Theorem recexprlem1ssu 7290

Description: The upper cut of one is a subset of the upper cut of 𝐴 ·_P 𝐵. Lemma for recexpr 7294. (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 7212	. . . 4 ⊢ (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
2	1	abeq2i 2205	. . 3 ⊢ (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
3		prop 7131	. . . . . 6 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
4		prmuloc2 7223	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))
5	3, 4	sylan 278	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))
6		prnminu 7145	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) → ∃𝑧 ∈ (2nd ‘𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
7	3, 6	sylan 278	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) → ∃𝑧 ∈ (2nd ‘𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
8	7	ad2ant2rl 496	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))) → ∃𝑧 ∈ (2nd ‘𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
9		simp3 948	. . . . . . . . . . 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 972	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣 ∈ (1st ‘𝐴))
11		elprnql 7137	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → 𝑣 ∈ Q)
12	3, 11	sylan 278	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝑣 ∈ (1st ‘𝐴)) → 𝑣 ∈ Q)
13	12	ad2ant2r 494	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴))) → 𝑣 ∈ Q)
14	13	3adant3 966	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣 ∈ Q)
15		simp1r 971	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd ‘𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 1Q <Q 𝑤)
16		ltrelnq 7021	. . . . . . . . . . . . . . . . . 18 ⊢ <Q ⊆ (Q × Q)
17	16	brel 4519	. . . . . . . . . . . . . . . . 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 7048	. . . . . . . . . . . . . . . 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 7040	. . . . . . . . . . . . . . 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 1181	. . . . . . . . . . . . . 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 7049	. . . . . . . . . . . . . . . 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 5706	. . . . . . . . . . . . . 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 7045	. . . . . . . . . . . . . . 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 2131	. . . . . . . . . . . . 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 2163	. . . . . . . . . . . 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 7077	. . . . . . . . . . . . 13 ⊢ (𝑧 <Q (𝑣 ·Q 𝑤) → (*Q‘(𝑣 ·Q 𝑤)) <Q (*Q‘𝑧))
33		ltmnqg 7057	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ ·Q 𝑓) <Q (ℎ ·Q 𝑔)))
34	33	adantl 272	. . . . . . . . . . . . . 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 7032	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → (𝑣 ·Q 𝑤) ∈ Q)
36	14, 19, 35	syl2anc 404	. . . . . . . . . . . . . . 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 7048	. . . . . . . . . . . . . . 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 4519	. . . . . . . . . . . . . . . . 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 7048	. . . . . . . . . . . . . . 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 7039	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
45	44	adantl 272	. . . . . . . . . . . . . 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 5852	. . . . . . . . . . . . 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 7022	. . . . . . . . . . . . . . . . 17 ⊢ 1Q ∈ Q
49		mulidnq 7045	. . . . . . . . . . . . . . . . 17 ⊢ (1Q ∈ Q → (1Q ·Q 1Q) = 1Q)
50	48, 49	ax-mp 7	. . . . . . . . . . . . . . . 16 ⊢ (1Q ·Q 1Q) = 1Q
51		mulcomnqg 7039	. . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)))
53		recidnq 7049	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = 1Q)
54	52, 53	eqtr3d 2129	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ·Q 𝑤) ∈ Q → ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) = 1Q)
55	54, 24	oveqan12d 5709	. . . . . . . . . . . . . . . . . 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 404	. . . . . . . . . . . . . . . . 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 7040	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 ·Q 𝑔) ·Q ℎ) = (𝑓 ·Q (𝑔 ·Q ℎ)))
58	57	adantl 272	. . . . . . . . . . . . . . . . . 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 7032	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) ∈ Q)
60	59	adantl 272	. . . . . . . . . . . . . . . . . 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 5867	. . . . . . . . . . . . . . . . 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 2129	. . . . . . . . . . . . . . . 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 2142	. . . . . . . . . . . . . . 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 5836	. . . . . . . . . . . . . . . 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 5836	. . . . . . . . . . . . . . . 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 7047	. . . . . . . . . . . . . . . 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 404	. . . . . . . . . . . . . . 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 2163	. . . . . . . . . . . . 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 3870	. . . . . . . . . . . . . . . 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 5340	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → (*Q‘𝑦) = (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)))
73	72	eleq1d 2163	. . . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . . 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 2721	. . . . . . . . . . . . . 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 7279	. . . . . . . . . . . . 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 290	. . . . . . . . . . 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 425	. . . . . . . . . 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 7045	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = 𝑤)
83		mulcomnqg 7039	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ Q ∧ 1Q ∈ Q) → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
84	48, 83	mpan2 417	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Q → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
85	82, 84	eqtr3d 2129	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ Q → 𝑤 = (1Q ·Q 𝑤))
86	85	adantl 272	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → 𝑤 = (1Q ·Q 𝑤))
87		recidnq 7049	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ Q → (𝑧 ·Q (*Q‘𝑧)) = 1Q)
88	87	oveq1d 5705	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ Q → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
89	88	adantr 271	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
90		mulassnqg 7040	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ Q ∧ (*Q‘𝑧) ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
91	42, 90	syl3an2 1215	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
92	91	3anidm12 1238	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑧 ·Q (*Q‘𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
93	86, 89, 92	3eqtr2d 2133	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
94	41, 19, 93	syl2anc 404	. . . . . . . . . 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 5698	. . . . . . . . . . . 12 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑧 ·Q 𝑥) = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤)))
96	95	eqeq2d 2106	. . . . . . . . . . 11 ⊢ (𝑥 = ((*Q‘𝑧) ·Q 𝑤) → (𝑤 = (𝑧 ·Q 𝑥) ↔ 𝑤 = (𝑧 ·Q ((*Q‘𝑧) ·Q 𝑤))))
97	96	rspcev 2736	. . . . . . . . . 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 404	. . . . . . . . 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 1148	. . . . . . . 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 2486	. . . . . . 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 7288	. . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
102		df-imp 7125	. . . . . . . . . 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 7169	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
104	101, 103	mpdan 413	. . . . . . . 8 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)))
105	104	ad2antrr 473	. . . . . . 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 2507	. . . 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 3045	1 ⊢ (𝐴 ∈ P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 927   = wceq 1296  ∃wex 1433   ∈ wcel 1445  {cab 2081  ∃wrex 2371   ⊆ wss 3013  ⟨cop 3469   class class class wbr 3867  ‘cfv 5049  (class class class)co 5690  1^st c1st 5947  2^nd c2nd 5948  Qcnq 6936  1_Qc1q 6937   ·_Q cmq 6939  *_Qcrq 6940   <_Q cltq 6941  Pcnp 6947  1_Pc1p 6948   ·_P cmp 6950

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-2o 6220  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-rq 7008  df-ltnqqs 7009  df-enq0 7080  df-nq0 7081  df-0nq0 7082  df-plq0 7083  df-mq0 7084  df-inp 7122  df-i1p 7123  df-imp 7125

This theorem is referenced by:  recexprlemex  7293