Theorem recexprlem1ssl 7570

Description: The lower cut of one is a subset of the lower cut of 𝐴 ·_P 𝐵. Lemma for recexpr 7575. (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 7492	. . . 4 ⊢ (1st ‘1P) = {𝑤 ∣ 𝑤 <Q 1Q}
2	1	abeq2i 2276	. . 3 ⊢ (𝑤 ∈ (1st ‘1P) ↔ 𝑤 <Q 1Q)
3		rec1nq 7332	. . . . . . 7 ⊢ (*Q‘1Q) = 1Q
4		ltrnqi 7358	. . . . . . 7 ⊢ (𝑤 <Q 1Q → (*Q‘1Q) <Q (*Q‘𝑤))
5	3, 4	eqbrtrrid 4017	. . . . . 6 ⊢ (𝑤 <Q 1Q → 1Q <Q (*Q‘𝑤))
6		prop 7412	. . . . . . 7 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
7		prmuloc2 7504	. . . . . . 7 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 1Q <Q (*Q‘𝑤)) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))
8	6, 7	sylan 281	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 1Q <Q (*Q‘𝑤)) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))
9	5, 8	sylan2 284	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑤 <Q 1Q) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))
10		prnmaxl 7425	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → ∃𝑧 ∈ (1st ‘𝐴)𝑣 <Q 𝑧)
11	6, 10	sylan 281	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → ∃𝑧 ∈ (1st ‘𝐴)𝑣 <Q 𝑧)
12	11	ad2ant2r 501	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → ∃𝑧 ∈ (1st ‘𝐴)𝑣 <Q 𝑧)
13		elprnql 7418	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → 𝑣 ∈ Q)
14	6, 13	sylan 281	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → 𝑣 ∈ Q)
15	14	ad2ant2r 501	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → 𝑣 ∈ Q)
16	15	3adant3 1007	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑣 ∈ Q)
17		simp1r 1012	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑤 <Q 1Q)
18		ltrelnq 7302	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
19	18	brel 4655	. . . . . . . . . . . . 13 ⊢ (𝑤 <Q 1Q → (𝑤 ∈ Q ∧ 1Q ∈ Q))
20	19	simpld 111	. . . . . . . . . . . 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 989	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑣 <Q 𝑧)
23		simp2r 1014	. . . . . . . . . . 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 109	. . . . . . . . . . . 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 7358	. . . . . . . . . . . . . 14 ⊢ (𝑣 <Q 𝑧 → (*Q‘𝑧) <Q (*Q‘𝑣))
26		ltmnqg 7338	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ ·Q 𝑓) <Q (ℎ ·Q 𝑔)))
27	26	adantl 275	. . . . . . . . . . . . . . 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 521	. . . . . . . . . . . . . . . 16 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → 𝑣 <Q 𝑧)
29	18	brel 4655	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 <Q 𝑧 → (𝑣 ∈ Q ∧ 𝑧 ∈ Q))
30	29	simprd 113	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 <Q 𝑧 → 𝑧 ∈ Q)
31		recclnq 7329	. . . . . . . . . . . . . . . 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 7329	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ Q → (*Q‘𝑣) ∈ Q)
34	33	ad2antrr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (*Q‘𝑣) ∈ Q)
35		simplr 520	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → 𝑤 ∈ Q)
36		mulcomnqg 7320	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
37	36	adantl 275	. . . . . . . . . . . . . . 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 6007	. . . . . . . . . . . . . 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	syl5ib 153	. . . . . . . . . . . . 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 7320	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 ∈ Q ∧ (*Q‘𝑣) ∈ Q) → (𝑣 ·Q (*Q‘𝑣)) = ((*Q‘𝑣) ·Q 𝑣))
41	33, 40	mpdan 418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ Q → (𝑣 ·Q (*Q‘𝑣)) = ((*Q‘𝑣) ·Q 𝑣))
42		recidnq 7330	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ Q → (𝑣 ·Q (*Q‘𝑣)) = 1Q)
43	41, 42	eqtr3d 2200	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ Q → ((*Q‘𝑣) ·Q 𝑣) = 1Q)
44		recidnq 7330	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ Q → (𝑤 ·Q (*Q‘𝑤)) = 1Q)
45	43, 44	oveqan12d 5860	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → (((*Q‘𝑣) ·Q 𝑣) ·Q (𝑤 ·Q (*Q‘𝑤))) = (1Q ·Q 1Q))
46	45	adantr 274	. . . . . . . . . . . . . . . . . 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 519	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → 𝑣 ∈ Q)
48		mulassnqg 7321	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 ·Q 𝑔) ·Q ℎ) = (𝑓 ·Q (𝑔 ·Q ℎ)))
49	48	adantl 275	. . . . . . . . . . . . . . . . . . 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 7329	. . . . . . . . . . . . . . . . . . . 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 7313	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) ∈ Q)
53	52	adantl 275	. . . . . . . . . . . . . . . . . . 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 6022	. . . . . . . . . . . . . . . . . 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 2200	. . . . . . . . . . . . . . . . 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 7303	. . . . . . . . . . . . . . . . . 18 ⊢ 1Q ∈ Q
57		mulidnq 7326	. . . . . . . . . . . . . . . . . 18 ⊢ (1Q ∈ Q → (1Q ·Q 1Q) = 1Q)
58	56, 57	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (1Q ·Q 1Q) = 1Q
59	55, 58	eqtr3di 2213	. . . . . . . . . . . . . . . 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 7313	. . . . . . . . . . . . . . . . . . 19 ⊢ (((*Q‘𝑣) ∈ Q ∧ 𝑤 ∈ Q) → ((*Q‘𝑣) ·Q 𝑤) ∈ Q)
61	33, 60	sylan 281	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → ((*Q‘𝑣) ·Q 𝑤) ∈ Q)
62		mulclnq 7313	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ Q ∧ (*Q‘𝑤) ∈ Q) → (𝑣 ·Q (*Q‘𝑤)) ∈ Q)
63	50, 62	sylan2 284	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → (𝑣 ·Q (*Q‘𝑤)) ∈ Q)
64		recmulnqg 7328	. . . . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . . 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 166	. . . . . . . . . . . . . . 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 2234	. . . . . . . . . . . . . 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 157	. . . . . . . . . . . . 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 3985	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ((*Q‘𝑣) ·Q 𝑤) → (((*Q‘𝑧) ·Q 𝑤) <Q 𝑦 ↔ ((*Q‘𝑧) ·Q 𝑤) <Q ((*Q‘𝑣) ·Q 𝑤)))
71		fveq2 5485	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ((*Q‘𝑣) ·Q 𝑤) → (*Q‘𝑦) = (*Q‘((*Q‘𝑣) ·Q 𝑤)))
72	71	eleq1d 2234	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ((*Q‘𝑣) ·Q 𝑤) → ((*Q‘𝑦) ∈ (2nd ‘𝐴) ↔ (*Q‘((*Q‘𝑣) ·Q 𝑤)) ∈ (2nd ‘𝐴)))
73	70, 72	anbi12d 465	. . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . 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 7559	. . . . . . . . . . . . . . 15 ⊢ (((*Q‘𝑧) ·Q 𝑤) ∈ (1st ‘𝐵) ↔ ∃𝑦(((*Q‘𝑧) ·Q 𝑤) <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))
78	75, 77	syl6ibr 161	. . . . . . . . . . . . . 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 274	. . . . . . . . . . . . 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 293	. . . . . . . . . . . 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 1229	. . . . . . . . . 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 1010	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑧 ∈ Q)
84		mulidnq 7326	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = 𝑤)
85		mulcomnqg 7320	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ Q ∧ 1Q ∈ Q) → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
86	56, 85	mpan2 422	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
87	84, 86	eqtr3d 2200	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ Q → 𝑤 = (1Q ·Q 𝑤))
88	87	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → 𝑤 = (1Q ·Q 𝑤))
89		recidnq 7330	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ Q → (𝑧 ·Q (*Q‘𝑧)) = 1Q)
90	89	oveq1d 5856	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ Q → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
91	90	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
92		mulassnqg 7321	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ Q ∧ (*Q‘𝑧) ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
93	31, 92	syl3an2 1262	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
94	93	3anidm12 1285	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
95	88, 91, 94	3eqtr2d 2204	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
96	83, 21, 95	syl2anc 409	. . . . . . . . . 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 5849	. . . . . . . . . . . 12 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑧 ·Q 𝑥) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
98	97	eqeq2d 2177	. . . . . . . . . . 11 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑤 = (𝑧 ·Q 𝑥) ↔ 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤))))
99	98	rspcev 2829	. . . . . . . . . 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 409	. . . . . . . . 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 1195	. . . . . . . 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 2566	. . . . . . 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 7569	. . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
104		df-imp 7406	. . . . . . . . . 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 7449	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st ‘𝐴)∃𝑥 ∈ (1st ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
106	103, 105	mpdan 418	. . . . . . . 8 ⊢ (𝐴 ∈ P → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st ‘𝐴)∃𝑥 ∈ (1st ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
107	106	ad2antrr 480	. . . . . . 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 168	. . . . . 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 2587	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑤 <Q 1Q) → 𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)))
111	110	ex 114	. . 3 ⊢ (𝐴 ∈ P → (𝑤 <Q 1Q → 𝑤 ∈ (1st ‘(𝐴 ·P 𝐵))))
112	2, 111	syl5bi 151	. 2 ⊢ (𝐴 ∈ P → (𝑤 ∈ (1st ‘1P) → 𝑤 ∈ (1st ‘(𝐴 ·P 𝐵))))
113	112	ssrdv 3147	1 ⊢ (𝐴 ∈ P → (1st ‘1P) ⊆ (1st ‘(𝐴 ·P 𝐵)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 968   = wceq 1343  ∃wex 1480   ∈ wcel 2136  {cab 2151  ∃wrex 2444   ⊆ wss 3115  ⟨cop 3578   class class class wbr 3981  ‘cfv 5187  (class class class)co 5841  1^st c1st 6103  2^nd c2nd 6104  Qcnq 7217  1_Qc1q 7218   ·_Q cmq 7220  *_Qcrq 7221   <_Q cltq 7222  Pcnp 7228  1_Pc1p 7229   ·_P cmp 7231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-ral 2448  df-rex 2449  df-reu 2450  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-eprel 4266  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-irdg 6334  df-1o 6380  df-2o 6381  df-oadd 6384  df-omul 6385  df-er 6497  df-ec 6499  df-qs 6503  df-ni 7241  df-pli 7242  df-mi 7243  df-lti 7244  df-plpq 7281  df-mpq 7282  df-enq 7284  df-nqqs 7285  df-plqqs 7286  df-mqqs 7287  df-1nqqs 7288  df-rq 7289  df-ltnqqs 7290  df-enq0 7361  df-nq0 7362  df-0nq0 7363  df-plq0 7364  df-mq0 7365  df-inp 7403  df-i1p 7404  df-imp 7406

This theorem is referenced by:  recexprlemex  7574