Theorem recexprlem1ssu 6732

Description: The upper cut of one is a subset of the upper cut of 𝐴 ·_P 𝐵. Lemma for recexpr 6736. (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 6654	. . . 4 ⊢ (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
2	1	abeq2i 2148	. . 3 ⊢ (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
3		prop 6573	. . . . . 6 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
4		prmuloc2 6665	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))
5	3, 4	sylan 267	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))
6		prnminu 6587	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) → ∃𝑧 ∈ (2nd ‘𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
7	3, 6	sylan 267	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) → ∃𝑧 ∈ (2nd ‘𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
8	7	ad2ant2rl 480	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))) → ∃𝑧 ∈ (2nd ‘𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
9		simp3 906	. . . . . . . . . . 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 930	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣 ∈ (1st ‘𝐴))
11		elprnql 6579	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → 𝑣 ∈ Q)
12	3, 11	sylan 267	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → 𝑣 ∈ Q)
13	12	ad2ant2r 478	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))) → 𝑣 ∈ Q)
14	13	3adant3 924	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣 ∈ Q)
15		simp1r 929	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 1Q <Q 𝑤)
16		ltrelnq 6463	. . . . . . . . . . . . . . . . . 18 ⊢ <Q ⊆ (Q × Q)
17	16	brel 4392	. . . . . . . . . . . . . . . . 17 ⊢ (1Q <Q 𝑤 → (1Q ∈ Q ∧ 𝑤 ∈ Q))
18	17	simprd 107	. . . . . . . . . . . . . . . 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 6490	. . . . . . . . . . . . . . . 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 6482	. . . . . . . . . . . . . . 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 1135	. . . . . . . . . . . . . 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 6491	. . . . . . . . . . . . . . . 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 5528	. . . . . . . . . . . . . 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 6487	. . . . . . . . . . . . . . 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 2076	. . . . . . . . . . . . 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 2106	. . . . . . . . . . . 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 156	. . . . . . . . . . 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 6519	. . . . . . . . . . . . 13 ⊢ (𝑧 <Q (𝑣 ·Q 𝑤) → (*Q‘(𝑣 ·Q 𝑤)) <Q (*Q‘𝑧))
33		ltmnqg 6499	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ ·Q 𝑓) <Q (ℎ ·Q 𝑔)))
34	33	adantl 262	. . . . . . . . . . . . . 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 6474	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → (𝑣 ·Q 𝑤) ∈ Q)
36	14, 19, 35	syl2anc 391	. . . . . . . . . . . . . . 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 6490	. . . . . . . . . . . . . . 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 4392	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 <Q (𝑣 ·Q 𝑤) → (𝑧 ∈ Q ∧ (𝑣 ·Q 𝑤) ∈ Q))
40	39	simpld 105	. . . . . . . . . . . . . . . 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 6490	. . . . . . . . . . . . . . 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 6481	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
45	44	adantl 262	. . . . . . . . . . . . . 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 5670	. . . . . . . . . . . . 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 143	. . . . . . . . . . . 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 6464	. . . . . . . . . . . . . . . . 17 ⊢ 1Q ∈ Q
49		mulidnq 6487	. . . . . . . . . . . . . . . . 17 ⊢ (1Q ∈ Q → (1Q ·Q 1Q) = 1Q)
50	48, 49	ax-mp 7	. . . . . . . . . . . . . . . 16 ⊢ (1Q ·Q 1Q) = 1Q
51		mulcomnqg 6481	. . . . . . . . . . . . . . . . . . . . 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 398	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)))
53		recidnq 6491	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = 1Q)
54	52, 53	eqtr3d 2074	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) = 1Q)
55	54, 24	oveqan12d 5531	. . . . . . . . . . . . . . . . . 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 391	. . . . . . . . . . . . . . . . 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 6482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 ·Q 𝑔) ·Q ℎ) = (𝑓 ·Q (𝑔 ·Q ℎ)))
58	57	adantl 262	. . . . . . . . . . . . . . . . . 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 6474	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) ∈ Q)
60	59	adantl 262	. . . . . . . . . . . . . . . . . 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 5685	. . . . . . . . . . . . . . . . 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 2074	. . . . . . . . . . . . . . . 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 2087	. . . . . . . . . . . . . . 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 5654	. . . . . . . . . . . . . . . 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 5654	. . . . . . . . . . . . . . . 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 6489	. . . . . . . . . . . . . . . 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 391	. . . . . . . . . . . . . . 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 156	. . . . . . . . . . . . . 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 2106	. . . . . . . . . . . . 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 147	. . . . . . . . . . . 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 3767	. . . . . . . . . . . . . . . 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 5178	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → (*Q‘𝑦) = (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)))
73	72	eleq1d 2106	. . . . . . . . . . . . . . . 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 442	. . . . . . . . . . . . . . 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 2641	. . . . . . . . . . . . . 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 6721	. . . . . . . . . . . . 13 ⊢ (((*Q‘𝑧) ·Q 𝑤) ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q ((*Q‘𝑧) ·Q 𝑤) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)))
79	76, 78	syl6ibr 151	. . . . . . . . . . . 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 279	. . . . . . . . . . 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 409	. . . . . . . . . 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 6487	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = 𝑤)
83		mulcomnqg 6481	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ Q ∧ 1Q ∈ Q) → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
84	48, 83	mpan2 401	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
85	82, 84	eqtr3d 2074	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ Q → 𝑤 = (1Q ·Q 𝑤))
86	85	adantl 262	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → 𝑤 = (1Q ·Q 𝑤))
87		recidnq 6491	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ Q → (𝑧 ·Q (*Q‘𝑧)) = 1Q)
88	87	oveq1d 5527	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ Q → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
89	88	adantr 261	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
90		mulassnqg 6482	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ Q ∧ (*Q‘𝑧) ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
91	42, 90	syl3an2 1169	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
92	91	3anidm12 1192	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
93	86, 89, 92	3eqtr2d 2078	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
94	41, 19, 93	syl2anc 391	. . . . . . . . . 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 5520	. . . . . . . . . . . 12 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑧 ·Q 𝑥) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
96	95	eqeq2d 2051	. . . . . . . . . . 11 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑤 = (𝑧 ·Q 𝑥) ↔ 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤))))
97	96	rspcev 2656	. . . . . . . . . 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 391	. . . . . . . . 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 1106	. . . . . . . 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 2420	. . . . . . 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 6730	. . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
102		df-imp 6567	. . . . . . . . . 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 6611	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
104	101, 103	mpdan 398	. . . . . . . 8 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
105	104	ad2antrr 457	. . . . . . 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 158	. . . . . 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 2437	. . . 4 ⊢ ((𝐴 ∈ P ∧ 1Q <Q 𝑤) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)))
109	108	ex 108	. . 3 ⊢ (𝐴 ∈ P → (1Q <Q 𝑤 → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
110	2, 109	syl5bi 141	. 2 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘1P) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
111	110	ssrdv 2951	1 ⊢ (𝐴 ∈ P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  ∃wrex 2307   ⊆ wss 2917  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  (class class class)co 5512  1^st c1st 5765  2^nd c2nd 5766  Qcnq 6378  1_Qc1q 6379   ·_Q cmq 6381  *_Qcrq 6382   <_Q cltq 6383  Pcnp 6389  1_Pc1p 6390   ·_P cmp 6392

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-imp 6567

This theorem is referenced by:  recexprlemex  6735