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Theorem 1idprl 7410
Description: Lemma for 1idpr 7412. (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 3117 . . . . . 6 (1st ‘1P) ⊆ (1st ‘1P)
2 rexss 3164 . . . . . 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 5 . . . . 5 (∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔) ↔ ∃𝑔 ∈ (1st ‘1P)(𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔)))
4 1pr 7374 . . . . . . . . . . 11 1PP
5 prop 7295 . . . . . . . . . . . 12 (1PP → ⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P)
6 elprnql 7301 . . . . . . . . . . . 12 ((⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P𝑔 ∈ (1st ‘1P)) → 𝑔Q)
75, 6sylan 281 . . . . . . . . . . 11 ((1PP𝑔 ∈ (1st ‘1P)) → 𝑔Q)
84, 7mpan 420 . . . . . . . . . 10 (𝑔 ∈ (1st ‘1P) → 𝑔Q)
9 prop 7295 . . . . . . . . . . . 12 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
10 elprnql 7301 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → 𝑓Q)
119, 10sylan 281 . . . . . . . . . . 11 ((𝐴P𝑓 ∈ (1st𝐴)) → 𝑓Q)
12 breq1 3932 . . . . . . . . . . . . 13 (𝑥 = (𝑓 ·Q 𝑔) → (𝑥 <Q 𝑓 ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
13123ad2ant3 1004 . . . . . . . . . . . 12 ((𝑓Q𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓 ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
14 1prl 7375 . . . . . . . . . . . . . . 15 (1st ‘1P) = {𝑔𝑔 <Q 1Q}
1514abeq2i 2250 . . . . . . . . . . . . . 14 (𝑔 ∈ (1st ‘1P) ↔ 𝑔 <Q 1Q)
16 1nq 7186 . . . . . . . . . . . . . . . . 17 1QQ
17 ltmnqg 7221 . . . . . . . . . . . . . . . . 17 ((𝑔Q ∧ 1QQ𝑓Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q)))
1816, 17mp3an2 1303 . . . . . . . . . . . . . . . 16 ((𝑔Q𝑓Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q)))
1918ancoms 266 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q)))
20 mulidnq 7209 . . . . . . . . . . . . . . . . 17 (𝑓Q → (𝑓 ·Q 1Q) = 𝑓)
2120breq2d 3941 . . . . . . . . . . . . . . . 16 (𝑓Q → ((𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q) ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
2221adantr 274 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔Q) → ((𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q) ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
2319, 22bitrd 187 . . . . . . . . . . . . . 14 ((𝑓Q𝑔Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
2415, 23syl5rbb 192 . . . . . . . . . . . . 13 ((𝑓Q𝑔Q) → ((𝑓 ·Q 𝑔) <Q 𝑓𝑔 ∈ (1st ‘1P)))
25243adant3 1001 . . . . . . . . . . . 12 ((𝑓Q𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → ((𝑓 ·Q 𝑔) <Q 𝑓𝑔 ∈ (1st ‘1P)))
2613, 25bitrd 187 . . . . . . . . . . 11 ((𝑓Q𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P)))
2711, 26syl3an1 1249 . . . . . . . . . 10 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P)))
288, 27syl3an2 1250 . . . . . . . . 9 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P)))
29283expia 1183 . . . . . . . 8 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st ‘1P)) → (𝑥 = (𝑓 ·Q 𝑔) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P))))
3029pm5.32rd 446 . . . . . . 7 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st ‘1P)) → ((𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ (𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔))))
3130rexbidva 2434 . . . . . 6 ((𝐴P𝑓 ∈ (1st𝐴)) → (∃𝑔 ∈ (1st ‘1P)(𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ ∃𝑔 ∈ (1st ‘1P)(𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔))))
32 r19.42v 2588 . . . . . 6 (∃𝑔 ∈ (1st ‘1P)(𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ (𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)))
3331, 32bitr3di 194 . . . . 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 191 . . . 4 ((𝐴P𝑓 ∈ (1st𝐴)) → (∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔) ↔ (𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
3534rexbidva 2434 . . 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 7289 . . . . 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 7196 . . . . 5 ((𝑢Q𝑣Q) → (𝑢 ·Q 𝑣) ∈ Q)
3836, 37genpelvl 7332 . . . 4 ((𝐴P ∧ 1PP) → (𝑥 ∈ (1st ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)))
394, 38mpan2 421 . . 3 (𝐴P → (𝑥 ∈ (1st ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)))
40 prnmaxl 7308 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → ∃𝑓 ∈ (1st𝐴)𝑥 <Q 𝑓)
419, 40sylan 281 . . . . . 6 ((𝐴P𝑥 ∈ (1st𝐴)) → ∃𝑓 ∈ (1st𝐴)𝑥 <Q 𝑓)
42 ltrelnq 7185 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
4342brel 4591 . . . . . . . . . . . 12 (𝑥 <Q 𝑓 → (𝑥Q𝑓Q))
44 ltmnqg 7221 . . . . . . . . . . . . . . . 16 ((𝑦Q𝑧Q𝑤Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
4544adantl 275 . . . . . . . . . . . . . . 15 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q𝑤Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
46 simpl 108 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → 𝑥Q)
47 simpr 109 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → 𝑓Q)
48 recclnq 7212 . . . . . . . . . . . . . . . 16 (𝑓Q → (*Q𝑓) ∈ Q)
4948adantl 275 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → (*Q𝑓) ∈ Q)
50 mulcomnqg 7203 . . . . . . . . . . . . . . . 16 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5150adantl 275 . . . . . . . . . . . . . . 15 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5245, 46, 47, 49, 51caovord2d 5940 . . . . . . . . . . . . . 14 ((𝑥Q𝑓Q) → (𝑥 <Q 𝑓 ↔ (𝑥 ·Q (*Q𝑓)) <Q (𝑓 ·Q (*Q𝑓))))
53 recidnq 7213 . . . . . . . . . . . . . . . 16 (𝑓Q → (𝑓 ·Q (*Q𝑓)) = 1Q)
5453breq2d 3941 . . . . . . . . . . . . . . 15 (𝑓Q → ((𝑥 ·Q (*Q𝑓)) <Q (𝑓 ·Q (*Q𝑓)) ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5554adantl 275 . . . . . . . . . . . . . 14 ((𝑥Q𝑓Q) → ((𝑥 ·Q (*Q𝑓)) <Q (𝑓 ·Q (*Q𝑓)) ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5652, 55bitrd 187 . . . . . . . . . . . . 13 ((𝑥Q𝑓Q) → (𝑥 <Q 𝑓 ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5756biimpd 143 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5843, 57mpcom 36 . . . . . . . . . . 11 (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) <Q 1Q)
59 mulclnq 7196 . . . . . . . . . . . . . 14 ((𝑥Q ∧ (*Q𝑓) ∈ Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
6048, 59sylan2 284 . . . . . . . . . . . . 13 ((𝑥Q𝑓Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
6143, 60syl 14 . . . . . . . . . . . 12 (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) ∈ Q)
62 breq1 3932 . . . . . . . . . . . . 13 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑔 <Q 1Q ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
6362, 14elab2g 2831 . . . . . . . . . . . 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 166 . . . . . . . . . 10 (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) ∈ (1st ‘1P))
66 mulassnqg 7204 . . . . . . . . . . . . . 14 ((𝑦Q𝑧Q𝑤Q) → ((𝑦 ·Q 𝑧) ·Q 𝑤) = (𝑦 ·Q (𝑧 ·Q 𝑤)))
6766adantl 275 . . . . . . . . . . . . 13 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q𝑤Q)) → ((𝑦 ·Q 𝑧) ·Q 𝑤) = (𝑦 ·Q (𝑧 ·Q 𝑤)))
6847, 46, 49, 51, 67caov12d 5952 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑓 ·Q (𝑥 ·Q (*Q𝑓))) = (𝑥 ·Q (𝑓 ·Q (*Q𝑓))))
6953oveq2d 5790 . . . . . . . . . . . . 13 (𝑓Q → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
7069adantl 275 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
71 mulidnq 7209 . . . . . . . . . . . . 13 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
7271adantr 274 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q 1Q) = 𝑥)
7368, 70, 723eqtrrd 2177 . . . . . . . . . . 11 ((𝑥Q𝑓Q) → 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
7443, 73syl 14 . . . . . . . . . 10 (𝑥 <Q 𝑓𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
75 oveq2 5782 . . . . . . . . . . . 12 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑓 ·Q 𝑔) = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
7675eqeq2d 2151 . . . . . . . . . . 11 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑥 = (𝑓 ·Q 𝑔) ↔ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓)))))
7776rspcev 2789 . . . . . . . . . 10 (((𝑥 ·Q (*Q𝑓)) ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓)))) → ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))
7865, 74, 77syl2anc 408 . . . . . . . . 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 323 . . . . . . 7 (𝑓 ∈ (1st𝐴) → (𝑥 <Q 𝑓 → (𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
8180reximia 2527 . . . . . 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 114 . . . 4 (𝐴P → (𝑥 ∈ (1st𝐴) → ∃𝑓 ∈ (1st𝐴)(𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
84 prcdnql 7304 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → (𝑥 <Q 𝑓𝑥 ∈ (1st𝐴)))
859, 84sylan 281 . . . . . 6 ((𝐴P𝑓 ∈ (1st𝐴)) → (𝑥 <Q 𝑓𝑥 ∈ (1st𝐴)))
8685adantrd 277 . . . . 5 ((𝐴P𝑓 ∈ (1st𝐴)) → ((𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)) → 𝑥 ∈ (1st𝐴)))
8786rexlimdva 2549 . . . 4 (𝐴P → (∃𝑓 ∈ (1st𝐴)(𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)) → 𝑥 ∈ (1st𝐴)))
8883, 87impbid 128 . . 3 (𝐴P → (𝑥 ∈ (1st𝐴) ↔ ∃𝑓 ∈ (1st𝐴)(𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
8935, 39, 883bitr4d 219 . 2 (𝐴P → (𝑥 ∈ (1st ‘(𝐴 ·P 1P)) ↔ 𝑥 ∈ (1st𝐴)))
9089eqrdv 2137 1 (𝐴P → (1st ‘(𝐴 ·P 1P)) = (1st𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962   = wceq 1331  wcel 1480  wrex 2417  wss 3071  cop 3530   class class class wbr 3929  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7100  1Qc1q 7101   ·Q cmq 7103  *Qcrq 7104   <Q cltq 7105  Pcnp 7111  1Pc1p 7112   ·P cmp 7114
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 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-inp 7286  df-i1p 7287  df-imp 7289
This theorem is referenced by:  1idpr  7412
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