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Theorem 1idprl 7586

Description: Lemma for 1idpr 7588. (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 3175	. . . . . 6 ⊢ (1^st ‘1_P) ⊆ (1^st ‘1_P)
2		rexss 3222	. . . . . 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 7550	. . . . . . . . . . 11 ⊢ 1_P ∈ P
5		prop 7471	. . . . . . . . . . . 12 ⊢ (1_P ∈ P → ⟨(1^st ‘1_P), (2^nd ‘1_P)⟩ ∈ P)
6		elprnql 7477	. . . . . . . . . . . 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 7471	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
10		elprnql 7477	. . . . . . . . . . . 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 4005	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑓 ·_Q 𝑔) → (𝑥 <_Q 𝑓 ↔ (𝑓 ·_Q 𝑔) <_Q 𝑓))
13	12	3ad2ant3 1020	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ 𝑥 = (𝑓 ·_Q 𝑔)) → (𝑥 <_Q 𝑓 ↔ (𝑓 ·_Q 𝑔) <_Q 𝑓))
14		1prl 7551	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘1_P) = {𝑔 ∣ 𝑔 <_Q 1_Q}
15	14	abeq2i 2288	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (1^st ‘1_P) ↔ 𝑔 <_Q 1_Q)
16		1nq 7362	. . . . . . . . . . . . . . . . 17 ⊢ 1_Q ∈ Q
17		ltmnqg 7397	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 ∈ Q ∧ 1_Q ∈ Q ∧ 𝑓 ∈ Q) → (𝑔 <_Q 1_Q ↔ (𝑓 ·_Q 𝑔) <_Q (𝑓 ·_Q 1_Q)))
18	16, 17	mp3an2 1325	. . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ Q → (𝑓 ·_Q 1_Q) = 𝑓)
21	20	breq2d 4014	. . . . . . . . . . . . . . . 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 1017	. . . . . . . . . . . 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 1271	. . . . . . . . . 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 1272	. . . . . . . . 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 1205	. . . . . . . 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 2474	. . . . . 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 2634	. . . . . 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 2474	. . 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 7465	. . . . 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 7372	. . . . 5 ⊢ ((𝑢 ∈ Q ∧ 𝑣 ∈ Q) → (𝑢 ·_Q 𝑣) ∈ Q)
38	36, 37	genpelvl 7508	. . . 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 7484	. . . . . . 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 7361	. . . . . . . . . . . . 13 ⊢ <_Q ⊆ (Q × Q)
43	42	brel 4677	. . . . . . . . . . . 12 ⊢ (𝑥 <_Q 𝑓 → (𝑥 ∈ Q ∧ 𝑓 ∈ Q))
44		ltmnqg 7397	. . . . . . . . . . . . . . . 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 7388	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (*_Q‘𝑓) ∈ Q)
49	48	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (*_Q‘𝑓) ∈ Q)
50		mulcomnqg 7379	. . . . . . . . . . . . . . . 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 6041	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 <_Q 𝑓 ↔ (𝑥 ·_Q (_Q‘𝑓)) <_Q (𝑓 ·_Q (_Q‘𝑓))))
53		recidnq 7389	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (𝑓 ·_Q (*_Q‘𝑓)) = 1_Q)
54	53	breq2d 4014	. . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . 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 4005	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑥 ·_Q (_Q‘𝑓)) → (𝑔 <_Q 1_Q ↔ (𝑥 ·_Q (_Q‘𝑓)) <_Q 1_Q))
63	62, 14	elab2g 2884	. . . . . . . . . . . 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 7380	. . . . . . . . . . . . . 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 6053	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑓 ·_Q (𝑥 ·_Q (_Q‘𝑓))) = (𝑥 ·_Q (𝑓 ·_Q (_Q‘𝑓))))
69	53	oveq2d 5888	. . . . . . . . . . . . 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 7385	. . . . . . . . . . . . 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 2215	. . . . . . . . . . 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 5880	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑥 ·_Q (_Q‘𝑓)) → (𝑓 ·_Q 𝑔) = (𝑓 ·_Q (𝑥 ·_Q (_Q‘𝑓))))
76	75	eqeq2d 2189	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑥 ·_Q (_Q‘𝑓)) → (𝑥 = (𝑓 ·_Q 𝑔) ↔ 𝑥 = (𝑓 ·_Q (𝑥 ·_Q (_Q‘𝑓)))))
77	76	rspcev 2841	. . . . . . . . . 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 2572	. . . . . 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 7480	. . . . . . 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 2594	. . . 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 2175	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 978 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 ⊆ wss 3129 ⟨cop 3595 class class class wbr 4002 ‘cfv 5215 (class class class)co 5872 1^st c1st 6136 2^nd c2nd 6137 Qcnq 7276 1_Qc1q 7277 ·_Q cmq 7279 *_Qcrq 7280 <_Q cltq 7281 Pcnp 7287 1_Pc1p 7288 ·_P cmp 7290

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-eprel 4288 df-id 4292 df-po 4295 df-iso 4296 df-iord 4365 df-on 4367 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-recs 6303 df-irdg 6368 df-1o 6414 df-oadd 6418 df-omul 6419 df-er 6532 df-ec 6534 df-qs 6538 df-ni 7300 df-pli 7301 df-mi 7302 df-lti 7303 df-plpq 7340 df-mpq 7341 df-enq 7343 df-nqqs 7344 df-plqqs 7345 df-mqqs 7346 df-1nqqs 7347 df-rq 7348 df-ltnqqs 7349 df-inp 7462 df-i1p 7463 df-imp 7465

This theorem is referenced by: 1idpr 7588

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