝑧 ∈ (1st ‘𝐴)∃𝑥 ∈ (1st ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
103	76	recexprlempr 7622	. . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
104		df-imp 7459	. . . . . . . . . 10 ⊢ ·P = (𝑦 ∈ P, 𝑤 ∈ P ↦ ⟨{𝑢 ∈ Q ∣ ∃𝑓 ∈ Q ∃𝑔 ∈ Q (𝑓 ∈ (1st ‘𝑦) ∧ 𝑔 ∈ (1st ‘𝑤) ∧ 𝑢 = (𝑓 ·Q 𝑔))}, {𝑢 ∈ Q ∣ ∃𝑓 ∈ Q ∃𝑔 ∈ Q (𝑓 ∈ (2nd ‘𝑦) ∧ 𝑔 ∈ (2nd ‘𝑤) ∧ 𝑢 = (𝑓 ·Q 𝑔))}⟩)
105	104, 52	genpelvl 7502	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st ‘𝐴)∃𝑥 ∈ (1st ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
106	103, 105	mpdan 421	. . . . . . . 8 ⊢ (𝐴 ∈ P → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st ‘𝐴)∃𝑥 ∈ (1st ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
107	106	ad2antrr 488	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st ‘𝐴)∃𝑥 ∈ (1st ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
108	102, 107	sylibrd 169	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (∃𝑧 ∈ (1st ‘𝐴)𝑣 <Q 𝑧 → 𝑤 ∈ (1st ‘(𝐴 ·P 𝐵))))
109	12, 108	mpd 13	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → 𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)))
110	9, 109	rexlimddv 2599	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑤 <Q 1Q) → 𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)))
111	110	ex 115	. . 3 ⊢ (𝐴 ∈ P → (𝑤 <Q 1Q → 𝑤 ∈ (1st ‘(𝐴 ·P 𝐵))))
112	2, 111	biimtrid 152	. 2 ⊢ (𝐴 ∈ P → (𝑤 ∈ (1st ‘1P) → 𝑤 ∈ (1st ‘(𝐴 ·P 𝐵))))
113	112	ssrdv 3161	1 ⊢ (𝐴 ∈ P → (1st ‘1P) ⊆ (1st ‘(𝐴 ·P 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353  ∃wex 1492   ∈ wcel 2148  {cab 2163  ∃wrex 2456   ⊆ wss 3129  ⟨cop 3594   class class class wbr 4000  ‘cfv 5212  (class class class)co 5869  1^st c1st 6133  2^nd c2nd 6134  Qcnq 7270  1_Qc1q 7271   ·_Q cmq 7273  *_Qcrq 7274   <_Q cltq 7275  Pcnp 7281  1_Pc1p 7282   ·_P cmp 7284

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-i1p 7457  df-imp 7459

This theorem is referenced by:  recexprlemex  7627