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Theorem 1idprl 7246

Description: Lemma for 1idpr 7248. (Contributed by Jim Kingdon, 13-Dec-2019.)

**Assertion**
Ref	Expression
1idprl	⊢ (𝐴 ∈ P → (1^st ‘(𝐴 ·_P 1_P)) = (1^st ‘𝐴))

Proof of Theorem 1idprl Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3059	. . . . . 6 ⊢ (1^st ‘1_P) ⊆ (1^st ‘1_P)
2		rexss 3103	. . . . . 6 ⊢ ((1^st ‘1_P) ⊆ (1^st ‘1_P) → (∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔) ↔ ∃𝑔 ∈ (1^st ‘1_P)(𝑔 ∈ (1^st ‘1_P) ∧ 𝑥 = (𝑓 ·_Q 𝑔))))
3	1, 2	ax-mp 7	. . . . 5 ⊢ (∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔) ↔ ∃𝑔 ∈ (1^st ‘1_P)(𝑔 ∈ (1^st ‘1_P) ∧ 𝑥 = (𝑓 ·_Q 𝑔)))
4		r19.42v 2538	. . . . . 6 ⊢ (∃𝑔 ∈ (1^st ‘1_P)(𝑥 <_Q 𝑓 ∧ 𝑥 = (𝑓 ·_Q 𝑔)) ↔ (𝑥 <_Q 𝑓 ∧ ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔)))
5		1pr 7210	. . . . . . . . . . 11 ⊢ 1_P ∈ P
6		prop 7131	. . . . . . . . . . . 12 ⊢ (1_P ∈ P → ⟨(1^st ‘1_P), (2^nd ‘1_P)⟩ ∈ P)
7		elprnql 7137	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘1_P), (2^nd ‘1_P)⟩ ∈ P ∧ 𝑔 ∈ (1^st ‘1_P)) → 𝑔 ∈ Q)
8	6, 7	sylan 278	. . . . . . . . . . 11 ⊢ ((1_P ∈ P ∧ 𝑔 ∈ (1^st ‘1_P)) → 𝑔 ∈ Q)
9	5, 8	mpan 416	. . . . . . . . . 10 ⊢ (𝑔 ∈ (1^st ‘1_P) → 𝑔 ∈ Q)
10		prop 7131	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
11		elprnql 7137	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
12	10, 11	sylan 278	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
13		breq1 3870	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑓 ·_Q 𝑔) → (𝑥 <_Q 𝑓 ↔ (𝑓 ·_Q 𝑔) <_Q 𝑓))
14	13	3ad2ant3 969	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ 𝑥 = (𝑓 ·_Q 𝑔)) → (𝑥 <_Q 𝑓 ↔ (𝑓 ·_Q 𝑔) <_Q 𝑓))
15		1prl 7211	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘1_P) = {𝑔 ∣ 𝑔 <_Q 1_Q}
16	15	abeq2i 2205	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (1^st ‘1_P) ↔ 𝑔 <_Q 1_Q)
17		1nq 7022	. . . . . . . . . . . . . . . . 17 ⊢ 1_Q ∈ Q
18		ltmnqg 7057	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 ∈ Q ∧ 1_Q ∈ Q ∧ 𝑓 ∈ Q) → (𝑔 <_Q 1_Q ↔ (𝑓 ·_Q 𝑔) <_Q (𝑓 ·_Q 1_Q)))
19	17, 18	mp3an2 1268	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ Q ∧ 𝑓 ∈ Q) → (𝑔 <_Q 1_Q ↔ (𝑓 ·_Q 𝑔) <_Q (𝑓 ·_Q 1_Q)))
20	19	ancoms 265	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑔 <_Q 1_Q ↔ (𝑓 ·_Q 𝑔) <_Q (𝑓 ·_Q 1_Q)))
21		mulidnq 7045	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ Q → (𝑓 ·_Q 1_Q) = 𝑓)
22	21	breq2d 3879	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → ((𝑓 ·_Q 𝑔) <_Q (𝑓 ·_Q 1_Q) ↔ (𝑓 ·_Q 𝑔) <_Q 𝑓))
23	22	adantr 271	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → ((𝑓 ·_Q 𝑔) <_Q (𝑓 ·_Q 1_Q) ↔ (𝑓 ·_Q 𝑔) <_Q 𝑓))
24	20, 23	bitrd 187	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑔 <_Q 1_Q ↔ (𝑓 ·_Q 𝑔) <_Q 𝑓))
25	16, 24	syl5rbb 192	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → ((𝑓 ·_Q 𝑔) <_Q 𝑓 ↔ 𝑔 ∈ (1^st ‘1_P)))
26	25	3adant3 966	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ 𝑥 = (𝑓 ·_Q 𝑔)) → ((𝑓 ·_Q 𝑔) <_Q 𝑓 ↔ 𝑔 ∈ (1^st ‘1_P)))
27	14, 26	bitrd 187	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ 𝑥 = (𝑓 ·_Q 𝑔)) → (𝑥 <_Q 𝑓 ↔ 𝑔 ∈ (1^st ‘1_P)))
28	12, 27	syl3an1 1214	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) ∧ 𝑔 ∈ Q ∧ 𝑥 = (𝑓 ·_Q 𝑔)) → (𝑥 <_Q 𝑓 ↔ 𝑔 ∈ (1^st ‘1_P)))
29	9, 28	syl3an2 1215	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) ∧ 𝑔 ∈ (1^st ‘1_P) ∧ 𝑥 = (𝑓 ·_Q 𝑔)) → (𝑥 <_Q 𝑓 ↔ 𝑔 ∈ (1^st ‘1_P)))
30	29	3expia 1148	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) ∧ 𝑔 ∈ (1^st ‘1_P)) → (𝑥 = (𝑓 ·_Q 𝑔) → (𝑥 <_Q 𝑓 ↔ 𝑔 ∈ (1^st ‘1_P))))
31	30	pm5.32rd 440	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) ∧ 𝑔 ∈ (1^st ‘1_P)) → ((𝑥 <_Q 𝑓 ∧ 𝑥 = (𝑓 ·_Q 𝑔)) ↔ (𝑔 ∈ (1^st ‘1_P) ∧ 𝑥 = (𝑓 ·_Q 𝑔))))
32	31	rexbidva 2388	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → (∃𝑔 ∈ (1^st ‘1_P)(𝑥 <_Q 𝑓 ∧ 𝑥 = (𝑓 ·_Q 𝑔)) ↔ ∃𝑔 ∈ (1^st ‘1_P)(𝑔 ∈ (1^st ‘1_P) ∧ 𝑥 = (𝑓 ·_Q 𝑔))))
33	4, 32	syl5rbbr 194	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → (∃𝑔 ∈ (1^st ‘1_P)(𝑔 ∈ (1^st ‘1_P) ∧ 𝑥 = (𝑓 ·_Q 𝑔)) ↔ (𝑥 <_Q 𝑓 ∧ ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔))))
34	3, 33	syl5bb 191	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → (∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔) ↔ (𝑥 <_Q 𝑓 ∧ ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔))))
35	34	rexbidva 2388	. . 3 ⊢ (𝐴 ∈ P → (∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔) ↔ ∃𝑓 ∈ (1^st ‘𝐴)(𝑥 <_Q 𝑓 ∧ ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔))))
36		df-imp 7125	. . . . 5 ⊢ ·_P = (𝑦 ∈ P, 𝑧 ∈ P ↦ ⟨{𝑤 ∈ Q ∣ ∃𝑢 ∈ Q ∃𝑣 ∈ Q (𝑢 ∈ (1^st ‘𝑦) ∧ 𝑣 ∈ (1^st ‘𝑧) ∧ 𝑤 = (𝑢 ·_Q 𝑣))}, {𝑤 ∈ Q ∣ ∃𝑢 ∈ Q ∃𝑣 ∈ Q (𝑢 ∈ (2^nd ‘𝑦) ∧ 𝑣 ∈ (2^nd ‘𝑧) ∧ 𝑤 = (𝑢 ·_Q 𝑣))}⟩)
37		mulclnq 7032	. . . . 5 ⊢ ((𝑢 ∈ Q ∧ 𝑣 ∈ Q) → (𝑢 ·_Q 𝑣) ∈ Q)
38	36, 37	genpelvl 7168	. . . 4 ⊢ ((𝐴 ∈ P ∧ 1_P ∈ P) → (𝑥 ∈ (1^st ‘(𝐴 ·_P 1_P)) ↔ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔)))
39	5, 38	mpan2 417	. . 3 ⊢ (𝐴 ∈ P → (𝑥 ∈ (1^st ‘(𝐴 ·_P 1_P)) ↔ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔)))
40		prnmaxl 7144	. . . . . . 7 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → ∃𝑓 ∈ (1^st ‘𝐴)𝑥 <_Q 𝑓)
41	10, 40	sylan 278	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → ∃𝑓 ∈ (1^st ‘𝐴)𝑥 <_Q 𝑓)
42		ltrelnq 7021	. . . . . . . . . . . . 13 ⊢ <_Q ⊆ (Q × Q)
43	42	brel 4519	. . . . . . . . . . . 12 ⊢ (𝑥 <_Q 𝑓 → (𝑥 ∈ Q ∧ 𝑓 ∈ Q))
44		ltmnqg 7057	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑦 <_Q 𝑧 ↔ (𝑤 ·_Q 𝑦) <_Q (𝑤 ·_Q 𝑧)))
45	44	adantl 272	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ Q ∧ 𝑓 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q)) → (𝑦 <_Q 𝑧 ↔ (𝑤 ·_Q 𝑦) <_Q (𝑤 ·_Q 𝑧)))
46		simpl 108	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → 𝑥 ∈ Q)
47		simpr 109	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → 𝑓 ∈ Q)
48		recclnq 7048	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (*_Q‘𝑓) ∈ Q)
49	48	adantl 272	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (*_Q‘𝑓) ∈ Q)
50		mulcomnqg 7039	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·_Q 𝑧) = (𝑧 ·_Q 𝑦))
51	50	adantl 272	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ Q ∧ 𝑓 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q)) → (𝑦 ·_Q 𝑧) = (𝑧 ·_Q 𝑦))
52	45, 46, 47, 49, 51	caovord2d 5852	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 <_Q 𝑓 ↔ (𝑥 ·_Q (_Q‘𝑓)) <_Q (𝑓 ·_Q (_Q‘𝑓))))
53		recidnq 7049	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (𝑓 ·_Q (*_Q‘𝑓)) = 1_Q)
54	53	breq2d 3879	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ Q → ((𝑥 ·_Q (_Q‘𝑓)) <_Q (𝑓 ·_Q (_Q‘𝑓)) ↔ (𝑥 ·_Q (*_Q‘𝑓)) <_Q 1_Q))
55	54	adantl 272	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → ((𝑥 ·_Q (_Q‘𝑓)) <_Q (𝑓 ·_Q (_Q‘𝑓)) ↔ (𝑥 ·_Q (*_Q‘𝑓)) <_Q 1_Q))
56	52, 55	bitrd 187	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 <_Q 𝑓 ↔ (𝑥 ·_Q (*_Q‘𝑓)) <_Q 1_Q))
57	56	biimpd 143	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 <_Q 𝑓 → (𝑥 ·_Q (*_Q‘𝑓)) <_Q 1_Q))
58	43, 57	mpcom 36	. . . . . . . . . . 11 ⊢ (𝑥 <_Q 𝑓 → (𝑥 ·_Q (*_Q‘𝑓)) <_Q 1_Q)
59		mulclnq 7032	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ (_Q‘𝑓) ∈ Q) → (𝑥 ·_Q (_Q‘𝑓)) ∈ Q)
60	48, 59	sylan2 281	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 ·_Q (*_Q‘𝑓)) ∈ Q)
61	43, 60	syl 14	. . . . . . . . . . . 12 ⊢ (𝑥 <_Q 𝑓 → (𝑥 ·_Q (*_Q‘𝑓)) ∈ Q)
62		breq1 3870	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑥 ·_Q (_Q‘𝑓)) → (𝑔 <_Q 1_Q ↔ (𝑥 ·_Q (_Q‘𝑓)) <_Q 1_Q))
63	62, 15	elab2g 2776	. . . . . . . . . . . 12 ⊢ ((𝑥 ·_Q (_Q‘𝑓)) ∈ Q → ((𝑥 ·_Q (_Q‘𝑓)) ∈ (1^st ‘1_P) ↔ (𝑥 ·_Q (*_Q‘𝑓)) <_Q 1_Q))
64	61, 63	syl 14	. . . . . . . . . . 11 ⊢ (𝑥 <_Q 𝑓 → ((𝑥 ·_Q (_Q‘𝑓)) ∈ (1^st ‘1_P) ↔ (𝑥 ·_Q (_Q‘𝑓)) <_Q 1_Q))
65	58, 64	mpbird 166	. . . . . . . . . 10 ⊢ (𝑥 <_Q 𝑓 → (𝑥 ·_Q (*_Q‘𝑓)) ∈ (1^st ‘1_P))
66		mulassnqg 7040	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑦 ·_Q 𝑧) ·_Q 𝑤) = (𝑦 ·_Q (𝑧 ·_Q 𝑤)))
67	66	adantl 272	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ Q ∧ 𝑓 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q)) → ((𝑦 ·_Q 𝑧) ·_Q 𝑤) = (𝑦 ·_Q (𝑧 ·_Q 𝑤)))
68	47, 46, 49, 51, 67	caov12d 5864	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑓 ·_Q (𝑥 ·_Q (_Q‘𝑓))) = (𝑥 ·_Q (𝑓 ·_Q (_Q‘𝑓))))
69	53	oveq2d 5706	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ Q → (𝑥 ·_Q (𝑓 ·_Q (*_Q‘𝑓))) = (𝑥 ·_Q 1_Q))
70	69	adantl 272	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 ·_Q (𝑓 ·_Q (*_Q‘𝑓))) = (𝑥 ·_Q 1_Q))
71		mulidnq 7045	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Q → (𝑥 ·_Q 1_Q) = 𝑥)
72	71	adantr 271	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 ·_Q 1_Q) = 𝑥)
73	68, 70, 72	3eqtrrd 2132	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → 𝑥 = (𝑓 ·_Q (𝑥 ·_Q (*_Q‘𝑓))))
74	43, 73	syl 14	. . . . . . . . . 10 ⊢ (𝑥 <_Q 𝑓 → 𝑥 = (𝑓 ·_Q (𝑥 ·_Q (*_Q‘𝑓))))
75		oveq2 5698	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑥 ·_Q (_Q‘𝑓)) → (𝑓 ·_Q 𝑔) = (𝑓 ·_Q (𝑥 ·_Q (_Q‘𝑓))))
76	75	eqeq2d 2106	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑥 ·_Q (_Q‘𝑓)) → (𝑥 = (𝑓 ·_Q 𝑔) ↔ 𝑥 = (𝑓 ·_Q (𝑥 ·_Q (_Q‘𝑓)))))
77	76	rspcev 2736	. . . . . . . . . 10 ⊢ (((𝑥 ·_Q (_Q‘𝑓)) ∈ (1^st ‘1_P) ∧ 𝑥 = (𝑓 ·_Q (𝑥 ·_Q (_Q‘𝑓)))) → ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔))
78	65, 74, 77	syl2anc 404	. . . . . . . . 9 ⊢ (𝑥 <_Q 𝑓 → ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔))
79	78	a1i 9	. . . . . . . 8 ⊢ (𝑓 ∈ (1^st ‘𝐴) → (𝑥 <_Q 𝑓 → ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔)))
80	79	ancld 319	. . . . . . 7 ⊢ (𝑓 ∈ (1^st ‘𝐴) → (𝑥 <_Q 𝑓 → (𝑥 <_Q 𝑓 ∧ ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔))))
81	80	reximia 2480	. . . . . 6 ⊢ (∃𝑓 ∈ (1^st ‘𝐴)𝑥 <_Q 𝑓 → ∃𝑓 ∈ (1^st ‘𝐴)(𝑥 <_Q 𝑓 ∧ ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔)))
82	41, 81	syl 14	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → ∃𝑓 ∈ (1^st ‘𝐴)(𝑥 <_Q 𝑓 ∧ ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔)))
83	82	ex 114	. . . 4 ⊢ (𝐴 ∈ P → (𝑥 ∈ (1^st ‘𝐴) → ∃𝑓 ∈ (1^st ‘𝐴)(𝑥 <_Q 𝑓 ∧ ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔))))
84		prcdnql 7140	. . . . . . 7 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘𝐴)))
85	10, 84	sylan 278	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘𝐴)))
86	85	adantrd 274	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → ((𝑥 <_Q 𝑓 ∧ ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔)) → 𝑥 ∈ (1^st ‘𝐴)))
87	86	rexlimdva 2502	. . . 4 ⊢ (𝐴 ∈ P → (∃𝑓 ∈ (1^st ‘𝐴)(𝑥 <_Q 𝑓 ∧ ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔)) → 𝑥 ∈ (1^st ‘𝐴)))
88	83, 87	impbid 128	. . 3 ⊢ (𝐴 ∈ P → (𝑥 ∈ (1^st ‘𝐴) ↔ ∃𝑓 ∈ (1^st ‘𝐴)(𝑥 <_Q 𝑓 ∧ ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔))))
89	35, 39, 88	3bitr4d 219	. 2 ⊢ (𝐴 ∈ P → (𝑥 ∈ (1^st ‘(𝐴 ·_P 1_P)) ↔ 𝑥 ∈ (1^st ‘𝐴)))
90	89	eqrdv 2093	1 ⊢ (𝐴 ∈ P → (1^st ‘(𝐴 ·_P 1_P)) = (1^st ‘𝐴))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 927 = wceq 1296 ∈ wcel 1445 ∃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-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-inp 7122 df-i1p 7123 df-imp 7125

This theorem is referenced by: 1idpr 7248

W3C validator