Theorem recexprlem1ssu 7854

Description: The upper cut of one is a subset of the upper cut of 𝐴 ·_P 𝐵. Lemma for recexpr 7858. (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
recexprlem1ssu	⊢ (𝐴 ∈ P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))

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

Proof of Theorem recexprlem1ssu

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

Step	Hyp	Ref	Expression
1		1pru 7776	. . . 4 ⊢ (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
2	1	abeq2i 2342	. . 3 ⊢ (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
3		prop 7695	. . . . . 6 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
4		prmuloc2 7787	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))
5	3, 4	sylan 283	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))
6		prnminu 7709	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) → ∃𝑧 ∈ (2nd ‘𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
7	3, 6	sylan 283	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) → ∃𝑧 ∈ (2nd ‘𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
8	7	ad2ant2rl 511	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))) → ∃𝑧 ∈ (2nd ‘𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
9		simp3 1025	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑧 <Q (𝑣 ·Q 𝑤))
10		simp2l 1049	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣 ∈ (1st ‘𝐴))
11		elprnql 7701	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → 𝑣 ∈ Q)
12	3, 11	sylan 283	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → 𝑣 ∈ Q)
13	12	ad2ant2r 509	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))) → 𝑣 ∈ Q)
14	13	3adant3 1043	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣 ∈ Q)
15		simp1r 1048	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 1Q <Q 𝑤)
16		ltrelnq 7585	. . . . . . . . . . . . . . . . . 18 ⊢ <Q ⊆ (Q × Q)
17	16	brel 4778	. . . . . . . . . . . . . . . . 17 ⊢ (1Q <Q 𝑤 → (1Q ∈ Q ∧ 𝑤 ∈ Q))
18	17	simprd 114	. . . . . . . . . . . . . . . 16 ⊢ (1Q <Q 𝑤 → 𝑤 ∈ Q)
19	15, 18	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑤 ∈ Q)
20		recclnq 7612	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ Q → (*Q‘𝑤) ∈ Q)
21	19, 20	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (*Q‘𝑤) ∈ Q)
22		mulassnqg 7604	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q ∧ (*Q‘𝑤) ∈ Q) → ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤)) = (𝑣 ·Q (𝑤 ·Q (*Q‘𝑤))))
23	14, 19, 21, 22	syl3anc 1273	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤)) = (𝑣 ·Q (𝑤 ·Q (*Q‘𝑤))))
24		recidnq 7613	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ Q → (𝑤 ·Q (*Q‘𝑤)) = 1Q)
25	19, 24	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (𝑤 ·Q (*Q‘𝑤)) = 1Q)
26	25	oveq2d 6034	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (𝑣 ·Q (𝑤 ·Q (*Q‘𝑤))) = (𝑣 ·Q 1Q))
27		mulidnq 7609	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ Q → (𝑣 ·Q 1Q) = 𝑣)
28	14, 27	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (𝑣 ·Q 1Q) = 𝑣)
29	23, 26, 28	3eqtrd 2268	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤)) = 𝑣)
30	29	eleq1d 2300	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤)) ∈ (1st ‘𝐴) ↔ 𝑣 ∈ (1st ‘𝐴)))
31	10, 30	mpbird 167	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤)) ∈ (1st ‘𝐴))
32		ltrnqi 7641	. . . . . . . . . . . . 13 ⊢ (𝑧 <Q (𝑣 ·Q 𝑤) → (*Q‘(𝑣 ·Q 𝑤)) <Q (*Q‘𝑧))
33		ltmnqg 7621	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ ·Q 𝑓) <Q (ℎ ·Q 𝑔)))
34	33	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <Q 𝑔 ↔ (ℎ ·Q 𝑓) <Q (ℎ ·Q 𝑔)))
35		mulclnq 7596	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → (𝑣 ·Q 𝑤) ∈ Q)
36	14, 19, 35	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (𝑣 ·Q 𝑤) ∈ Q)
37		recclnq 7612	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → (*Q‘(𝑣 ·Q 𝑤)) ∈ Q)
38	36, 37	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (*Q‘(𝑣 ·Q 𝑤)) ∈ Q)
39	16	brel 4778	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 <Q (𝑣 ·Q 𝑤) → (𝑧 ∈ Q ∧ (𝑣 ·Q 𝑤) ∈ Q))
40	39	simpld 112	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 <Q (𝑣 ·Q 𝑤) → 𝑧 ∈ Q)
41	9, 40	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑧 ∈ Q)
42		recclnq 7612	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ Q → (*Q‘𝑧) ∈ Q)
43	41, 42	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (*Q‘𝑧) ∈ Q)
44		mulcomnqg 7603	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
45	44	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
46	34, 38, 43, 19, 45	caovord2d 6192	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((*Q‘(𝑣 ·Q 𝑤)) <Q (*Q‘𝑧) ↔ ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) <Q ((*Q‘𝑧) ·Q 𝑤)))
47	32, 46	imbitrid 154	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (𝑧 <Q (𝑣 ·Q 𝑤) → ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) <Q ((*Q‘𝑧) ·Q 𝑤)))
48		mulcomnqg 7603	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑣 ·Q 𝑤) ∈ Q ∧ (*Q‘(𝑣 ·Q 𝑤)) ∈ Q) → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)))
49	37, 48	mpdan 421	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)))
50		recidnq 7613	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = 1Q)
51	49, 50	eqtr3d 2266	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) = 1Q)
52	51, 24	oveqan12d 6037	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑣 ·Q 𝑤) ∈ Q ∧ 𝑤 ∈ Q) → (((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) ·Q (𝑤 ·Q (*Q‘𝑤))) = (1Q ·Q 1Q))
53	36, 19, 52	syl2anc 411	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) ·Q (𝑤 ·Q (*Q‘𝑤))) = (1Q ·Q 1Q))
54		mulassnqg 7604	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 ·Q 𝑔) ·Q ℎ) = (𝑓 ·Q (𝑔 ·Q ℎ)))
55	54	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → ((𝑓 ·Q 𝑔) ·Q ℎ) = (𝑓 ·Q (𝑔 ·Q ℎ)))
56		mulclnq 7596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) ∈ Q)
57	56	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 ·Q 𝑔) ∈ Q)
58	38, 36, 19, 45, 55, 21, 57	caov4d 6207	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) ·Q (𝑤 ·Q (*Q‘𝑤))) = (((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ·Q ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤))))
59	53, 58	eqtr3d 2266	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (1Q ·Q 1Q) = (((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ·Q ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤))))
60		1nq 7586	. . . . . . . . . . . . . . . . 17 ⊢ 1Q ∈ Q
61		mulidnq 7609	. . . . . . . . . . . . . . . . 17 ⊢ (1Q ∈ Q → (1Q ·Q 1Q) = 1Q)
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (1Q ·Q 1Q) = 1Q
63	59, 62	eqtr3di 2279	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ·Q ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤))) = 1Q)
64	57, 38, 19	caovcld 6176	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ∈ Q)
65	57, 36, 21	caovcld 6176	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤)) ∈ Q)
66		recmulnqg 7611	. . . . . . . . . . . . . . . 16 ⊢ ((((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ∈ Q ∧ ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤)) ∈ Q) → ((*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) = ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤)) ↔ (((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ·Q ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤))) = 1Q))
67	64, 65, 66	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) = ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤)) ↔ (((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ·Q ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤))) = 1Q))
68	63, 67	mpbird 167	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) = ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤)))
69	68	eleq1d 2300	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st ‘𝐴) ↔ ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤)) ∈ (1st ‘𝐴)))
70	69	biimprd 158	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤)) ∈ (1st ‘𝐴) → (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st ‘𝐴)))
71		breq1 4091	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → (𝑦 <Q ((*Q‘𝑧) ·Q 𝑤) ↔ ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) <Q ((*Q‘𝑧) ·Q 𝑤)))
72		fveq2 5639	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → (*Q‘𝑦) = (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)))
73	72	eleq1d 2300	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → ((*Q‘𝑦) ∈ (1st ‘𝐴) ↔ (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st ‘𝐴)))
74	71, 73	anbi12d 473	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → ((𝑦 <Q ((*Q‘𝑧) ·Q 𝑤) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) ↔ (((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) <Q ((*Q‘𝑧) ·Q 𝑤) ∧ (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st ‘𝐴))))
75	74	spcegv 2894	. . . . . . . . . . . . . 14 ⊢ (((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ∈ Q → ((((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) <Q ((*Q‘𝑧) ·Q 𝑤) ∧ (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st ‘𝐴)) → ∃𝑦(𝑦 <Q ((*Q‘𝑧) ·Q 𝑤) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))))
76	64, 75	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) <Q ((*Q‘𝑧) ·Q 𝑤) ∧ (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st ‘𝐴)) → ∃𝑦(𝑦 <Q ((*Q‘𝑧) ·Q 𝑤) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))))
77		recexpr.1	. . . . . . . . . . . . . 14 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩
78	77	recexprlemelu 7843	. . . . . . . . . . . . 13 ⊢ (((*Q‘𝑧) ·Q 𝑤) ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q ((*Q‘𝑧) ·Q 𝑤) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)))
79	76, 78	imbitrrdi 162	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) <Q ((*Q‘𝑧) ·Q 𝑤) ∧ (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st ‘𝐴)) → ((*Q‘𝑧) ·Q 𝑤) ∈ (2nd ‘𝐵)))
80	47, 70, 79	syl2and 295	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑧 <Q (𝑣 ·Q 𝑤) ∧ ((𝑣 ·Q 𝑤) ·Q (*Q‘𝑤)) ∈ (1st ‘𝐴)) → ((*Q‘𝑧) ·Q 𝑤) ∈ (2nd ‘𝐵)))
81	9, 31, 80	mp2and 433	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((*Q‘𝑧) ·Q 𝑤) ∈ (2nd ‘𝐵))
82		mulidnq 7609	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = 𝑤)
83		mulcomnqg 7603	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ Q ∧ 1Q ∈ Q) → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
84	60, 83	mpan2 425	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
85	82, 84	eqtr3d 2266	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ Q → 𝑤 = (1Q ·Q 𝑤))
86	85	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → 𝑤 = (1Q ·Q 𝑤))
87		recidnq 7613	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ Q → (𝑧 ·Q (*Q‘𝑧)) = 1Q)
88	87	oveq1d 6033	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ Q → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
89	88	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
90		mulassnqg 7604	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ Q ∧ (*Q‘𝑧) ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
91	42, 90	syl3an2 1307	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
92	91	3anidm12 1331	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
93	86, 89, 92	3eqtr2d 2270	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
94	41, 19, 93	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
95		oveq2 6026	. . . . . . . . . . . 12 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑧 ·Q 𝑥) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
96	95	eqeq2d 2243	. . . . . . . . . . 11 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑤 = (𝑧 ·Q 𝑥) ↔ 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤))))
97	96	rspcev 2910	. . . . . . . . . 10 ⊢ ((((*Q‘𝑧) ·Q 𝑤) ∈ (2nd ‘𝐵) ∧ 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤))) → ∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥))
98	81, 94, 97	syl2anc 411	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥))
99	98	3expia 1231	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))) → (𝑧 <Q (𝑣 ·Q 𝑤) → ∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
100	99	reximdv 2633	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))) → (∃𝑧 ∈ (2nd ‘𝐴)𝑧 <Q (𝑣 ·Q 𝑤) → ∃𝑧 ∈ (2nd ‘𝐴)∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
101	77	recexprlempr 7852	. . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
102		df-imp 7689	. . . . . . . . . 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))}⟩)
103	102, 56	genpelvu 7733	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
104	101, 103	mpdan 421	. . . . . . . 8 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
105	104	ad2antrr 488	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
106	100, 105	sylibrd 169	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))) → (∃𝑧 ∈ (2nd ‘𝐴)𝑧 <Q (𝑣 ·Q 𝑤) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
107	8, 106	mpd 13	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)))
108	5, 107	rexlimddv 2655	. . . 4 ⊢ ((𝐴 ∈ P ∧ 1Q <Q 𝑤) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)))
109	108	ex 115	. . 3 ⊢ (𝐴 ∈ P → (1Q <Q 𝑤 → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
110	2, 109	biimtrid 152	. 2 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘1P) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
111	110	ssrdv 3233	1 ⊢ (𝐴 ∈ P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397  ∃wex 1540   ∈ wcel 2202  {cab 2217  ∃wrex 2511   ⊆ 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  1_Qc1q 7501   ·_Q cmq 7503  *_Qcrq 7504   <_Q cltq 7505  Pcnp 7511  1_Pc1p 7512   ·_P cmp 7514

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-po 4393  df-iso 4394  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-2o 6583  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-rq 7572  df-ltnqqs 7573  df-enq0 7644  df-nq0 7645  df-0nq0 7646  df-plq0 7647  df-mq0 7648  df-inp 7686  df-i1p 7687  df-imp 7689

This theorem is referenced by:  recexprlemex  7857