Theorem recexprlem1ssl 7465

Description: The lower cut of one is a subset of the lower cut of 𝐴 ·_P 𝐵. Lemma for recexpr 7470. (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 7387	. . . 4 ⊢ (1st ‘1P) = {𝑤 ∣ 𝑤 <Q 1Q}
2	1	abeq2i 2251	. . 3 ⊢ (𝑤 ∈ (1st ‘1P) ↔ 𝑤 <Q 1Q)
3		rec1nq 7227	. . . . . . 7 ⊢ (*Q‘1Q) = 1Q
4		ltrnqi 7253	. . . . . . 7 ⊢ (𝑤 <Q 1Q → (*Q‘1Q) <Q (*Q‘𝑤))
5	3, 4	eqbrtrrid 3972	. . . . . 6 ⊢ (𝑤 <Q 1Q → 1Q <Q (*Q‘𝑤))
6		prop 7307	. . . . . . 7 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
7		prmuloc2 7399	. . . . . . 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 7320	. . . . . . . 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 7313	. . . . . . . . . . . . . 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 1002	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑣 ∈ Q)
17		simp1r 1007	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑤 <Q 1Q)
18		ltrelnq 7197	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
19	18	brel 4599	. . . . . . . . . . . . 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 984	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑣 <Q 𝑧)
23		simp2r 1009	. . . . . . . . . . 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 7253	. . . . . . . . . . . . . 14 ⊢ (𝑣 <Q 𝑧 → (*Q‘𝑧) <Q (*Q‘𝑣))
26		ltmnqg 7233	. . . . . . . . . . . . . . . 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 4599	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 <Q 𝑧 → (𝑣 ∈ Q ∧ 𝑧 ∈ Q))
30	29	simprd 113	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 <Q 𝑧 → 𝑧 ∈ Q)
31		recclnq 7224	. . . . . . . . . . . . . . . 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 7224	. . . . . . . . . . . . . . . 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 7215	. . . . . . . . . . . . . . . 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 5948	. . . . . . . . . . . . . 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		1nq 7198	. . . . . . . . . . . . . . . . . 18 ⊢ 1Q ∈ Q
41		mulidnq 7221	. . . . . . . . . . . . . . . . . 18 ⊢ (1Q ∈ Q → (1Q ·Q 1Q) = 1Q)
42	40, 41	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (1Q ·Q 1Q) = 1Q
43		mulcomnqg 7215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 ∈ Q ∧ (*Q‘𝑣) ∈ Q) → (𝑣 ·Q (*Q‘𝑣)) = ((*Q‘𝑣) ·Q 𝑣))
44	33, 43	mpdan 418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ Q → (𝑣 ·Q (*Q‘𝑣)) = ((*Q‘𝑣) ·Q 𝑣))
45		recidnq 7225	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ Q → (𝑣 ·Q (*Q‘𝑣)) = 1Q)
46	44, 45	eqtr3d 2175	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ Q → ((*Q‘𝑣) ·Q 𝑣) = 1Q)
47		recidnq 7225	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ Q → (𝑤 ·Q (*Q‘𝑤)) = 1Q)
48	46, 47	oveqan12d 5801	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → (((*Q‘𝑣) ·Q 𝑣) ·Q (𝑤 ·Q (*Q‘𝑤))) = (1Q ·Q 1Q))
49	48	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))
50		simpll 519	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → 𝑣 ∈ Q)
51		mulassnqg 7216	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 ·Q 𝑔) ·Q ℎ) = (𝑓 ·Q (𝑔 ·Q ℎ)))
52	51	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)))
53		recclnq 7224	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ Q → (*Q‘𝑤) ∈ Q)
54	35, 53	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) → (*Q‘𝑤) ∈ Q)
55		mulclnq 7208	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) ∈ Q)
56	55	adantl 275	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑣 ∈ Q ∧ 𝑤 ∈ Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 ·Q 𝑔) ∈ Q)
57	34, 50, 35, 37, 52, 54, 56	caov4d 5963	. . . . . . . . . . . . . . . . . 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))))
58	49, 57	eqtr3d 2175	. . . . . . . . . . . . . . . . 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))))
59	42, 58	syl5reqr 2188	. . . . . . . . . . . . . . . 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 7208	. . . . . . . . . . . . . . . . . . 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 7208	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ Q ∧ (*Q‘𝑤) ∈ Q) → (𝑣 ·Q (*Q‘𝑤)) ∈ Q)
63	53, 62	sylan2 284	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → (𝑣 ·Q (*Q‘𝑤)) ∈ Q)
64		recmulnqg 7223	. . . . . . . . . . . . . . . . . 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 2209	. . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ((*Q‘𝑣) ·Q 𝑤) → (((*Q‘𝑧) ·Q 𝑤) <Q 𝑦 ↔ ((*Q‘𝑧) ·Q 𝑤) <Q ((*Q‘𝑣) ·Q 𝑤)))
71		fveq2 5429	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ((*Q‘𝑣) ·Q 𝑤) → (*Q‘𝑦) = (*Q‘((*Q‘𝑣) ·Q 𝑤)))
72	71	eleq1d 2209	. . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . 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 7454	. . . . . . . . . . . . . . 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 1218	. . . . . . . . . 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 1005	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q (*Q‘𝑤)) ∈ (2nd ‘𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑧 ∈ Q)
84		mulidnq 7221	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = 𝑤)
85		mulcomnqg 7215	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ Q ∧ 1Q ∈ Q) → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
86	40, 85	mpan2 422	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
87	84, 86	eqtr3d 2175	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ Q → 𝑤 = (1Q ·Q 𝑤))
88	87	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → 𝑤 = (1Q ·Q 𝑤))
89		recidnq 7225	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ Q → (𝑧 ·Q (*Q‘𝑧)) = 1Q)
90	89	oveq1d 5797	. . . . . . . . . . . . 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 7216	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ Q ∧ (*Q‘𝑧) ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
93	31, 92	syl3an2 1251	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
94	93	3anidm12 1274	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
95	88, 91, 94	3eqtr2d 2179	. . . . . . . . . . 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 5790	. . . . . . . . . . . 12 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑧 ·Q 𝑥) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
98	97	eqeq2d 2152	. . . . . . . . . . 11 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑤 = (𝑧 ·Q 𝑥) ↔ 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤))))
99	98	rspcev 2793	. . . . . . . . . 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 1184	. . . . . . . 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 2536	. . . . . . 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 7464	. . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
104		df-imp 7301	. . . . . . . . . 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, 55	genpelvl 7344	. . . . . . . . 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 2557	. . . 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 3108	1 ⊢ (𝐴 ∈ P → (1st ‘1P) ⊆ (1st ‘(𝐴 ·P 𝐵)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1332  ∃wex 1469   ∈ wcel 1481  {cab 2126  ∃wrex 2418   ⊆ wss 3076  ⟨cop 3535   class class class wbr 3937  ‘cfv 5131  (class class class)co 5782  1^st c1st 6044  2^nd c2nd 6045  Qcnq 7112  1_Qc1q 7113   ·_Q cmq 7115  *_Qcrq 7116   <_Q cltq 7117  Pcnp 7123  1_Pc1p 7124   ·_P cmp 7126

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-i1p 7299  df-imp 7301

This theorem is referenced by:  recexprlemex  7469