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Theorem 1idprl 6746
Description: Lemma for 1idpr 6748. (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 2992 . . . . . 6 (1st ‘1P) ⊆ (1st ‘1P)
2 rexss 3035 . . . . . 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 2484 . . . . . 6 (∃𝑔 ∈ (1st ‘1P)(𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ (𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)))
5 1pr 6710 . . . . . . . . . . 11 1PP
6 prop 6631 . . . . . . . . . . . 12 (1PP → ⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P)
7 elprnql 6637 . . . . . . . . . . . 12 ((⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P𝑔 ∈ (1st ‘1P)) → 𝑔Q)
86, 7sylan 271 . . . . . . . . . . 11 ((1PP𝑔 ∈ (1st ‘1P)) → 𝑔Q)
95, 8mpan 408 . . . . . . . . . 10 (𝑔 ∈ (1st ‘1P) → 𝑔Q)
10 prop 6631 . . . . . . . . . . . 12 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
11 elprnql 6637 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → 𝑓Q)
1210, 11sylan 271 . . . . . . . . . . 11 ((𝐴P𝑓 ∈ (1st𝐴)) → 𝑓Q)
13 breq1 3795 . . . . . . . . . . . . 13 (𝑥 = (𝑓 ·Q 𝑔) → (𝑥 <Q 𝑓 ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
14133ad2ant3 938 . . . . . . . . . . . 12 ((𝑓Q𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓 ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
15 1prl 6711 . . . . . . . . . . . . . . 15 (1st ‘1P) = {𝑔𝑔 <Q 1Q}
1615abeq2i 2164 . . . . . . . . . . . . . 14 (𝑔 ∈ (1st ‘1P) ↔ 𝑔 <Q 1Q)
17 1nq 6522 . . . . . . . . . . . . . . . . 17 1QQ
18 ltmnqg 6557 . . . . . . . . . . . . . . . . 17 ((𝑔Q ∧ 1QQ𝑓Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q)))
1917, 18mp3an2 1231 . . . . . . . . . . . . . . . 16 ((𝑔Q𝑓Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q)))
2019ancoms 259 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q)))
21 mulidnq 6545 . . . . . . . . . . . . . . . . 17 (𝑓Q → (𝑓 ·Q 1Q) = 𝑓)
2221breq2d 3804 . . . . . . . . . . . . . . . 16 (𝑓Q → ((𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q) ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
2322adantr 265 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔Q) → ((𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q) ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
2420, 23bitrd 181 . . . . . . . . . . . . . 14 ((𝑓Q𝑔Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
2516, 24syl5rbb 186 . . . . . . . . . . . . 13 ((𝑓Q𝑔Q) → ((𝑓 ·Q 𝑔) <Q 𝑓𝑔 ∈ (1st ‘1P)))
26253adant3 935 . . . . . . . . . . . 12 ((𝑓Q𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → ((𝑓 ·Q 𝑔) <Q 𝑓𝑔 ∈ (1st ‘1P)))
2714, 26bitrd 181 . . . . . . . . . . 11 ((𝑓Q𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P)))
2812, 27syl3an1 1179 . . . . . . . . . 10 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P)))
299, 28syl3an2 1180 . . . . . . . . 9 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P)))
30293expia 1117 . . . . . . . 8 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st ‘1P)) → (𝑥 = (𝑓 ·Q 𝑔) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P))))
3130pm5.32rd 432 . . . . . . 7 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st ‘1P)) → ((𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ (𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔))))
3231rexbidva 2340 . . . . . 6 ((𝐴P𝑓 ∈ (1st𝐴)) → (∃𝑔 ∈ (1st ‘1P)(𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ ∃𝑔 ∈ (1st ‘1P)(𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔))))
334, 32syl5rbbr 188 . . . . 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 185 . . . 4 ((𝐴P𝑓 ∈ (1st𝐴)) → (∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔) ↔ (𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
3534rexbidva 2340 . . 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 6625 . . . . 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 6532 . . . . 5 ((𝑢Q𝑣Q) → (𝑢 ·Q 𝑣) ∈ Q)
3836, 37genpelvl 6668 . . . 4 ((𝐴P ∧ 1PP) → (𝑥 ∈ (1st ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)))
395, 38mpan2 409 . . 3 (𝐴P → (𝑥 ∈ (1st ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)))
40 prnmaxl 6644 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → ∃𝑓 ∈ (1st𝐴)𝑥 <Q 𝑓)
4110, 40sylan 271 . . . . . 6 ((𝐴P𝑥 ∈ (1st𝐴)) → ∃𝑓 ∈ (1st𝐴)𝑥 <Q 𝑓)
42 ltrelnq 6521 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
4342brel 4420 . . . . . . . . . . . 12 (𝑥 <Q 𝑓 → (𝑥Q𝑓Q))
44 ltmnqg 6557 . . . . . . . . . . . . . . . 16 ((𝑦Q𝑧Q𝑤Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
4544adantl 266 . . . . . . . . . . . . . . 15 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q𝑤Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
46 simpl 106 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → 𝑥Q)
47 simpr 107 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → 𝑓Q)
48 recclnq 6548 . . . . . . . . . . . . . . . 16 (𝑓Q → (*Q𝑓) ∈ Q)
4948adantl 266 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → (*Q𝑓) ∈ Q)
50 mulcomnqg 6539 . . . . . . . . . . . . . . . 16 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5150adantl 266 . . . . . . . . . . . . . . 15 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5245, 46, 47, 49, 51caovord2d 5698 . . . . . . . . . . . . . 14 ((𝑥Q𝑓Q) → (𝑥 <Q 𝑓 ↔ (𝑥 ·Q (*Q𝑓)) <Q (𝑓 ·Q (*Q𝑓))))
53 recidnq 6549 . . . . . . . . . . . . . . . 16 (𝑓Q → (𝑓 ·Q (*Q𝑓)) = 1Q)
5453breq2d 3804 . . . . . . . . . . . . . . 15 (𝑓Q → ((𝑥 ·Q (*Q𝑓)) <Q (𝑓 ·Q (*Q𝑓)) ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5554adantl 266 . . . . . . . . . . . . . 14 ((𝑥Q𝑓Q) → ((𝑥 ·Q (*Q𝑓)) <Q (𝑓 ·Q (*Q𝑓)) ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5652, 55bitrd 181 . . . . . . . . . . . . 13 ((𝑥Q𝑓Q) → (𝑥 <Q 𝑓 ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5756biimpd 136 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5843, 57mpcom 36 . . . . . . . . . . 11 (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) <Q 1Q)
59 mulclnq 6532 . . . . . . . . . . . . . 14 ((𝑥Q ∧ (*Q𝑓) ∈ Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
6048, 59sylan2 274 . . . . . . . . . . . . 13 ((𝑥Q𝑓Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
6143, 60syl 14 . . . . . . . . . . . 12 (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) ∈ Q)
62 breq1 3795 . . . . . . . . . . . . 13 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑔 <Q 1Q ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
6362, 15elab2g 2712 . . . . . . . . . . . 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 160 . . . . . . . . . 10 (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) ∈ (1st ‘1P))
66 mulassnqg 6540 . . . . . . . . . . . . . 14 ((𝑦Q𝑧Q𝑤Q) → ((𝑦 ·Q 𝑧) ·Q 𝑤) = (𝑦 ·Q (𝑧 ·Q 𝑤)))
6766adantl 266 . . . . . . . . . . . . 13 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q𝑤Q)) → ((𝑦 ·Q 𝑧) ·Q 𝑤) = (𝑦 ·Q (𝑧 ·Q 𝑤)))
6847, 46, 49, 51, 67caov12d 5710 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑓 ·Q (𝑥 ·Q (*Q𝑓))) = (𝑥 ·Q (𝑓 ·Q (*Q𝑓))))
6953oveq2d 5556 . . . . . . . . . . . . 13 (𝑓Q → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
7069adantl 266 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
71 mulidnq 6545 . . . . . . . . . . . . 13 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
7271adantr 265 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q 1Q) = 𝑥)
7368, 70, 723eqtrrd 2093 . . . . . . . . . . 11 ((𝑥Q𝑓Q) → 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
7443, 73syl 14 . . . . . . . . . 10 (𝑥 <Q 𝑓𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
75 oveq2 5548 . . . . . . . . . . . 12 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑓 ·Q 𝑔) = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
7675eqeq2d 2067 . . . . . . . . . . 11 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑥 = (𝑓 ·Q 𝑔) ↔ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓)))))
7776rspcev 2673 . . . . . . . . . 10 (((𝑥 ·Q (*Q𝑓)) ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓)))) → ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))
7865, 74, 77syl2anc 397 . . . . . . . . 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 312 . . . . . . 7 (𝑓 ∈ (1st𝐴) → (𝑥 <Q 𝑓 → (𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
8180reximia 2431 . . . . . 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 112 . . . 4 (𝐴P → (𝑥 ∈ (1st𝐴) → ∃𝑓 ∈ (1st𝐴)(𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
84 prcdnql 6640 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → (𝑥 <Q 𝑓𝑥 ∈ (1st𝐴)))
8510, 84sylan 271 . . . . . 6 ((𝐴P𝑓 ∈ (1st𝐴)) → (𝑥 <Q 𝑓𝑥 ∈ (1st𝐴)))
8685adantrd 268 . . . . 5 ((𝐴P𝑓 ∈ (1st𝐴)) → ((𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)) → 𝑥 ∈ (1st𝐴)))
8786rexlimdva 2450 . . . 4 (𝐴P → (∃𝑓 ∈ (1st𝐴)(𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)) → 𝑥 ∈ (1st𝐴)))
8883, 87impbid 124 . . 3 (𝐴P → (𝑥 ∈ (1st𝐴) ↔ ∃𝑓 ∈ (1st𝐴)(𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
8935, 39, 883bitr4d 213 . 2 (𝐴P → (𝑥 ∈ (1st ‘(𝐴 ·P 1P)) ↔ 𝑥 ∈ (1st𝐴)))
9089eqrdv 2054 1 (𝐴P → (1st ‘(𝐴 ·P 1P)) = (1st𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  wrex 2324  wss 2945  cop 3406   class class class wbr 3792  cfv 4930  (class class class)co 5540  1st c1st 5793  2nd c2nd 5794  Qcnq 6436  1Qc1q 6437   ·Q cmq 6439  *Qcrq 6440   <Q cltq 6441  Pcnp 6447  1Pc1p 6448   ·P cmp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-inp 6622  df-i1p 6623  df-imp 6625
This theorem is referenced by:  1idpr  6748
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