Theorem 1idpru 7948

Description: Lemma for 1idpr 7949. (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 3268	. . . . . 6 ⊢ (2nd ‘1P) ⊆ (2nd ‘1P)
2		rexss 3315	. . . . . 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 7911	. . . . . . . . . . 11 ⊢ 1P ∈ P
5		prop 7832	. . . . . . . . . . . 12 ⊢ (1P ∈ P → ⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P)
6		elprnqu 7839	. . . . . . . . . . . 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 428	. . . . . . . . . 10 ⊢ (ℎ ∈ (2nd ‘1P) → ℎ ∈ Q)
9		prop 7832	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
10		elprnqu 7839	. . . . . . . . . . . 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 4129	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑓 ·Q ℎ) → (𝑓 <Q 𝑥 ↔ 𝑓 <Q (𝑓 ·Q ℎ)))
13	12	3ad2ant3 1051	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ ℎ ∈ Q ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q 𝑥 ↔ 𝑓 <Q (𝑓 ·Q ℎ)))
14		1pru 7913	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘1P) = {ℎ ∣ 1Q <Q ℎ}
15	14	abeq2i 2349	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (2nd ‘1P) ↔ 1Q <Q ℎ)
16		1nq 7723	. . . . . . . . . . . . . . . . 17 ⊢ 1Q ∈ Q
17		ltmnqg 7758	. . . . . . . . . . . . . . . . 17 ⊢ ((1Q ∈ Q ∧ ℎ ∈ Q ∧ 𝑓 ∈ Q) → (1Q <Q ℎ ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q ℎ)))
18	16, 17	mp3an1 1365	. . . . . . . . . . . . . . . 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 7746	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ Q → (𝑓 ·Q 1Q) = 𝑓)
21	20	breq1d 4135	. . . . . . . . . . . . . . . 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 1048	. . . . . . . . . . . 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 1311	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ ℎ ∈ Q ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q 𝑥 ↔ ℎ ∈ (2nd ‘1P)))
28	8, 27	syl3an2 1312	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ ℎ ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ℎ)) → (𝑓 <Q 𝑥 ↔ ℎ ∈ (2nd ‘1P)))
29	28	3expia 1236	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ ℎ ∈ (2nd ‘1P)) → (𝑥 = (𝑓 ·Q ℎ) → (𝑓 <Q 𝑥 ↔ ℎ ∈ (2nd ‘1P))))
30	29	pm5.32rd 455	. . . . . . 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 2547	. . . . . 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 2708	. . . . . 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 2547	. . 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 7826	. . . . 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 7733	. . . . 5 ⊢ ((𝑢 ∈ Q ∧ 𝑣 ∈ Q) → (𝑢 ·Q 𝑣) ∈ Q)
38	36, 37	genpelvu 7870	. . . 4 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ)))
39	4, 38	mpan2 429	. . 3 ⊢ (𝐴 ∈ P → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ℎ)))
40		prnminu 7846	. . . . . . 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 7722	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
43	42	brel 4822	. . . . . . . . . . . . 13 ⊢ (𝑓 <Q 𝑥 → (𝑓 ∈ Q ∧ 𝑥 ∈ Q))
44	43	ancomd 267	. . . . . . . . . . . 12 ⊢ (𝑓 <Q 𝑥 → (𝑥 ∈ Q ∧ 𝑓 ∈ Q))
45		ltmnqg 7758	. . . . . . . . . . . . . . . 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 7749	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (*Q‘𝑓) ∈ Q)
50	49	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (*Q‘𝑓) ∈ Q)
51		mulcomnqg 7740	. . . . . . . . . . . . . . . 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 6249	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑓 <Q 𝑥 ↔ (𝑓 ·Q (*Q‘𝑓)) <Q (𝑥 ·Q (*Q‘𝑓))))
54		recidnq 7750	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ Q → (𝑓 ·Q (*Q‘𝑓)) = 1Q)
55	54	breq1d 4135	. . . . . . . . . . . . . . 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 7733	. . . . . . . . . . . . 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 4129	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝑥 ·Q (*Q‘𝑓)) → (1Q <Q ℎ ↔ 1Q <Q (𝑥 ·Q (*Q‘𝑓))))
63	62, 14	elab2g 2973	. . . . . . . . . . . 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 7741	. . . . . . . . . . . . . 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 6261	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑓 ·Q (𝑥 ·Q (*Q‘𝑓))) = (𝑥 ·Q (𝑓 ·Q (*Q‘𝑓))))
69	54	oveq2d 6091	. . . . . . . . . . . . 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 7746	. . . . . . . . . . . . 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 2276	. . . . . . . . . . 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 6083	. . . . . . . . . . . 12 ⊢ (ℎ = (𝑥 ·Q (*Q‘𝑓)) → (𝑓 ·Q ℎ) = (𝑓 ·Q (𝑥 ·Q (*Q‘𝑓))))
76	75	eqeq2d 2250	. . . . . . . . . . 11 ⊢ (ℎ = (𝑥 ·Q (*Q‘𝑓)) → (𝑥 = (𝑓 ·Q ℎ) ↔ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q‘𝑓)))))
77	76	rspcev 2929	. . . . . . . . . 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 415	. . . . . . . . 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 2645	. . . . . 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 7842	. . . . . . 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 2668	. . . 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 2236	1 ⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 1P)) = (2nd ‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∃wrex 2529   ⊆ wss 3220  ⟨cop 3708   class class class wbr 4125  ‘cfv 5372  (class class class)co 6075  1^st c1st 6362  2^nd c2nd 6363  Qcnq 7637  1_Qc1q 7638   ·_Q cmq 7640  *_Qcrq 7641   <_Q cltq 7642  Pcnp 7648  1_Pc1p 7649   ·_P cmp 7651

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-inp 7823  df-i1p 7824  df-imp 7826

This theorem is referenced by:  1idpr  7949