Theorem 1idpru 7906

Description: Lemma for 1idpr 7907. (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 3258	. . . . . 6 ⊢ (2nd ‘1P) ⊆ (2nd ‘1P)
2		rexss 3305	. . . . . 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 7869	. . . . . . . . . . 11 ⊢ 1P ∈ P
5		prop 7790	. . . . . . . . . . . 12 ⊢ (1P ∈ P → ⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P)
6		elprnqu 7797	. . . . . . . . . . . 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 7790	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
10		elprnqu 7797	. . . . . . . . . . . 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 4113	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑓 ·Q ℎ) → (𝑓 <Q 𝑥 ↔ 𝑓 <Q (𝑓 ·Q ℎ)))
13	12	3ad2ant3 1047	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q 𝑥 ↔ 𝑓 <Q (𝑓 ·Q ℎ)))
14		1pru 7871	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘1P) = {ℎ ∣ 1Q <Q ℎ}
15	14	abeq2i 2343	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (2nd ‘1P) ↔ 1Q <Q ℎ)
16		1nq 7681	. . . . . . . . . . . . . . . . 17 ⊢ 1Q ∈ Q
17		ltmnqg 7716	. . . . . . . . . . . . . . . . 17 ⊢ ((1Q ∈ Q ∧ ℎ ∈ Q ∧ 𝑓 ∈ Q) → (1Q <Q ℎ ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q ℎ)))
18	16, 17	mp3an1 1361	. . . . . . . . . . . . . . . 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 7704	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ Q → (𝑓 ·Q 1Q) = 𝑓)
21	20	breq1d 4119	. . . . . . . . . . . . . . . 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 1044	. . . . . . . . . . . 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 1307	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ ℎ ∈ Q ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q 𝑥 ↔ ℎ ∈ (2nd ‘1P)))
28	8, 27	syl3an2 1308	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ ℎ ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q 𝑥 ↔ ℎ ∈ (2nd ‘1P)))
29	28	3expia 1232	. . . . . . . 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 2539	. . . . . 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 2700	. . . . . 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 2539	. . 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 7784	. . . . 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 7691	. . . . 5 ⊢ ((𝑢 ∈ Q ∧ 𝑣 ∈ Q) → (𝑢 ·Q 𝑣) ∈ Q)
38	36, 37	genpelvu 7828	. . . 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 7804	. . . . . . 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 7680	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
43	42	brel 4802	. . . . . . . . . . . . 13 ⊢ (𝑓 <Q 𝑥 → (𝑓 ∈ Q ∧ 𝑥 ∈ Q))
44	43	ancomd 267	. . . . . . . . . . . 12 ⊢ (𝑓 <Q 𝑥 → (𝑥 ∈ Q ∧ 𝑓 ∈ Q))
45		ltmnqg 7716	. . . . . . . . . . . . . . . 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 7707	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (*Q‘𝑓) ∈ Q)
50	49	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (*Q‘𝑓) ∈ Q)
51		mulcomnqg 7698	. . . . . . . . . . . . . . . 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 6224	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑓 <Q 𝑥 ↔ (𝑓 ·Q (*Q‘𝑓)) <Q (𝑥 ·Q (*Q‘𝑓))))
54		recidnq 7708	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (𝑓 ·Q (*Q‘𝑓)) = 1Q)
55	54	breq1d 4119	. . . . . . . . . . . . . . 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 7691	. . . . . . . . . . . . 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 4113	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝑥 ·Q (*Q‘𝑓)) → (1Q <Q ℎ ↔ 1Q <Q (𝑥 ·Q (*Q‘𝑓))))
63	62, 14	elab2g 2964	. . . . . . . . . . . 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 7699	. . . . . . . . . . . . . 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 6236	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑓 ·Q (𝑥 ·Q (*Q‘𝑓))) = (𝑥 ·Q (𝑓 ·Q (*Q‘𝑓))))
69	54	oveq2d 6066	. . . . . . . . . . . . 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 7704	. . . . . . . . . . . . 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 2270	. . . . . . . . . . 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 6058	. . . . . . . . . . . 12 ⊢ (ℎ = (𝑥 ·Q (*Q‘𝑓)) → (𝑓 ·Q ℎ) = (𝑓 ·Q (𝑥 ·Q (*Q‘𝑓))))
76	75	eqeq2d 2244	. . . . . . . . . . 11 ⊢ (ℎ = (𝑥 ·Q (*Q‘𝑓)) → (𝑥 = (𝑓 ·Q ℎ) ↔ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q‘𝑓)))))
77	76	rspcev 2921	. . . . . . . . . 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 2637	. . . . . 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 7800	. . . . . . 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 2660	. . . 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 2230	1 ⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 1P)) = (2nd ‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  ∃wrex 2521   ⊆ wss 3211  ⟨cop 3692   class class class wbr 4109  ‘cfv 5352  (class class class)co 6050  1^st c1st 6332  2^nd c2nd 6333  Qcnq 7595  1_Qc1q 7596   ·_Q cmq 7598  *_Qcrq 7599   <_Q cltq 7600  Pcnp 7606  1_Pc1p 7607   ·_P cmp 7609

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-inp 7781  df-i1p 7782  df-imp 7784

This theorem is referenced by:  1idpr  7907