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Theorem 1idprl 7096
Description: Lemma for 1idpr 7098. (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 3033 . . . . . 6 (1st ‘1P) ⊆ (1st ‘1P)
2 rexss 3077 . . . . . 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 2520 . . . . . 6 (∃𝑔 ∈ (1st ‘1P)(𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ (𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)))
5 1pr 7060 . . . . . . . . . . 11 1PP
6 prop 6981 . . . . . . . . . . . 12 (1PP → ⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P)
7 elprnql 6987 . . . . . . . . . . . 12 ((⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P𝑔 ∈ (1st ‘1P)) → 𝑔Q)
86, 7sylan 277 . . . . . . . . . . 11 ((1PP𝑔 ∈ (1st ‘1P)) → 𝑔Q)
95, 8mpan 415 . . . . . . . . . 10 (𝑔 ∈ (1st ‘1P) → 𝑔Q)
10 prop 6981 . . . . . . . . . . . 12 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
11 elprnql 6987 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → 𝑓Q)
1210, 11sylan 277 . . . . . . . . . . 11 ((𝐴P𝑓 ∈ (1st𝐴)) → 𝑓Q)
13 breq1 3825 . . . . . . . . . . . . 13 (𝑥 = (𝑓 ·Q 𝑔) → (𝑥 <Q 𝑓 ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
14133ad2ant3 964 . . . . . . . . . . . 12 ((𝑓Q𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓 ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
15 1prl 7061 . . . . . . . . . . . . . . 15 (1st ‘1P) = {𝑔𝑔 <Q 1Q}
1615abeq2i 2195 . . . . . . . . . . . . . 14 (𝑔 ∈ (1st ‘1P) ↔ 𝑔 <Q 1Q)
17 1nq 6872 . . . . . . . . . . . . . . . . 17 1QQ
18 ltmnqg 6907 . . . . . . . . . . . . . . . . 17 ((𝑔Q ∧ 1QQ𝑓Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q)))
1917, 18mp3an2 1259 . . . . . . . . . . . . . . . 16 ((𝑔Q𝑓Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q)))
2019ancoms 264 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q)))
21 mulidnq 6895 . . . . . . . . . . . . . . . . 17 (𝑓Q → (𝑓 ·Q 1Q) = 𝑓)
2221breq2d 3834 . . . . . . . . . . . . . . . 16 (𝑓Q → ((𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q) ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
2322adantr 270 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔Q) → ((𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q) ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
2420, 23bitrd 186 . . . . . . . . . . . . . 14 ((𝑓Q𝑔Q) → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
2516, 24syl5rbb 191 . . . . . . . . . . . . 13 ((𝑓Q𝑔Q) → ((𝑓 ·Q 𝑔) <Q 𝑓𝑔 ∈ (1st ‘1P)))
26253adant3 961 . . . . . . . . . . . 12 ((𝑓Q𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → ((𝑓 ·Q 𝑔) <Q 𝑓𝑔 ∈ (1st ‘1P)))
2714, 26bitrd 186 . . . . . . . . . . 11 ((𝑓Q𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P)))
2812, 27syl3an1 1205 . . . . . . . . . 10 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔Q𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P)))
299, 28syl3an2 1206 . . . . . . . . 9 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P)))
30293expia 1143 . . . . . . . 8 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st ‘1P)) → (𝑥 = (𝑓 ·Q 𝑔) → (𝑥 <Q 𝑓𝑔 ∈ (1st ‘1P))))
3130pm5.32rd 439 . . . . . . 7 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st ‘1P)) → ((𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ (𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔))))
3231rexbidva 2373 . . . . . 6 ((𝐴P𝑓 ∈ (1st𝐴)) → (∃𝑔 ∈ (1st ‘1P)(𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ ∃𝑔 ∈ (1st ‘1P)(𝑔 ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q 𝑔))))
334, 32syl5rbbr 193 . . . . 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 190 . . . 4 ((𝐴P𝑓 ∈ (1st𝐴)) → (∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔) ↔ (𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
3534rexbidva 2373 . . 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 6975 . . . . 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 6882 . . . . 5 ((𝑢Q𝑣Q) → (𝑢 ·Q 𝑣) ∈ Q)
3836, 37genpelvl 7018 . . . 4 ((𝐴P ∧ 1PP) → (𝑥 ∈ (1st ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)))
395, 38mpan2 416 . . 3 (𝐴P → (𝑥 ∈ (1st ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)))
40 prnmaxl 6994 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → ∃𝑓 ∈ (1st𝐴)𝑥 <Q 𝑓)
4110, 40sylan 277 . . . . . 6 ((𝐴P𝑥 ∈ (1st𝐴)) → ∃𝑓 ∈ (1st𝐴)𝑥 <Q 𝑓)
42 ltrelnq 6871 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
4342brel 4460 . . . . . . . . . . . 12 (𝑥 <Q 𝑓 → (𝑥Q𝑓Q))
44 ltmnqg 6907 . . . . . . . . . . . . . . . 16 ((𝑦Q𝑧Q𝑤Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
4544adantl 271 . . . . . . . . . . . . . . 15 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q𝑤Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
46 simpl 107 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → 𝑥Q)
47 simpr 108 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → 𝑓Q)
48 recclnq 6898 . . . . . . . . . . . . . . . 16 (𝑓Q → (*Q𝑓) ∈ Q)
4948adantl 271 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → (*Q𝑓) ∈ Q)
50 mulcomnqg 6889 . . . . . . . . . . . . . . . 16 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5150adantl 271 . . . . . . . . . . . . . . 15 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5245, 46, 47, 49, 51caovord2d 5773 . . . . . . . . . . . . . 14 ((𝑥Q𝑓Q) → (𝑥 <Q 𝑓 ↔ (𝑥 ·Q (*Q𝑓)) <Q (𝑓 ·Q (*Q𝑓))))
53 recidnq 6899 . . . . . . . . . . . . . . . 16 (𝑓Q → (𝑓 ·Q (*Q𝑓)) = 1Q)
5453breq2d 3834 . . . . . . . . . . . . . . 15 (𝑓Q → ((𝑥 ·Q (*Q𝑓)) <Q (𝑓 ·Q (*Q𝑓)) ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5554adantl 271 . . . . . . . . . . . . . 14 ((𝑥Q𝑓Q) → ((𝑥 ·Q (*Q𝑓)) <Q (𝑓 ·Q (*Q𝑓)) ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5652, 55bitrd 186 . . . . . . . . . . . . 13 ((𝑥Q𝑓Q) → (𝑥 <Q 𝑓 ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5756biimpd 142 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) <Q 1Q))
5843, 57mpcom 36 . . . . . . . . . . 11 (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) <Q 1Q)
59 mulclnq 6882 . . . . . . . . . . . . . 14 ((𝑥Q ∧ (*Q𝑓) ∈ Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
6048, 59sylan2 280 . . . . . . . . . . . . 13 ((𝑥Q𝑓Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
6143, 60syl 14 . . . . . . . . . . . 12 (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) ∈ Q)
62 breq1 3825 . . . . . . . . . . . . 13 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑔 <Q 1Q ↔ (𝑥 ·Q (*Q𝑓)) <Q 1Q))
6362, 15elab2g 2753 . . . . . . . . . . . 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 165 . . . . . . . . . 10 (𝑥 <Q 𝑓 → (𝑥 ·Q (*Q𝑓)) ∈ (1st ‘1P))
66 mulassnqg 6890 . . . . . . . . . . . . . 14 ((𝑦Q𝑧Q𝑤Q) → ((𝑦 ·Q 𝑧) ·Q 𝑤) = (𝑦 ·Q (𝑧 ·Q 𝑤)))
6766adantl 271 . . . . . . . . . . . . 13 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q𝑤Q)) → ((𝑦 ·Q 𝑧) ·Q 𝑤) = (𝑦 ·Q (𝑧 ·Q 𝑤)))
6847, 46, 49, 51, 67caov12d 5785 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑓 ·Q (𝑥 ·Q (*Q𝑓))) = (𝑥 ·Q (𝑓 ·Q (*Q𝑓))))
6953oveq2d 5631 . . . . . . . . . . . . 13 (𝑓Q → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
7069adantl 271 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
71 mulidnq 6895 . . . . . . . . . . . . 13 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
7271adantr 270 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q 1Q) = 𝑥)
7368, 70, 723eqtrrd 2122 . . . . . . . . . . 11 ((𝑥Q𝑓Q) → 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
7443, 73syl 14 . . . . . . . . . 10 (𝑥 <Q 𝑓𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
75 oveq2 5623 . . . . . . . . . . . 12 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑓 ·Q 𝑔) = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
7675eqeq2d 2096 . . . . . . . . . . 11 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑥 = (𝑓 ·Q 𝑔) ↔ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓)))))
7776rspcev 2715 . . . . . . . . . 10 (((𝑥 ·Q (*Q𝑓)) ∈ (1st ‘1P) ∧ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓)))) → ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))
7865, 74, 77syl2anc 403 . . . . . . . . 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 318 . . . . . . 7 (𝑓 ∈ (1st𝐴) → (𝑥 <Q 𝑓 → (𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
8180reximia 2464 . . . . . 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 113 . . . 4 (𝐴P → (𝑥 ∈ (1st𝐴) → ∃𝑓 ∈ (1st𝐴)(𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
84 prcdnql 6990 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → (𝑥 <Q 𝑓𝑥 ∈ (1st𝐴)))
8510, 84sylan 277 . . . . . 6 ((𝐴P𝑓 ∈ (1st𝐴)) → (𝑥 <Q 𝑓𝑥 ∈ (1st𝐴)))
8685adantrd 273 . . . . 5 ((𝐴P𝑓 ∈ (1st𝐴)) → ((𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)) → 𝑥 ∈ (1st𝐴)))
8786rexlimdva 2485 . . . 4 (𝐴P → (∃𝑓 ∈ (1st𝐴)(𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔)) → 𝑥 ∈ (1st𝐴)))
8883, 87impbid 127 . . 3 (𝐴P → (𝑥 ∈ (1st𝐴) ↔ ∃𝑓 ∈ (1st𝐴)(𝑥 <Q 𝑓 ∧ ∃𝑔 ∈ (1st ‘1P)𝑥 = (𝑓 ·Q 𝑔))))
8935, 39, 883bitr4d 218 . 2 (𝐴P → (𝑥 ∈ (1st ‘(𝐴 ·P 1P)) ↔ 𝑥 ∈ (1st𝐴)))
9089eqrdv 2083 1 (𝐴P → (1st ‘(𝐴 ·P 1P)) = (1st𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 922   = wceq 1287  wcel 1436  wrex 2356  wss 2988  cop 3434   class class class wbr 3822  cfv 4983  (class class class)co 5615  1st c1st 5868  2nd c2nd 5869  Qcnq 6786  1Qc1q 6787   ·Q cmq 6789  *Qcrq 6790   <Q cltq 6791  Pcnp 6797  1Pc1p 6798   ·P cmp 6800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-eprel 4092  df-id 4096  df-po 4099  df-iso 4100  df-iord 4169  df-on 4171  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-irdg 6091  df-1o 6137  df-oadd 6141  df-omul 6142  df-er 6246  df-ec 6248  df-qs 6252  df-ni 6810  df-pli 6811  df-mi 6812  df-lti 6813  df-plpq 6850  df-mpq 6851  df-enq 6853  df-nqqs 6854  df-plqqs 6855  df-mqqs 6856  df-1nqqs 6857  df-rq 6858  df-ltnqqs 6859  df-inp 6972  df-i1p 6973  df-imp 6975
This theorem is referenced by:  1idpr  7098
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