Theorem recexprlem1ssu 7571

Description: The upper cut of one is a subset of the upper 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
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 7493	. . . 4 ⊢ (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
2	1	abeq2i 2276	. . 3 ⊢ (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
3		prop 7412	. . . . . 6 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
4		prmuloc2 7504	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))
5	3, 4	sylan 281	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))
6		prnminu 7426	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) → ∃𝑧 ∈ (2nd ‘𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
7	3, 6	sylan 281	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) → ∃𝑧 ∈ (2nd ‘𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
8	7	ad2ant2rl 503	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))) → ∃𝑧 ∈ (2nd ‘𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
9		simp3 989	. . . . . . . . . . 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 1013	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣 ∈ (1st ‘𝐴))
11		elprnql 7418	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → 𝑣 ∈ Q)
12	3, 11	sylan 281	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → 𝑣 ∈ Q)
13	12	ad2ant2r 501	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))) → 𝑣 ∈ Q)
14	13	3adant3 1007	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣 ∈ Q)
15		simp1r 1012	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 1Q <Q 𝑤)
16		ltrelnq 7302	. . . . . . . . . . . . . . . . . 18 ⊢ <Q ⊆ (Q × Q)
17	16	brel 4655	. . . . . . . . . . . . . . . . 17 ⊢ (1Q <Q 𝑤 → (1Q ∈ Q ∧ 𝑤 ∈ Q))
18	17	simprd 113	. . . . . . . . . . . . . . . 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 7329	. . . . . . . . . . . . . . . 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 7321	. . . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . 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 7330	. . . . . . . . . . . . . . . 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 5857	. . . . . . . . . . . . . 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 7326	. . . . . . . . . . . . . . 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 2202	. . . . . . . . . . . . 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 2234	. . . . . . . . . . . 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 166	. . . . . . . . . . 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 7358	. . . . . . . . . . . . 13 ⊢ (𝑧 <Q (𝑣 ·Q 𝑤) → (*Q‘(𝑣 ·Q 𝑤)) <Q (*Q‘𝑧))
33		ltmnqg 7338	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ ·Q 𝑓) <Q (ℎ ·Q 𝑔)))
34	33	adantl 275	. . . . . . . . . . . . . 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 7313	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → (𝑣 ·Q 𝑤) ∈ Q)
36	14, 19, 35	syl2anc 409	. . . . . . . . . . . . . . 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 7329	. . . . . . . . . . . . . . 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 4655	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 <Q (𝑣 ·Q 𝑤) → (𝑧 ∈ Q ∧ (𝑣 ·Q 𝑤) ∈ Q))
40	39	simpld 111	. . . . . . . . . . . . . . . 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 7329	. . . . . . . . . . . . . . 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 7320	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
45	44	adantl 275	. . . . . . . . . . . . . 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 6007	. . . . . . . . . . . . 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	syl5ib 153	. . . . . . . . . . . 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 7320	. . . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)))
50		recidnq 7330	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = 1Q)
51	49, 50	eqtr3d 2200	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) = 1Q)
52	51, 24	oveqan12d 5860	. . . . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . . . . 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 7321	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 ·Q 𝑔) ·Q ℎ) = (𝑓 ·Q (𝑔 ·Q ℎ)))
55	54	adantl 275	. . . . . . . . . . . . . . . . . 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 7313	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) ∈ Q)
57	56	adantl 275	. . . . . . . . . . . . . . . . . 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 6022	. . . . . . . . . . . . . . . . 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 2200	. . . . . . . . . . . . . . . 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 7303	. . . . . . . . . . . . . . . . 17 ⊢ 1Q ∈ Q
61		mulidnq 7326	. . . . . . . . . . . . . . . . 17 ⊢ (1Q ∈ Q → (1Q ·Q 1Q) = 1Q)
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (1Q ·Q 1Q) = 1Q
63	59, 62	eqtr3di 2213	. . . . . . . . . . . . . . 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 5991	. . . . . . . . . . . . . . . 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 5991	. . . . . . . . . . . . . . . 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 7328	. . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . . 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 166	. . . . . . . . . . . . . 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 2234	. . . . . . . . . . . . 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 157	. . . . . . . . . . . 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 3984	. . . . . . . . . . . . . . . 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 5485	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → (*Q‘𝑦) = (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)))
73	72	eleq1d 2234	. . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . 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 7560	. . . . . . . . . . . . 13 ⊢ (((*Q‘𝑧) ·Q 𝑤) ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q ((*Q‘𝑧) ·Q 𝑤) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)))
79	76, 78	syl6ibr 161	. . . . . . . . . . . 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 293	. . . . . . . . . . 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 430	. . . . . . . . . 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 7326	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = 𝑤)
83		mulcomnqg 7320	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ Q ∧ 1Q ∈ Q) → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
84	60, 83	mpan2 422	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
85	82, 84	eqtr3d 2200	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ Q → 𝑤 = (1Q ·Q 𝑤))
86	85	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → 𝑤 = (1Q ·Q 𝑤))
87		recidnq 7330	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ Q → (𝑧 ·Q (*Q‘𝑧)) = 1Q)
88	87	oveq1d 5856	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ Q → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
89	88	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
90		mulassnqg 7321	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ Q ∧ (*Q‘𝑧) ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
91	42, 90	syl3an2 1262	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
92	91	3anidm12 1285	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
93	86, 89, 92	3eqtr2d 2204	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
94	41, 19, 93	syl2anc 409	. . . . . . . . . 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 5849	. . . . . . . . . . . 12 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑧 ·Q 𝑥) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
96	95	eqeq2d 2177	. . . . . . . . . . 11 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑤 = (𝑧 ·Q 𝑥) ↔ 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤))))
97	96	rspcev 2829	. . . . . . . . . 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 409	. . . . . . . . 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 1195	. . . . . . . 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 2566	. . . . . . 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 7569	. . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
102		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))}⟩)
103	102, 56	genpelvu 7450	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
104	101, 103	mpdan 418	. . . . . . . 8 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
105	104	ad2antrr 480	. . . . . . 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 168	. . . . . 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 2587	. . . 4 ⊢ ((𝐴 ∈ P ∧ 1Q <Q 𝑤) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)))
109	108	ex 114	. . 3 ⊢ (𝐴 ∈ P → (1Q <Q 𝑤 → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
110	2, 109	syl5bi 151	. 2 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘1P) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
111	110	ssrdv 3147	1 ⊢ (𝐴 ∈ P → (2nd ‘1P) ⊆ (2nd ‘(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