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Theorem 1idprl 7657

Description: Lemma for 1idpr 7659. (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 3203	. . . . . 6 ⊢ (1^st ‘1_P) ⊆ (1^st ‘1_P)
2		rexss 3250	. . . . . 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		1pr 7621	. . . . . . . . . . 11 ⊢ 1_P ∈ P
5		prop 7542	. . . . . . . . . . . 12 ⊢ (1_P ∈ P → ⟨(1^st ‘1_P), (2^nd ‘1_P)⟩ ∈ P)
6		elprnql 7548	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘1_P), (2^nd ‘1_P)⟩ ∈ P ∧ 𝑔 ∈ (1^st ‘1_P)) → 𝑔 ∈ Q)
7	5, 6	sylan 283	. . . . . . . . . . 11 ⊢ ((1_P ∈ P ∧ 𝑔 ∈ (1^st ‘1_P)) → 𝑔 ∈ Q)
8	4, 7	mpan 424	. . . . . . . . . 10 ⊢ (𝑔 ∈ (1^st ‘1_P) → 𝑔 ∈ Q)
9		prop 7542	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
10		elprnql 7548	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
11	9, 10	sylan 283	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
12		breq1 4036	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑓 ·_Q 𝑔) → (𝑥 <_Q 𝑓 ↔ (𝑓 ·_Q 𝑔) <_Q 𝑓))
13	12	3ad2ant3 1022	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ 𝑥 = (𝑓 ·_Q 𝑔)) → (𝑥 <_Q 𝑓 ↔ (𝑓 ·_Q 𝑔) <_Q 𝑓))
14		1prl 7622	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘1_P) = {𝑔 ∣ 𝑔 <_Q 1_Q}
15	14	abeq2i 2307	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (1^st ‘1_P) ↔ 𝑔 <_Q 1_Q)
16		1nq 7433	. . . . . . . . . . . . . . . . 17 ⊢ 1_Q ∈ Q
17		ltmnqg 7468	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 ∈ Q ∧ 1_Q ∈ Q ∧ 𝑓 ∈ Q) → (𝑔 <_Q 1_Q ↔ (𝑓 ·_Q 𝑔) <_Q (𝑓 ·_Q 1_Q)))
18	16, 17	mp3an2 1336	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ Q ∧ 𝑓 ∈ Q) → (𝑔 <_Q 1_Q ↔ (𝑓 ·_Q 𝑔) <_Q (𝑓 ·_Q 1_Q)))
19	18	ancoms 268	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑔 <_Q 1_Q ↔ (𝑓 ·_Q 𝑔) <_Q (𝑓 ·_Q 1_Q)))
20		mulidnq 7456	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ Q → (𝑓 ·_Q 1_Q) = 𝑓)
21	20	breq2d 4045	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → ((𝑓 ·_Q 𝑔) <_Q (𝑓 ·_Q 1_Q) ↔ (𝑓 ·_Q 𝑔) <_Q 𝑓))
22	21	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → ((𝑓 ·_Q 𝑔) <_Q (𝑓 ·_Q 1_Q) ↔ (𝑓 ·_Q 𝑔) <_Q 𝑓))
23	19, 22	bitrd 188	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑔 <_Q 1_Q ↔ (𝑓 ·_Q 𝑔) <_Q 𝑓))
24	15, 23	bitr2id 193	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → ((𝑓 ·_Q 𝑔) <_Q 𝑓 ↔ 𝑔 ∈ (1^st ‘1_P)))
25	24	3adant3 1019	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ 𝑥 = (𝑓 ·_Q 𝑔)) → ((𝑓 ·_Q 𝑔) <_Q 𝑓 ↔ 𝑔 ∈ (1^st ‘1_P)))
26	13, 25	bitrd 188	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ 𝑥 = (𝑓 ·_Q 𝑔)) → (𝑥 <_Q 𝑓 ↔ 𝑔 ∈ (1^st ‘1_P)))
27	11, 26	syl3an1 1282	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) ∧ 𝑔 ∈ Q ∧ 𝑥 = (𝑓 ·_Q 𝑔)) → (𝑥 <_Q 𝑓 ↔ 𝑔 ∈ (1^st ‘1_P)))
28	8, 27	syl3an2 1283	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) ∧ 𝑔 ∈ (1^st ‘1_P) ∧ 𝑥 = (𝑓 ·_Q 𝑔)) → (𝑥 <_Q 𝑓 ↔ 𝑔 ∈ (1^st ‘1_P)))
29	28	3expia 1207	. . . . . . . 8 ⊢ (((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	pm5.32rd 451	. . . . . . 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))))
31	30	rexbidva 2494	. . . . . 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))))
32		r19.42v 2654	. . . . . 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)))
33	31, 32	bitr3di 195	. . . . 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	bitrid 192	. . . 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 2494	. . 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 7536	. . . . 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 7443	. . . . 5 ⊢ ((𝑢 ∈ Q ∧ 𝑣 ∈ Q) → (𝑢 ·_Q 𝑣) ∈ Q)
38	36, 37	genpelvl 7579	. . . 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	4, 38	mpan2 425	. . 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 7555	. . . . . . 7 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → ∃𝑓 ∈ (1^st ‘𝐴)𝑥 <_Q 𝑓)
41	9, 40	sylan 283	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → ∃𝑓 ∈ (1^st ‘𝐴)𝑥 <_Q 𝑓)
42		ltrelnq 7432	. . . . . . . . . . . . 13 ⊢ <_Q ⊆ (Q × Q)
43	42	brel 4715	. . . . . . . . . . . 12 ⊢ (𝑥 <_Q 𝑓 → (𝑥 ∈ Q ∧ 𝑓 ∈ Q))
44		ltmnqg 7468	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑦 <_Q 𝑧 ↔ (𝑤 ·_Q 𝑦) <_Q (𝑤 ·_Q 𝑧)))
45	44	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ Q ∧ 𝑓 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q)) → (𝑦 <_Q 𝑧 ↔ (𝑤 ·_Q 𝑦) <_Q (𝑤 ·_Q 𝑧)))
46		simpl 109	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → 𝑥 ∈ Q)
47		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → 𝑓 ∈ Q)
48		recclnq 7459	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (*_Q‘𝑓) ∈ Q)
49	48	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (*_Q‘𝑓) ∈ Q)
50		mulcomnqg 7450	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·_Q 𝑧) = (𝑧 ·_Q 𝑦))
51	50	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ Q ∧ 𝑓 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q)) → (𝑦 ·_Q 𝑧) = (𝑧 ·_Q 𝑦))
52	45, 46, 47, 49, 51	caovord2d 6093	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 <_Q 𝑓 ↔ (𝑥 ·_Q (_Q‘𝑓)) <_Q (𝑓 ·_Q (_Q‘𝑓))))
53		recidnq 7460	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (𝑓 ·_Q (*_Q‘𝑓)) = 1_Q)
54	53	breq2d 4045	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ Q → ((𝑥 ·_Q (_Q‘𝑓)) <_Q (𝑓 ·_Q (_Q‘𝑓)) ↔ (𝑥 ·_Q (*_Q‘𝑓)) <_Q 1_Q))
55	54	adantl 277	. . . . . . . . . . . . . 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 188	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 <_Q 𝑓 ↔ (𝑥 ·_Q (*_Q‘𝑓)) <_Q 1_Q))
57	56	biimpd 144	. . . . . . . . . . . 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 7443	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ (_Q‘𝑓) ∈ Q) → (𝑥 ·_Q (_Q‘𝑓)) ∈ Q)
60	48, 59	sylan2 286	. . . . . . . . . . . . 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 4036	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑥 ·_Q (_Q‘𝑓)) → (𝑔 <_Q 1_Q ↔ (𝑥 ·_Q (_Q‘𝑓)) <_Q 1_Q))
63	62, 14	elab2g 2911	. . . . . . . . . . . 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 167	. . . . . . . . . 10 ⊢ (𝑥 <_Q 𝑓 → (𝑥 ·_Q (*_Q‘𝑓)) ∈ (1^st ‘1_P))
66		mulassnqg 7451	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑦 ·_Q 𝑧) ·_Q 𝑤) = (𝑦 ·_Q (𝑧 ·_Q 𝑤)))
67	66	adantl 277	. . . . . . . . . . . . 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 6105	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑓 ·_Q (𝑥 ·_Q (_Q‘𝑓))) = (𝑥 ·_Q (𝑓 ·_Q (_Q‘𝑓))))
69	53	oveq2d 5938	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ Q → (𝑥 ·_Q (𝑓 ·_Q (*_Q‘𝑓))) = (𝑥 ·_Q 1_Q))
70	69	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 ·_Q (𝑓 ·_Q (*_Q‘𝑓))) = (𝑥 ·_Q 1_Q))
71		mulidnq 7456	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Q → (𝑥 ·_Q 1_Q) = 𝑥)
72	71	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 ·_Q 1_Q) = 𝑥)
73	68, 70, 72	3eqtrrd 2234	. . . . . . . . . . 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 5930	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑥 ·_Q (_Q‘𝑓)) → (𝑓 ·_Q 𝑔) = (𝑓 ·_Q (𝑥 ·_Q (_Q‘𝑓))))
76	75	eqeq2d 2208	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑥 ·_Q (_Q‘𝑓)) → (𝑥 = (𝑓 ·_Q 𝑔) ↔ 𝑥 = (𝑓 ·_Q (𝑥 ·_Q (_Q‘𝑓)))))
77	76	rspcev 2868	. . . . . . . . . 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 411	. . . . . . . . 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 325	. . . . . . 7 ⊢ (𝑓 ∈ (1^st ‘𝐴) → (𝑥 <_Q 𝑓 → (𝑥 <_Q 𝑓 ∧ ∃𝑔 ∈ (1^st ‘1_P)𝑥 = (𝑓 ·_Q 𝑔))))
81	80	reximia 2592	. . . . . 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 115	. . . 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 7551	. . . . . . 7 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘𝐴)))
85	9, 84	sylan 283	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘𝐴)))
86	85	adantrd 279	. . . . 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 2614	. . . 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 129	. . 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 220	. 2 ⊢ (𝐴 ∈ P → (𝑥 ∈ (1^st ‘(𝐴 ·_P 1_P)) ↔ 𝑥 ∈ (1^st ‘𝐴)))
90	89	eqrdv 2194	1 ⊢ (𝐴 ∈ P → (1^st ‘(𝐴 ·_P 1_P)) = (1^st ‘𝐴))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ wcel 2167 ∃wrex 2476 ⊆ wss 3157 ⟨cop 3625 class class class wbr 4033 ‘cfv 5258 (class class class)co 5922 1^st c1st 6196 2^nd c2nd 6197 Qcnq 7347 1_Qc1q 7348 ·_Q cmq 7350 *_Qcrq 7351 <_Q cltq 7352 Pcnp 7358 1_Pc1p 7359 ·_P cmp 7361

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-eprel 4324 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-1o 6474 df-oadd 6478 df-omul 6479 df-er 6592 df-ec 6594 df-qs 6598 df-ni 7371 df-pli 7372 df-mi 7373 df-lti 7374 df-plpq 7411 df-mpq 7412 df-enq 7414 df-nqqs 7415 df-plqqs 7416 df-mqqs 7417 df-1nqqs 7418 df-rq 7419 df-ltnqqs 7420 df-inp 7533 df-i1p 7534 df-imp 7536

This theorem is referenced by: 1idpr 7659

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