Theorem 1idpru 7392

Description: Lemma for 1idpr 7393. (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 3112	. . . . . 6 ⊢ (2nd ‘1P) ⊆ (2nd ‘1P)
2		rexss 3159	. . . . . 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		r19.42v 2586	. . . . . 6 ⊢ (∃ℎ ∈ (2nd ‘1P)(𝑓 <Q 𝑥 ∧ 𝑥 = (𝑓 ·Q ℎ)) ↔ (𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ)))
5		1pr 7355	. . . . . . . . . . 11 ⊢ 1P ∈ P
6		prop 7276	. . . . . . . . . . . 12 ⊢ (1P ∈ P → ⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P)
7		elprnqu 7283	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P ∧ ℎ ∈ (2nd ‘1P)) → ℎ ∈ Q)
8	6, 7	sylan 281	. . . . . . . . . . 11 ⊢ ((1P ∈ P ∧ ℎ ∈ (2nd ‘1P)) → ℎ ∈ Q)
9	5, 8	mpan 420	. . . . . . . . . 10 ⊢ (ℎ ∈ (2nd ‘1P) → ℎ ∈ Q)
10		prop 7276	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
11		elprnqu 7283	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → 𝑓 ∈ Q)
12	10, 11	sylan 281	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → 𝑓 ∈ Q)
13		breq2 3928	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑓 ·Q ℎ) → (𝑓 <Q 𝑥 ↔ 𝑓 <Q (𝑓 ·Q ℎ)))
14	13	3ad2ant3 1004	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q 𝑥 ↔ 𝑓 <Q (𝑓 ·Q ℎ)))
15		1pru 7357	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘1P) = {ℎ ∣ 1Q <Q ℎ}
16	15	abeq2i 2248	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (2nd ‘1P) ↔ 1Q <Q ℎ)
17		1nq 7167	. . . . . . . . . . . . . . . . 17 ⊢ 1Q ∈ Q
18		ltmnqg 7202	. . . . . . . . . . . . . . . . 17 ⊢ ((1Q ∈ Q ∧ ℎ ∈ Q ∧ 𝑓 ∈ Q) → (1Q <Q ℎ ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q ℎ)))
19	17, 18	mp3an1 1302	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ ∈ Q ∧ 𝑓 ∈ Q) → (1Q <Q ℎ ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q ℎ)))
20	19	ancoms 266	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q) → (1Q <Q ℎ ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q ℎ)))
21		mulidnq 7190	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ Q → (𝑓 ·Q 1Q) = 𝑓)
22	21	breq1d 3934	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → ((𝑓 ·Q 1Q) <Q (𝑓 ·Q ℎ) ↔ 𝑓 <Q (𝑓 ·Q ℎ)))
23	22	adantr 274	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 ·Q 1Q) <Q (𝑓 ·Q ℎ) ↔ 𝑓 <Q (𝑓 ·Q ℎ)))
24	20, 23	bitrd 187	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q) → (1Q <Q ℎ ↔ 𝑓 <Q (𝑓 ·Q ℎ)))
25	16, 24	syl5rbb 192	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q (𝑓 ·Q ℎ) ↔ ℎ ∈ (2nd ‘1P)))
26	25	3adant3 1001	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q (𝑓 ·Q ℎ) ↔ ℎ ∈ (2nd ‘1P)))
27	14, 26	bitrd 187	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q 𝑥 ↔ ℎ ∈ (2nd ‘1P)))
28	12, 27	syl3an1 1249	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ ℎ ∈ Q ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q 𝑥 ↔ ℎ ∈ (2nd ‘1P)))
29	9, 28	syl3an2 1250	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ ℎ ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q 𝑥 ↔ ℎ ∈ (2nd ‘1P)))
30	29	3expia 1183	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ ℎ ∈ (2nd ‘1P)) → (𝑥 = (𝑓 ·Q ℎ) → (𝑓 <Q 𝑥 ↔ ℎ ∈ (2nd ‘1P))))
31	30	pm5.32rd 446	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ ℎ ∈ (2nd ‘1P)) → ((𝑓 <Q 𝑥 ∧ 𝑥 = (𝑓 ·Q ℎ)) ↔ (ℎ ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ℎ))))
32	31	rexbidva 2432	. . . . . 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))))
33	4, 32	syl5rbbr 194	. . . . 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	syl5bb 191	. . . 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 2432	. . 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 7270	. . . . 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 7177	. . . . 5 ⊢ ((𝑢 ∈ Q ∧ 𝑣 ∈ Q) → (𝑢 ·Q 𝑣) ∈ Q)
38	36, 37	genpelvu 7314	. . . 4 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ)))
39	5, 38	mpan2 421	. . 3 ⊢ (𝐴 ∈ P → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ)))
40		prnminu 7290	. . . . . . 7 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (2nd ‘𝐴)) → ∃𝑓 ∈ (2nd ‘𝐴)𝑓 <Q 𝑥)
41	10, 40	sylan 281	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (2nd ‘𝐴)) → ∃𝑓 ∈ (2nd ‘𝐴)𝑓 <Q 𝑥)
42		ltrelnq 7166	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
43	42	brel 4586	. . . . . . . . . . . . 13 ⊢ (𝑓 <Q 𝑥 → (𝑓 ∈ Q ∧ 𝑥 ∈ Q))
44	43	ancomd 265	. . . . . . . . . . . 12 ⊢ (𝑓 <Q 𝑥 → (𝑥 ∈ Q ∧ 𝑓 ∈ Q))
45		ltmnqg 7202	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
46	45	adantl 275	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ Q ∧ 𝑓 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
47		simpr 109	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → 𝑓 ∈ Q)
48		simpl 108	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → 𝑥 ∈ Q)
49		recclnq 7193	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (*Q‘𝑓) ∈ Q)
50	49	adantl 275	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (*Q‘𝑓) ∈ Q)
51		mulcomnqg 7184	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
52	51	adantl 275	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ Q ∧ 𝑓 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
53	46, 47, 48, 50, 52	caovord2d 5933	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑓 <Q 𝑥 ↔ (𝑓 ·Q (*Q‘𝑓)) <Q (𝑥 ·Q (*Q‘𝑓))))
54		recidnq 7194	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (𝑓 ·Q (*Q‘𝑓)) = 1Q)
55	54	breq1d 3934	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ Q → ((𝑓 ·Q (*Q‘𝑓)) <Q (𝑥 ·Q (*Q‘𝑓)) ↔ 1Q <Q (𝑥 ·Q (*Q‘𝑓))))
56	55	adantl 275	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → ((𝑓 ·Q (*Q‘𝑓)) <Q (𝑥 ·Q (*Q‘𝑓)) ↔ 1Q <Q (𝑥 ·Q (*Q‘𝑓))))
57	53, 56	bitrd 187	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑓 <Q 𝑥 ↔ 1Q <Q (𝑥 ·Q (*Q‘𝑓))))
58	57	biimpd 143	. . . . . . . . . . . 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 7177	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Q ∧ (*Q‘𝑓) ∈ Q) → (𝑥 ·Q (*Q‘𝑓)) ∈ Q)
61	49, 60	sylan2 284	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 ·Q (*Q‘𝑓)) ∈ Q)
62		breq2 3928	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝑥 ·Q (*Q‘𝑓)) → (1Q <Q ℎ ↔ 1Q <Q (𝑥 ·Q (*Q‘𝑓))))
63	62, 15	elab2g 2826	. . . . . . . . . . . 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 166	. . . . . . . . . 10 ⊢ (𝑓 <Q 𝑥 → (𝑥 ·Q (*Q‘𝑓)) ∈ (2nd ‘1P))
66		mulassnqg 7185	. . . . . . . . . . . . . 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, 48, 50, 52, 67	caov12d 5945	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑓 ·Q (𝑥 ·Q (*Q‘𝑓))) = (𝑥 ·Q (𝑓 ·Q (*Q‘𝑓))))
69	54	oveq2d 5783	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ Q → (𝑥 ·Q (𝑓 ·Q (*Q‘𝑓))) = (𝑥 ·Q 1Q))
70	69	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 ·Q (𝑓 ·Q (*Q‘𝑓))) = (𝑥 ·Q 1Q))
71		mulidnq 7190	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Q → (𝑥 ·Q 1Q) = 𝑥)
72	71	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 ·Q 1Q) = 𝑥)
73	68, 70, 72	3eqtrrd 2175	. . . . . . . . . . 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 5775	. . . . . . . . . . . 12 ⊢ (ℎ = (𝑥 ·Q (*Q‘𝑓)) → (𝑓 ·Q ℎ) = (𝑓 ·Q (𝑥 ·Q (*Q‘𝑓))))
76	75	eqeq2d 2149	. . . . . . . . . . 11 ⊢ (ℎ = (𝑥 ·Q (*Q‘𝑓)) → (𝑥 = (𝑓 ·Q ℎ) ↔ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q‘𝑓)))))
77	76	rspcev 2784	. . . . . . . . . 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 408	. . . . . . . . 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 323	. . . . . . 7 ⊢ (𝑓 ∈ (2nd ‘𝐴) → (𝑓 <Q 𝑥 → (𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ))))
81	80	reximia 2525	. . . . . 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 114	. . . 4 ⊢ (𝐴 ∈ P → (𝑥 ∈ (2nd ‘𝐴) → ∃𝑓 ∈ (2nd ‘𝐴)(𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ))))
84		prcunqu 7286	. . . . . . 7 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘𝐴)))
85	10, 84	sylan 281	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘𝐴)))
86	85	adantrd 277	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → ((𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ)) → 𝑥 ∈ (2nd ‘𝐴)))
87	86	rexlimdva 2547	. . . 4 ⊢ (𝐴 ∈ P → (∃𝑓 ∈ (2nd ‘𝐴)(𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ)) → 𝑥 ∈ (2nd ‘𝐴)))
88	83, 87	impbid 128	. . 3 ⊢ (𝐴 ∈ P → (𝑥 ∈ (2nd ‘𝐴) ↔ ∃𝑓 ∈ (2nd ‘𝐴)(𝑓 <Q 𝑥 ∧ ∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ))))
89	35, 39, 88	3bitr4d 219	. 2 ⊢ (𝐴 ∈ P → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ 𝑥 ∈ (2nd ‘𝐴)))
90	89	eqrdv 2135	1 ⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 1P)) = (2nd ‘𝐴))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∃wrex 2415   ⊆ wss 3066  ⟨cop 3525   class class class wbr 3924  ‘cfv 5118  (class class class)co 5767  1^st c1st 6029  2^nd c2nd 6030  Qcnq 7081  1_Qc1q 7082   ·_Q cmq 7084  *_Qcrq 7085   <_Q cltq 7086  Pcnp 7092  1_Pc1p 7093   ·_P cmp 7095

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 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-inp 7267  df-i1p 7268  df-imp 7270

This theorem is referenced by:  1idpr  7393