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Theorem 1idprl 6686
Description: Lemma for 1idpr 6688. (Contributed by Jim Kingdon, 13-Dec-2019.)
Assertion
Ref Expression
1idprl (𝐴P → (1st ‘(𝐴 ·P 1P)) = (1st𝐴))

Proof of Theorem 1idprl
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 2964 . . . . . 6 (1st ‘1P) ⊆ (1st ‘1P)
2 rexss 3007 . . . . . 6 ((1st ‘1P) ⊆ (1st ‘1P) → (∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔) ↔ ∃𝑔 ∈ (1st ‘1P)(𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔))))
31, 2ax-mp 7 . . . . 5 (∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔) ↔ ∃𝑔 ∈ (1st ‘1P)(𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔)))
4 r19.42v 2467 . . . . . 6 (∃𝑔 ∈ (1st ‘1P)(𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ (𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)))
5 1pr 6650 . . . . . . . . . . 11 1PP
6 prop 6571 . . . . . . . . . . . 12 (1PP → ⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P)
7 elprnql 6577 . . . . . . . . . . . 12 ((⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P𝑔 ∈ (1st ‘1P)) → 𝑔Q)
86, 7sylan 267 . . . . . . . . . . 11 ((1PP𝑔 ∈ (1st ‘1P)) → 𝑔Q)
95, 8mpan 400 . . . . . . . . . 10 (𝑔 ∈ (1st ‘1P) → 𝑔Q)
10 prop 6571 . . . . . . . . . . . 12 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
11 elprnql 6577 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → 𝑓Q)
1210, 11sylan 267 . . . . . . . . . . 11 ((𝐴P𝑓 ∈ (1st𝐴)) → 𝑓Q)
13 breq1 3767 . . . . . . . . . . . . 13 (𝑥 = (𝑓 ·Q 𝑔) → (𝑥 <Q 𝑓 ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
14133ad2ant3 927 . . . . . . . . . . . 12 ((𝑓Q𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓 ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
15 1prl 6651 . . . . . . . . . . . . . . 15 (1st ‘1P) = {𝑔𝑔 <Q 1Q}
1615abeq2i 2148 . . . . . . . . . . . . . 14 (𝑔 ∈ (1st ‘1P) ↔ 𝑔 <Q 1Q)
17 1nq 6462 . . . . . . . . . . . . . . . . 17 1QQ
18 ltmnqg 6497 . . . . . . . . . . . . . . . . 17 ((𝑔Q ∧ 1QQ𝑓Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q)))
1917, 18mp3an2 1220 . . . . . . . . . . . . . . . 16 ((𝑔Q𝑓Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q)))
2019ancoms 255 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q)))
21 mulidnq 6485 . . . . . . . . . . . . . . . . 17 (𝑓Q → (𝑓 ·Q 1Q) = 𝑓)
2221breq2d 3776 . . . . . . . . . . . . . . . 16 (𝑓Q → ((𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q) ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
2322adantr 261 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔Q) → ((𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q) ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
2420, 23bitrd 177 . . . . . . . . . . . . . 14 ((𝑓Q𝑔Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
2516, 24syl5rbb 182 . . . . . . . . . . . . 13 ((𝑓Q𝑔Q) → ((𝑓 ·Q 𝑔) <Q 𝑓𝑔 ∈ (1st ‘1P)))
26253adant3 924 . . . . . . . . . . . 12 ((𝑓Q𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → ((𝑓 ·Q 𝑔) <Q 𝑓𝑔 ∈ (1st ‘1P)))
2714, 26bitrd 177 . . . . . . . . . . 11 ((𝑓Q𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P)))
2812, 27syl3an1 1168 . . . . . . . . . 10 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P)))
299, 28syl3an2 1169 . . . . . . . . 9 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P)))
30293expia 1106 . . . . . . . 8 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st ‘1P)) → (𝑥 = (𝑓 ·Q 𝑔) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P))))
3130pm5.32rd 424 . . . . . . 7 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st ‘1P)) → ((𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ (𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔))))
3231rexbidva 2323 . . . . . 6 ((𝐴P𝑓 ∈ (1st𝐴)) → (∃𝑔 ∈ (1st ‘1P)(𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ ∃𝑔 ∈ (1st ‘1P)(𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔))))
334, 32syl5rbbr 184 . . . . 5 ((𝐴P𝑓 ∈ (1st𝐴)) → (∃𝑔 ∈ (1st ‘1P)(𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔)) ↔ (𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
343, 33syl5bb 181 . . . 4 ((𝐴P𝑓 ∈ (1st𝐴)) → (∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔) ↔ (𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
3534rexbidva 2323 . . 3 (𝐴P → (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔) ↔ ∃𝑓 ∈ (1st𝐴)(𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
36 df-imp 6565 . . . . 5 ·P = (𝑦P, 𝑧P ↦ ⟨{𝑤Q ∣ ∃𝑢Q𝑣Q (𝑢 ∈ (1st𝑦) ∧ 𝑣 ∈ (1st𝑧) ∧ 𝑤 = (𝑢 ·Q 𝑣))}, {𝑤Q ∣ ∃𝑢Q𝑣Q (𝑢 ∈ (2nd𝑦) ∧ 𝑣 ∈ (2nd𝑧) ∧ 𝑤 = (𝑢 ·Q 𝑣))}⟩)
37 mulclnq 6472 . . . . 5 ((𝑢Q𝑣Q) → (𝑢 ·Q 𝑣) ∈ Q)
3836, 37genpelvl 6608 . . . 4 ((𝐴P ∧ 1PP) → (𝑥 ∈ (1st ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)))
395, 38mpan2 401 . . 3 (𝐴P → (𝑥 ∈ (1st ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)))
40 prnmaxl 6584 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → ∃𝑓 ∈ (1st𝐴)𝑥 <Q 𝑓)
4110, 40sylan 267 . . . . . 6 ((𝐴P𝑥 ∈ (1st𝐴)) → ∃𝑓 ∈ (1st𝐴)𝑥 <Q 𝑓)
42 ltrelnq 6461 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
4342brel 4392 . . . . . . . . . . . 12 (𝑥 <Q 𝑓 → (𝑥Q𝑓Q))
44 ltmnqg 6497 . . . . . . . . . . . . . . . 16 ((𝑦Q𝑧Q𝑤Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
4544adantl 262 . . . . . . . . . . . . . . 15 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q𝑤Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
46 simpl 102 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → 𝑥Q)
47 simpr 103 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → 𝑓Q)
48 recclnq 6488 . . . . . . . . . . . . . . . 16 (𝑓Q → (*Q𝑓) ∈ Q)
4948adantl 262 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → (*Q𝑓) ∈ Q)
50 mulcomnqg 6479 . . . . . . . . . . . . . . . 16 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5150adantl 262 . . . . . . . . . . . . . . 15 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5245, 46, 47, 49, 51caovord2d 5670 . . . . . . . . . . . . . 14 ((𝑥Q𝑓Q) → (𝑥 <Q 𝑓 ↔ (𝑥 ·Q (*Q𝑓)) <Q (𝑓 ·Q (*Q𝑓))))
53 recidnq 6489 . . . . . . . . . . . . . . . 16 (𝑓Q → (𝑓 ·Q (*Q𝑓)) = 1Q)
5453breq2d 3776 . . . . . . . . . . . . . . 15 (𝑓Q → ((𝑥 ·Q (*Q𝑓)) <Q (𝑓 ·Q (*Q𝑓)) ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5554adantl 262 . . . . . . . . . . . . . 14 ((𝑥Q𝑓Q) → ((𝑥 ·Q (*Q𝑓)) <Q (𝑓 ·Q (*Q𝑓)) ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5652, 55bitrd 177 . . . . . . . . . . . . 13 ((𝑥Q𝑓Q) → (𝑥 <Q 𝑓 ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5756biimpd 132 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5843, 57mpcom 32 . . . . . . . . . . 11 (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) <Q 1Q)
59 mulclnq 6472 . . . . . . . . . . . . . 14 ((𝑥Q ∧ (*Q𝑓) ∈ Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
6048, 59sylan2 270 . . . . . . . . . . . . 13 ((𝑥Q𝑓Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
6143, 60syl 14 . . . . . . . . . . . 12 (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) ∈ Q)
62 breq1 3767 . . . . . . . . . . . . 13 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑔 <Q 1Q ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
6362, 15elab2g 2689 . . . . . . . . . . . 12 ((𝑥 ·Q (*Q𝑓)) ∈ Q → ((𝑥 ·Q (*Q𝑓)) ∈ (1st ‘1P) ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
6461, 63syl 14 . . . . . . . . . . 11 (𝑥 <Q 𝑓 → ((𝑥 ·Q (*Q𝑓)) ∈ (1st ‘1P) ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
6558, 64mpbird 156 . . . . . . . . . 10 (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) ∈ (1st ‘1P))
66 mulassnqg 6480 . . . . . . . . . . . . . 14 ((𝑦Q𝑧Q𝑤Q) → ((𝑦 ·Q 𝑧) ·Q 𝑤) = (𝑦 ·Q (𝑧 ·Q 𝑤)))
6766adantl 262 . . . . . . . . . . . . 13 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q𝑤Q)) → ((𝑦 ·Q 𝑧) ·Q 𝑤) = (𝑦 ·Q (𝑧 ·Q 𝑤)))
6847, 46, 49, 51, 67caov12d 5682 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑓 ·Q (𝑥 ·Q (*Q𝑓))) = (𝑥 ·Q (𝑓 ·Q (*Q𝑓))))
6953oveq2d 5528 . . . . . . . . . . . . 13 (𝑓Q → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
7069adantl 262 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
71 mulidnq 6485 . . . . . . . . . . . . 13 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
7271adantr 261 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q 1Q) = 𝑥)
7368, 70, 723eqtrrd 2077 . . . . . . . . . . 11 ((𝑥Q𝑓Q) → 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
7443, 73syl 14 . . . . . . . . . 10 (𝑥 <Q 𝑓𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
75 oveq2 5520 . . . . . . . . . . . 12 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑓 ·Q 𝑔) = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
7675eqeq2d 2051 . . . . . . . . . . 11 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑥 = (𝑓 ·Q 𝑔) ↔ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓)))))
7776rspcev 2656 . . . . . . . . . 10 (((𝑥 ·Q (*Q𝑓)) ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓)))) → ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))
7865, 74, 77syl2anc 391 . . . . . . . . 9 (𝑥 <Q 𝑓 → ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))
7978a1i 9 . . . . . . . 8 (𝑓 ∈ (1st𝐴) → (𝑥 <Q 𝑓 → ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)))
8079ancld 308 . . . . . . 7 (𝑓 ∈ (1st𝐴) → (𝑥 <Q 𝑓 → (𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
8180reximia 2414 . . . . . 6 (∃𝑓 ∈ (1st𝐴)𝑥 <Q 𝑓 → ∃𝑓 ∈ (1st𝐴)(𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)))
8241, 81syl 14 . . . . 5 ((𝐴P𝑥 ∈ (1st𝐴)) → ∃𝑓 ∈ (1st𝐴)(𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)))
8382ex 108 . . . 4 (𝐴P → (𝑥 ∈ (1st𝐴) → ∃𝑓 ∈ (1st𝐴)(𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
84 prcdnql 6580 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → (𝑥 <Q 𝑓𝑥 ∈ (1st𝐴)))
8510, 84sylan 267 . . . . . 6 ((𝐴P𝑓 ∈ (1st𝐴)) → (𝑥 <Q 𝑓𝑥 ∈ (1st𝐴)))
8685adantrd 264 . . . . 5 ((𝐴P𝑓 ∈ (1st𝐴)) → ((𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)) → 𝑥 ∈ (1st𝐴)))
8786rexlimdva 2433 . . . 4 (𝐴P → (∃𝑓 ∈ (1st𝐴)(𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)) → 𝑥 ∈ (1st𝐴)))
8883, 87impbid 120 . . 3 (𝐴P → (𝑥 ∈ (1st𝐴) ↔ ∃𝑓 ∈ (1st𝐴)(𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
8935, 39, 883bitr4d 209 . 2 (𝐴P → (𝑥 ∈ (1st ‘(𝐴 ·P 1P)) ↔ 𝑥 ∈ (1st𝐴)))
9089eqrdv 2038 1 (𝐴P → (1st ‘(𝐴 ·P 1P)) = (1st𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wcel 1393  wrex 2307  wss 2917  cop 3378   class class class wbr 3764  cfv 4902  (class class class)co 5512  1st c1st 5765  2nd c2nd 5766  Qcnq 6376  1Qc1q 6377   ·Q cmq 6379  *Qcrq 6380   <Q cltq 6381  Pcnp 6387  1Pc1p 6388   ·P cmp 6390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-inp 6562  df-i1p 6563  df-imp 6565
This theorem is referenced by:  1idpr  6688
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