Theorem 1idpru 7789

Description: Lemma for 1idpr 7790. (Contributed by Jim Kingdon, 13-Dec-2019.)

Assertion

Ref	Expression
1idpru	⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 1P)) = (2nd ‘𝐴))

Proof of Theorem 1idpru

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3244	. . . . . 6 ⊢ (2nd ‘1P) ⊆ (2nd ‘1P)
2		rexss 3291	. . . . . 6 ⊢ ((2nd ‘1P) ⊆ (2nd ‘1P) → (∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ) ↔ ∃ℎ ∈ (2nd ‘1P)(ℎ ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ℎ))))
3	1, 2	ax-mp 5	. . . . 5 ⊢ (∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ) ↔ ∃ℎ ∈ (2nd ‘1P)(ℎ ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ℎ)))
4		1pr 7752	. . . . . . . . . . 11 ⊢ 1P ∈ P
5		prop 7673	. . . . . . . . . . . 12 ⊢ (1P ∈ P → ⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P)
6		elprnqu 7680	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P ∧ ℎ ∈ (2nd ‘1P)) → ℎ ∈ Q)
7	5, 6	sylan 283	. . . . . . . . . . 11 ⊢ ((1P ∈ P ∧ ℎ ∈ (2nd ‘1P)) → ℎ ∈ Q)
8	4, 7	mpan 424	. . . . . . . . . 10 ⊢ (ℎ ∈ (2nd ‘1P) → ℎ ∈ Q)
9		prop 7673	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
10		elprnqu 7680	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → 𝑓 ∈ Q)
11	9, 10	sylan 283	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → 𝑓 ∈ Q)
12		breq2 4087	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑓 ·Q ℎ) → (𝑓 <Q 𝑥 ↔ 𝑓 <Q (𝑓 ·Q ℎ)))
13	12	3ad2ant3 1044	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q 𝑥 ↔ 𝑓 <Q (𝑓 ·Q ℎ)))
14		1pru 7754	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘1P) = {ℎ ∣ 1Q <Q ℎ}
15	14	abeq2i 2340	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (2nd ‘1P) ↔ 1Q <Q ℎ)
16		1nq 7564	. . . . . . . . . . . . . . . . 17 ⊢ 1Q ∈ Q
17		ltmnqg 7599	. . . . . . . . . . . . . . . . 17 ⊢ ((1Q ∈ Q ∧ ℎ ∈ Q ∧ 𝑓 ∈ Q) → (1Q <Q ℎ ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q ℎ)))
18	16, 17	mp3an1 1358	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ ∈ Q ∧ 𝑓 ∈ Q) → (1Q <Q ℎ ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q ℎ)))
19	18	ancoms 268	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q) → (1Q <Q ℎ ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q ℎ)))
20		mulidnq 7587	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ Q → (𝑓 ·Q 1Q) = 𝑓)
21	20	breq1d 4093	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → ((𝑓 ·Q 1Q) <Q (𝑓 ·Q ℎ) ↔ 𝑓 <Q (𝑓 ·Q ℎ)))
22	21	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 ·Q 1Q) <Q (𝑓 ·Q ℎ) ↔ 𝑓 <Q (𝑓 ·Q ℎ)))
23	19, 22	bitrd 188	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q) → (1Q <Q ℎ ↔ 𝑓 <Q (𝑓 ·Q ℎ)))
24	15, 23	bitr2id 193	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q (𝑓 ·Q ℎ) ↔ ℎ ∈ (2nd ‘1P)))
25	24	3adant3 1041	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q (𝑓 ·Q ℎ) ↔ ℎ ∈ (2nd ‘1P)))
26	13, 25	bitrd 188	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q 𝑥 ↔ ℎ ∈ (2nd ‘1P)))
27	11, 26	syl3an1 1304	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ ℎ ∈ Q ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q 𝑥 ↔ ℎ ∈ (2nd ‘1P)))
28	8, 27	syl3an2 1305	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ ℎ ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q 𝑥 ↔ ℎ ∈ (2nd ‘1P)))
29	28	3expia 1229	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ ℎ ∈ (2nd ‘1P)) → (𝑥 = (𝑓 ·Q ℎ) → (𝑓 <Q 𝑥 ↔ ℎ ∈ (2nd ‘1P))))
30	29	pm5.32rd 451	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ ℎ ∈ (2nd ‘1P)) → ((𝑓 <Q 𝑥 ∧ 𝑥 = (𝑓 ·Q ℎ)) ↔ (ℎ ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ℎ))))
31	30	rexbidva 2527	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → (∃ℎ ∈ (2nd ‘1P)(𝑓 <Q 𝑥 ∧ 𝑥 = (𝑓 ·Q ℎ)) ↔ ∃ℎ ∈ (2nd ‘1P)(ℎ ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ℎ))))
32		r19.42v 2688	. . . . . 6 ⊢ (∃ℎ ∈ (2nd ‘1P)(𝑓 <Q 𝑥 ∧ 𝑥 = (𝑓 ·Q ℎ)) ↔ (𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ)))
33	31, 32	bitr3di 195	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → (∃ℎ ∈ (2nd ‘1P)(ℎ ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ℎ)) ↔ (𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ))))
34	3, 33	bitrid 192	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → (∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ) ↔ (𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ))))
35	34	rexbidva 2527	. . 3 ⊢ (𝐴 ∈ P → (∃𝑓 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ) ↔ ∃𝑓 ∈ (2nd ‘𝐴)(𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ))))
36		df-imp 7667	. . . . 5 ⊢ ·P = (𝑦 ∈ P, 𝑧 ∈ P ↦ ⟨{𝑤 ∈ Q ∣ ∃𝑢 ∈ Q ∃𝑣 ∈ Q (𝑢 ∈ (1st ‘𝑦) ∧ 𝑣 ∈ (1st ‘𝑧) ∧ 𝑤 = (𝑢 ·Q 𝑣))}, {𝑤 ∈ Q ∣ ∃𝑢 ∈ Q ∃𝑣 ∈ Q (𝑢 ∈ (2nd ‘𝑦) ∧ 𝑣 ∈ (2nd ‘𝑧) ∧ 𝑤 = (𝑢 ·Q 𝑣))}⟩)
37		mulclnq 7574	. . . . 5 ⊢ ((𝑢 ∈ Q ∧ 𝑣 ∈ Q) → (𝑢 ·Q 𝑣) ∈ Q)
38	36, 37	genpelvu 7711	. . . 4 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ)))
39	4, 38	mpan2 425	. . 3 ⊢ (𝐴 ∈ P → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ)))
40		prnminu 7687	. . . . . . 7 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (2nd ‘𝐴)) → ∃𝑓 ∈ (2nd ‘𝐴)𝑓 <Q 𝑥)
41	9, 40	sylan 283	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (2nd ‘𝐴)) → ∃𝑓 ∈ (2nd ‘𝐴)𝑓 <Q 𝑥)
42		ltrelnq 7563	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
43	42	brel 4771	. . . . . . . . . . . . 13 ⊢ (𝑓 <Q 𝑥 → (𝑓 ∈ Q ∧ 𝑥 ∈ Q))
44	43	ancomd 267	. . . . . . . . . . . 12 ⊢ (𝑓 <Q 𝑥 → (𝑥 ∈ Q ∧ 𝑓 ∈ Q))
45		ltmnqg 7599	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
46	45	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ Q ∧ 𝑓 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
47		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → 𝑓 ∈ Q)
48		simpl 109	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → 𝑥 ∈ Q)
49		recclnq 7590	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (*Q‘𝑓) ∈ Q)
50	49	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (*Q‘𝑓) ∈ Q)
51		mulcomnqg 7581	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
52	51	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ Q ∧ 𝑓 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
53	46, 47, 48, 50, 52	caovord2d 6181	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑓 <Q 𝑥 ↔ (𝑓 ·Q (*Q‘𝑓)) <Q (𝑥 ·Q (*Q‘𝑓))))
54		recidnq 7591	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (𝑓 ·Q (*Q‘𝑓)) = 1Q)
55	54	breq1d 4093	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ Q → ((𝑓 ·Q (*Q‘𝑓)) <Q (𝑥 ·Q (*Q‘𝑓)) ↔ 1Q <Q (𝑥 ·Q (*Q‘𝑓))))
56	55	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → ((𝑓 ·Q (*Q‘𝑓)) <Q (𝑥 ·Q (*Q‘𝑓)) ↔ 1Q <Q (𝑥 ·Q (*Q‘𝑓))))
57	53, 56	bitrd 188	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑓 <Q 𝑥 ↔ 1Q <Q (𝑥 ·Q (*Q‘𝑓))))
58	57	biimpd 144	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑓 <Q 𝑥 → 1Q <Q (𝑥 ·Q (*Q‘𝑓))))
59	44, 58	mpcom 36	. . . . . . . . . . 11 ⊢ (𝑓 <Q 𝑥 → 1Q <Q (𝑥 ·Q (*Q‘𝑓)))
60		mulclnq 7574	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Q ∧ (*Q‘𝑓) ∈ Q) → (𝑥 ·Q (*Q‘𝑓)) ∈ Q)
61	49, 60	sylan2 286	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 ·Q (*Q‘𝑓)) ∈ Q)
62		breq2 4087	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝑥 ·Q (*Q‘𝑓)) → (1Q <Q ℎ ↔ 1Q <Q (𝑥 ·Q (*Q‘𝑓))))
63	62, 14	elab2g 2950	. . . . . . . . . . . 12 ⊢ ((𝑥 ·Q (*Q‘𝑓)) ∈ Q → ((𝑥 ·Q (*Q‘𝑓)) ∈ (2nd ‘1P) ↔ 1Q <Q (𝑥 ·Q (*Q‘𝑓))))
64	44, 61, 63	3syl 17	. . . . . . . . . . 11 ⊢ (𝑓 <Q 𝑥 → ((𝑥 ·Q (*Q‘𝑓)) ∈ (2nd ‘1P) ↔ 1Q <Q (𝑥 ·Q (*Q‘𝑓))))
65	59, 64	mpbird 167	. . . . . . . . . 10 ⊢ (𝑓 <Q 𝑥 → (𝑥 ·Q (*Q‘𝑓)) ∈ (2nd ‘1P))
66		mulassnqg 7582	. . . . . . . . . . . . . 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, 48, 50, 52, 67	caov12d 6193	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑓 ·Q (𝑥 ·Q (*Q‘𝑓))) = (𝑥 ·Q (𝑓 ·Q (*Q‘𝑓))))
69	54	oveq2d 6023	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ Q → (𝑥 ·Q (𝑓 ·Q (*Q‘𝑓))) = (𝑥 ·Q 1Q))
70	69	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 ·Q (𝑓 ·Q (*Q‘𝑓))) = (𝑥 ·Q 1Q))
71		mulidnq 7587	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Q → (𝑥 ·Q 1Q) = 𝑥)
72	71	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 ·Q 1Q) = 𝑥)
73	68, 70, 72	3eqtrrd 2267	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q‘𝑓))))
74	44, 73	syl 14	. . . . . . . . . 10 ⊢ (𝑓 <Q 𝑥 → 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q‘𝑓))))
75		oveq2 6015	. . . . . . . . . . . 12 ⊢ (ℎ = (𝑥 ·Q (*Q‘𝑓)) → (𝑓 ·Q ℎ) = (𝑓 ·Q (𝑥 ·Q (*Q‘𝑓))))
76	75	eqeq2d 2241	. . . . . . . . . . 11 ⊢ (ℎ = (𝑥 ·Q (*Q‘𝑓)) → (𝑥 = (𝑓 ·Q ℎ) ↔ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q‘𝑓)))))
77	76	rspcev 2907	. . . . . . . . . 10 ⊢ (((𝑥 ·Q (*Q‘𝑓)) ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q‘𝑓)))) → ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ))
78	65, 74, 77	syl2anc 411	. . . . . . . . 9 ⊢ (𝑓 <Q 𝑥 → ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ))
79	78	a1i 9	. . . . . . . 8 ⊢ (𝑓 ∈ (2nd ‘𝐴) → (𝑓 <Q 𝑥 → ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ)))
80	79	ancld 325	. . . . . . 7 ⊢ (𝑓 ∈ (2nd ‘𝐴) → (𝑓 <Q 𝑥 → (𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ))))
81	80	reximia 2625	. . . . . 6 ⊢ (∃𝑓 ∈ (2nd ‘𝐴)𝑓 <Q 𝑥 → ∃𝑓 ∈ (2nd ‘𝐴)(𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ)))
82	41, 81	syl 14	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (2nd ‘𝐴)) → ∃𝑓 ∈ (2nd ‘𝐴)(𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ)))
83	82	ex 115	. . . 4 ⊢ (𝐴 ∈ P → (𝑥 ∈ (2nd ‘𝐴) → ∃𝑓 ∈ (2nd ‘𝐴)(𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ))))
84		prcunqu 7683	. . . . . . 7 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘𝐴)))
85	9, 84	sylan 283	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘𝐴)))
86	85	adantrd 279	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → ((𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ)) → 𝑥 ∈ (2nd ‘𝐴)))
87	86	rexlimdva 2648	. . . 4 ⊢ (𝐴 ∈ P → (∃𝑓 ∈ (2nd ‘𝐴)(𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ)) → 𝑥 ∈ (2nd ‘𝐴)))
88	83, 87	impbid 129	. . 3 ⊢ (𝐴 ∈ P → (𝑥 ∈ (2nd ‘𝐴) ↔ ∃𝑓 ∈ (2nd ‘𝐴)(𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ))))
89	35, 39, 88	3bitr4d 220	. 2 ⊢ (𝐴 ∈ P → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ 𝑥 ∈ (2nd ‘𝐴)))
90	89	eqrdv 2227	1 ⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 1P)) = (2nd ‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∃wrex 2509   ⊆ wss 3197  ⟨cop 3669   class class class wbr 4083  ‘cfv 5318  (class class class)co 6007  1^st c1st 6290  2^nd c2nd 6291  Qcnq 7478  1_Qc1q 7479   ·_Q cmq 7481  *_Qcrq 7482   <_Q cltq 7483  Pcnp 7489  1_Pc1p 7490   ·_P cmp 7492

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7502  df-pli 7503  df-mi 7504  df-lti 7505  df-plpq 7542  df-mpq 7543  df-enq 7545  df-nqqs 7546  df-plqqs 7547  df-mqqs 7548  df-1nqqs 7549  df-rq 7550  df-ltnqqs 7551  df-inp 7664  df-i1p 7665  df-imp 7667

This theorem is referenced by:  1idpr  7790