Theorem recexprlem1ssu 7554

Description: The upper cut of one is a subset of the upper cut of 𝐴 ·_P 𝐵. Lemma for recexpr 7558. (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 7476	. . . 4 ⊢ (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
2	1	abeq2i 2268	. . 3 ⊢ (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
3		prop 7395	. . . . . 6 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
4		prmuloc2 7487	. . . . . 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 7409	. . . . . . . 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 984	. . . . . . . . . . 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 1008	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣 ∈ (1st ‘𝐴))
11		elprnql 7401	. . . . . . . . . . . . . . . . . 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 1002	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣 ∈ Q)
15		simp1r 1007	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 1Q <Q 𝑤)
16		ltrelnq 7285	. . . . . . . . . . . . . . . . . 18 ⊢ <Q ⊆ (Q × Q)
17	16	brel 4638	. . . . . . . . . . . . . . . . 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 7312	. . . . . . . . . . . . . . . 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 7304	. . . . . . . . . . . . . . 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 1220	. . . . . . . . . . . . . 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 7313	. . . . . . . . . . . . . . . 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 5840	. . . . . . . . . . . . . 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 7309	. . . . . . . . . . . . . . 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 2194	. . . . . . . . . . . . 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 2226	. . . . . . . . . . . 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 7341	. . . . . . . . . . . . 13 ⊢ (𝑧 <Q (𝑣 ·Q 𝑤) → (*Q‘(𝑣 ·Q 𝑤)) <Q (*Q‘𝑧))
33		ltmnqg 7321	. . . . . . . . . . . . . . 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 7296	. . . . . . . . . . . . . . . 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 7312	. . . . . . . . . . . . . . 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 4638	. . . . . . . . . . . . . . . . 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 7312	. . . . . . . . . . . . . . 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 7303	. . . . . . . . . . . . . . 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 5990	. . . . . . . . . . . . 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		1nq 7286	. . . . . . . . . . . . . . . . 17 ⊢ 1Q ∈ Q
49		mulidnq 7309	. . . . . . . . . . . . . . . . 17 ⊢ (1Q ∈ Q → (1Q ·Q 1Q) = 1Q)
50	48, 49	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (1Q ·Q 1Q) = 1Q
51		mulcomnqg 7303	. . . . . . . . . . . . . . . . . . . . 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)))
52	37, 51	mpdan 418	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)))
53		recidnq 7313	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = 1Q)
54	52, 53	eqtr3d 2192	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) = 1Q)
55	54, 24	oveqan12d 5843	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑣 ·Q 𝑤) ∈ Q ∧ 𝑤 ∈ Q) → (((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) ·Q (𝑤 ·Q (*Q‘𝑤))) = (1Q ·Q 1Q))
56	36, 19, 55	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))
57		mulassnqg 7304	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 ·Q 𝑔) ·Q ℎ) = (𝑓 ·Q (𝑔 ·Q ℎ)))
58	57	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)))
59		mulclnq 7296	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) ∈ Q)
60	59	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)
61	38, 36, 19, 45, 58, 21, 60	caov4d 6005	. . . . . . . . . . . . . . . . 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))))
62	56, 61	eqtr3d 2192	. . . . . . . . . . . . . . . 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))))
63	50, 62	syl5reqr 2205	. . . . . . . . . . . . . . 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	60, 38, 19	caovcld 5974	. . . . . . . . . . . . . . . 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	60, 36, 21	caovcld 5974	. . . . . . . . . . . . . . . 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 7311	. . . . . . . . . . . . . . . 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 2226	. . . . . . . . . . . . 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 3968	. . . . . . . . . . . . . . . 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 5468	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → (*Q‘𝑦) = (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)))
73	72	eleq1d 2226	. . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . 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 7543	. . . . . . . . . . . . 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 7309	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = 𝑤)
83		mulcomnqg 7303	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ Q ∧ 1Q ∈ Q) → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
84	48, 83	mpan2 422	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
85	82, 84	eqtr3d 2192	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ Q → 𝑤 = (1Q ·Q 𝑤))
86	85	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → 𝑤 = (1Q ·Q 𝑤))
87		recidnq 7313	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ Q → (𝑧 ·Q (*Q‘𝑧)) = 1Q)
88	87	oveq1d 5839	. . . . . . . . . . . . 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 7304	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ Q ∧ (*Q‘𝑧) ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
91	42, 90	syl3an2 1254	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
92	91	3anidm12 1277	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
93	86, 89, 92	3eqtr2d 2196	. . . . . . . . . . 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 5832	. . . . . . . . . . . 12 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑧 ·Q 𝑥) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
96	95	eqeq2d 2169	. . . . . . . . . . 11 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑤 = (𝑧 ·Q 𝑥) ↔ 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤))))
97	96	rspcev 2816	. . . . . . . . . 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 1187	. . . . . . . 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 2558	. . . . . . 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 7552	. . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
102		df-imp 7389	. . . . . . . . . 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, 59	genpelvu 7433	. . . . . . . . 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 2579	. . . 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 3134	1 ⊢ (𝐴 ∈ P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1335  ∃wex 1472   ∈ wcel 2128  {cab 2143  ∃wrex 2436   ⊆ wss 3102  ⟨cop 3563   class class class wbr 3965  ‘cfv 5170  (class class class)co 5824  1^st c1st 6086  2^nd c2nd 6087  Qcnq 7200  1_Qc1q 7201   ·_Q cmq 7203  *_Qcrq 7204   <_Q cltq 7205  Pcnp 7211  1_Pc1p 7212   ·_P cmp 7214

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-i1p 7387  df-imp 7389

This theorem is referenced by:  recexprlemex  7557