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Theorem 1idprl 7398

Description: Lemma for 1idpr 7400. (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 3117	. . . . . 6 ⊢ (1^st ‘1_P) ⊆ (1^st ‘1_P)
2		rexss 3164	. . . . . 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 5	. . . . 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 2588	. . . . . 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 7362	. . . . . . . . . . 11 ⊢ 1_P ∈ P
6		prop 7283	. . . . . . . . . . . 12 ⊢ (1_P ∈ P → ⟨(1^st ‘1_P), (2^nd ‘1_P)⟩ ∈ P)
7		elprnql 7289	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘1_P), (2^nd ‘1_P)⟩ ∈ P ∧ 𝑔 ∈ (1^st ‘1_P)) → 𝑔 ∈ Q)
8	6, 7	sylan 281	. . . . . . . . . . 11 ⊢ ((1_P ∈ P ∧ 𝑔 ∈ (1^st ‘1_P)) → 𝑔 ∈ Q)
9	5, 8	mpan 420	. . . . . . . . . 10 ⊢ (𝑔 ∈ (1^st ‘1_P) → 𝑔 ∈ Q)
10		prop 7283	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
11		elprnql 7289	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
12	10, 11	sylan 281	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
13		breq1 3932	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑓 ·_Q 𝑔) → (𝑥 <_Q 𝑓 ↔ (𝑓 ·_Q 𝑔) <_Q 𝑓))
14	13	3ad2ant3 1004	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ 𝑥 = (𝑓 ·_Q 𝑔)) → (𝑥 <_Q 𝑓 ↔ (𝑓 ·_Q 𝑔) <_Q 𝑓))
15		1prl 7363	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘1_P) = {𝑔 ∣ 𝑔 <_Q 1_Q}
16	15	abeq2i 2250	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (1^st ‘1_P) ↔ 𝑔 <_Q 1_Q)
17		1nq 7174	. . . . . . . . . . . . . . . . 17 ⊢ 1_Q ∈ Q
18		ltmnqg 7209	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 ∈ Q ∧ 1_Q ∈ Q ∧ 𝑓 ∈ Q) → (𝑔 <_Q 1_Q ↔ (𝑓 ·_Q 𝑔) <_Q (𝑓 ·_Q 1_Q)))
19	17, 18	mp3an2 1303	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ Q ∧ 𝑓 ∈ Q) → (𝑔 <_Q 1_Q ↔ (𝑓 ·_Q 𝑔) <_Q (𝑓 ·_Q 1_Q)))
20	19	ancoms 266	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑔 <_Q 1_Q ↔ (𝑓 ·_Q 𝑔) <_Q (𝑓 ·_Q 1_Q)))
21		mulidnq 7197	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ Q → (𝑓 ·_Q 1_Q) = 𝑓)
22	21	breq2d 3941	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → ((𝑓 ·_Q 𝑔) <_Q (𝑓 ·_Q 1_Q) ↔ (𝑓 ·_Q 𝑔) <_Q 𝑓))
23	22	adantr 274	. . . . . . . . . . . . . . 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 1001	. . . . . . . . . . . 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 1249	. . . . . . . . . 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 1250	. . . . . . . . 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 1183	. . . . . . . 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 446	. . . . . . 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 2434	. . . . . 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 2434	. . 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 7277	. . . . 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 7184	. . . . 5 ⊢ ((𝑢 ∈ Q ∧ 𝑣 ∈ Q) → (𝑢 ·_Q 𝑣) ∈ Q)
38	36, 37	genpelvl 7320	. . . 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 421	. . 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 7296	. . . . . . 7 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → ∃𝑓 ∈ (1^st ‘𝐴)𝑥 <_Q 𝑓)
41	10, 40	sylan 281	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → ∃𝑓 ∈ (1^st ‘𝐴)𝑥 <_Q 𝑓)
42		ltrelnq 7173	. . . . . . . . . . . . 13 ⊢ <_Q ⊆ (Q × Q)
43	42	brel 4591	. . . . . . . . . . . 12 ⊢ (𝑥 <_Q 𝑓 → (𝑥 ∈ Q ∧ 𝑓 ∈ Q))
44		ltmnqg 7209	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑦 <_Q 𝑧 ↔ (𝑤 ·_Q 𝑦) <_Q (𝑤 ·_Q 𝑧)))
45	44	adantl 275	. . . . . . . . . . . . . . 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 7200	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (*_Q‘𝑓) ∈ Q)
49	48	adantl 275	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (*_Q‘𝑓) ∈ Q)
50		mulcomnqg 7191	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·_Q 𝑧) = (𝑧 ·_Q 𝑦))
51	50	adantl 275	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ Q ∧ 𝑓 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q)) → (𝑦 ·_Q 𝑧) = (𝑧 ·_Q 𝑦))
52	45, 46, 47, 49, 51	caovord2d 5940	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 <_Q 𝑓 ↔ (𝑥 ·_Q (_Q‘𝑓)) <_Q (𝑓 ·_Q (_Q‘𝑓))))
53		recidnq 7201	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (𝑓 ·_Q (*_Q‘𝑓)) = 1_Q)
54	53	breq2d 3941	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ Q → ((𝑥 ·_Q (_Q‘𝑓)) <_Q (𝑓 ·_Q (_Q‘𝑓)) ↔ (𝑥 ·_Q (*_Q‘𝑓)) <_Q 1_Q))
55	54	adantl 275	. . . . . . . . . . . . . 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 7184	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ (_Q‘𝑓) ∈ Q) → (𝑥 ·_Q (_Q‘𝑓)) ∈ Q)
60	48, 59	sylan2 284	. . . . . . . . . . . . 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 3932	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑥 ·_Q (_Q‘𝑓)) → (𝑔 <_Q 1_Q ↔ (𝑥 ·_Q (_Q‘𝑓)) <_Q 1_Q))
63	62, 15	elab2g 2831	. . . . . . . . . . . 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 7192	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑦 ·_Q 𝑧) ·_Q 𝑤) = (𝑦 ·_Q (𝑧 ·_Q 𝑤)))
67	66	adantl 275	. . . . . . . . . . . . 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 5952	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑓 ·_Q (𝑥 ·_Q (_Q‘𝑓))) = (𝑥 ·_Q (𝑓 ·_Q (_Q‘𝑓))))
69	53	oveq2d 5790	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ Q → (𝑥 ·_Q (𝑓 ·_Q (*_Q‘𝑓))) = (𝑥 ·_Q 1_Q))
70	69	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 ·_Q (𝑓 ·_Q (*_Q‘𝑓))) = (𝑥 ·_Q 1_Q))
71		mulidnq 7197	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Q → (𝑥 ·_Q 1_Q) = 𝑥)
72	71	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 ·_Q 1_Q) = 𝑥)
73	68, 70, 72	3eqtrrd 2177	. . . . . . . . . . 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 5782	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑥 ·_Q (_Q‘𝑓)) → (𝑓 ·_Q 𝑔) = (𝑓 ·_Q (𝑥 ·_Q (_Q‘𝑓))))
76	75	eqeq2d 2151	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑥 ·_Q (_Q‘𝑓)) → (𝑥 = (𝑓 ·_Q 𝑔) ↔ 𝑥 = (𝑓 ·_Q (𝑥 ·_Q (_Q‘𝑓)))))
77	76	rspcev 2789	. . . . . . . . . 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 408	. . . . . . . . 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 323	. . . . . . 7 ⊢ (𝑓 ∈ (1^st ‘𝐴) → (𝑥 <_Q 𝑓 → (𝑥 <_Q 𝑓 ∧ ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔))))
81	80	reximia 2527	. . . . . 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 7292	. . . . . . 7 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘𝐴)))
85	10, 84	sylan 281	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘𝐴)))
86	85	adantrd 277	. . . . 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 2549	. . . 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 2137	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 962 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 ⊆ wss 3071 ⟨cop 3530 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 1^st c1st 6036 2^nd c2nd 6037 Qcnq 7088 1_Qc1q 7089 ·_Q cmq 7091 *_Qcrq 7092 <_Q cltq 7093 Pcnp 7099 1_Pc1p 7100 ·_P cmp 7102

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-inp 7274 df-i1p 7275 df-imp 7277

This theorem is referenced by: 1idpr 7400

W3C validator