ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  1idpru GIF version

Theorem 1idpru 7540
Description: Lemma for 1idpr 7541. (Contributed by Jim Kingdon, 13-Dec-2019.)
Assertion
Ref Expression
1idpru (𝐴P → (2nd ‘(𝐴 ·P 1P)) = (2nd𝐴))

Proof of Theorem 1idpru
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3167 . . . . . 6 (2nd ‘1P) ⊆ (2nd ‘1P)
2 rexss 3214 . . . . . 6 ((2nd ‘1P) ⊆ (2nd ‘1P) → (∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ) ↔ ∃ ∈ (2nd ‘1P)( ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ))))
31, 2ax-mp 5 . . . . 5 (∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ) ↔ ∃ ∈ (2nd ‘1P)( ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q )))
4 1pr 7503 . . . . . . . . . . 11 1PP
5 prop 7424 . . . . . . . . . . . 12 (1PP → ⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P)
6 elprnqu 7431 . . . . . . . . . . . 12 ((⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P ∈ (2nd ‘1P)) → Q)
75, 6sylan 281 . . . . . . . . . . 11 ((1PP ∈ (2nd ‘1P)) → Q)
84, 7mpan 422 . . . . . . . . . 10 ( ∈ (2nd ‘1P) → Q)
9 prop 7424 . . . . . . . . . . . 12 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
10 elprnqu 7431 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
119, 10sylan 281 . . . . . . . . . . 11 ((𝐴P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
12 breq2 3991 . . . . . . . . . . . . 13 (𝑥 = (𝑓 ·Q ) → (𝑓 <Q 𝑥𝑓 <Q (𝑓 ·Q )))
13123ad2ant3 1015 . . . . . . . . . . . 12 ((𝑓QQ𝑥 = (𝑓 ·Q )) → (𝑓 <Q 𝑥𝑓 <Q (𝑓 ·Q )))
14 1pru 7505 . . . . . . . . . . . . . . 15 (2nd ‘1P) = { ∣ 1Q <Q }
1514abeq2i 2281 . . . . . . . . . . . . . 14 ( ∈ (2nd ‘1P) ↔ 1Q <Q )
16 1nq 7315 . . . . . . . . . . . . . . . . 17 1QQ
17 ltmnqg 7350 . . . . . . . . . . . . . . . . 17 ((1QQQ𝑓Q) → (1Q <Q ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q )))
1816, 17mp3an1 1319 . . . . . . . . . . . . . . . 16 ((Q𝑓Q) → (1Q <Q ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q )))
1918ancoms 266 . . . . . . . . . . . . . . 15 ((𝑓QQ) → (1Q <Q ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q )))
20 mulidnq 7338 . . . . . . . . . . . . . . . . 17 (𝑓Q → (𝑓 ·Q 1Q) = 𝑓)
2120breq1d 3997 . . . . . . . . . . . . . . . 16 (𝑓Q → ((𝑓 ·Q 1Q) <Q (𝑓 ·Q ) ↔ 𝑓 <Q (𝑓 ·Q )))
2221adantr 274 . . . . . . . . . . . . . . 15 ((𝑓QQ) → ((𝑓 ·Q 1Q) <Q (𝑓 ·Q ) ↔ 𝑓 <Q (𝑓 ·Q )))
2319, 22bitrd 187 . . . . . . . . . . . . . 14 ((𝑓QQ) → (1Q <Q 𝑓 <Q (𝑓 ·Q )))
2415, 23bitr2id 192 . . . . . . . . . . . . 13 ((𝑓QQ) → (𝑓 <Q (𝑓 ·Q ) ↔ ∈ (2nd ‘1P)))
25243adant3 1012 . . . . . . . . . . . 12 ((𝑓QQ𝑥 = (𝑓 ·Q )) → (𝑓 <Q (𝑓 ·Q ) ↔ ∈ (2nd ‘1P)))
2613, 25bitrd 187 . . . . . . . . . . 11 ((𝑓QQ𝑥 = (𝑓 ·Q )) → (𝑓 <Q 𝑥 ∈ (2nd ‘1P)))
2711, 26syl3an1 1266 . . . . . . . . . 10 (((𝐴P𝑓 ∈ (2nd𝐴)) ∧ Q𝑥 = (𝑓 ·Q )) → (𝑓 <Q 𝑥 ∈ (2nd ‘1P)))
288, 27syl3an2 1267 . . . . . . . . 9 (((𝐴P𝑓 ∈ (2nd𝐴)) ∧ ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q )) → (𝑓 <Q 𝑥 ∈ (2nd ‘1P)))
29283expia 1200 . . . . . . . 8 (((𝐴P𝑓 ∈ (2nd𝐴)) ∧ ∈ (2nd ‘1P)) → (𝑥 = (𝑓 ·Q ) → (𝑓 <Q 𝑥 ∈ (2nd ‘1P))))
3029pm5.32rd 448 . . . . . . 7 (((𝐴P𝑓 ∈ (2nd𝐴)) ∧ ∈ (2nd ‘1P)) → ((𝑓 <Q 𝑥𝑥 = (𝑓 ·Q )) ↔ ( ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ))))
3130rexbidva 2467 . . . . . 6 ((𝐴P𝑓 ∈ (2nd𝐴)) → (∃ ∈ (2nd ‘1P)(𝑓 <Q 𝑥𝑥 = (𝑓 ·Q )) ↔ ∃ ∈ (2nd ‘1P)( ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ))))
32 r19.42v 2627 . . . . . 6 (∃ ∈ (2nd ‘1P)(𝑓 <Q 𝑥𝑥 = (𝑓 ·Q )) ↔ (𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )))
3331, 32bitr3di 194 . . . . 5 ((𝐴P𝑓 ∈ (2nd𝐴)) → (∃ ∈ (2nd ‘1P)( ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q )) ↔ (𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))))
343, 33syl5bb 191 . . . 4 ((𝐴P𝑓 ∈ (2nd𝐴)) → (∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ) ↔ (𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))))
3534rexbidva 2467 . . 3 (𝐴P → (∃𝑓 ∈ (2nd𝐴)∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ) ↔ ∃𝑓 ∈ (2nd𝐴)(𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))))
36 df-imp 7418 . . . . 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 7325 . . . . 5 ((𝑢Q𝑣Q) → (𝑢 ·Q 𝑣) ∈ Q)
3836, 37genpelvu 7462 . . . 4 ((𝐴P ∧ 1PP) → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (2nd𝐴)∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )))
394, 38mpan2 423 . . 3 (𝐴P → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (2nd𝐴)∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )))
40 prnminu 7438 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (2nd𝐴)) → ∃𝑓 ∈ (2nd𝐴)𝑓 <Q 𝑥)
419, 40sylan 281 . . . . . 6 ((𝐴P𝑥 ∈ (2nd𝐴)) → ∃𝑓 ∈ (2nd𝐴)𝑓 <Q 𝑥)
42 ltrelnq 7314 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
4342brel 4661 . . . . . . . . . . . . 13 (𝑓 <Q 𝑥 → (𝑓Q𝑥Q))
4443ancomd 265 . . . . . . . . . . . 12 (𝑓 <Q 𝑥 → (𝑥Q𝑓Q))
45 ltmnqg 7350 . . . . . . . . . . . . . . . 16 ((𝑦Q𝑧Q𝑤Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
4645adantl 275 . . . . . . . . . . . . . . 15 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q𝑤Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
47 simpr 109 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → 𝑓Q)
48 simpl 108 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → 𝑥Q)
49 recclnq 7341 . . . . . . . . . . . . . . . 16 (𝑓Q → (*Q𝑓) ∈ Q)
5049adantl 275 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → (*Q𝑓) ∈ Q)
51 mulcomnqg 7332 . . . . . . . . . . . . . . . 16 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5251adantl 275 . . . . . . . . . . . . . . 15 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5346, 47, 48, 50, 52caovord2d 6019 . . . . . . . . . . . . . 14 ((𝑥Q𝑓Q) → (𝑓 <Q 𝑥 ↔ (𝑓 ·Q (*Q𝑓)) <Q (𝑥 ·Q (*Q𝑓))))
54 recidnq 7342 . . . . . . . . . . . . . . . 16 (𝑓Q → (𝑓 ·Q (*Q𝑓)) = 1Q)
5554breq1d 3997 . . . . . . . . . . . . . . 15 (𝑓Q → ((𝑓 ·Q (*Q𝑓)) <Q (𝑥 ·Q (*Q𝑓)) ↔ 1Q <Q (𝑥 ·Q (*Q𝑓))))
5655adantl 275 . . . . . . . . . . . . . 14 ((𝑥Q𝑓Q) → ((𝑓 ·Q (*Q𝑓)) <Q (𝑥 ·Q (*Q𝑓)) ↔ 1Q <Q (𝑥 ·Q (*Q𝑓))))
5753, 56bitrd 187 . . . . . . . . . . . . 13 ((𝑥Q𝑓Q) → (𝑓 <Q 𝑥 ↔ 1Q <Q (𝑥 ·Q (*Q𝑓))))
5857biimpd 143 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑓 <Q 𝑥 → 1Q <Q (𝑥 ·Q (*Q𝑓))))
5944, 58mpcom 36 . . . . . . . . . . 11 (𝑓 <Q 𝑥 → 1Q <Q (𝑥 ·Q (*Q𝑓)))
60 mulclnq 7325 . . . . . . . . . . . . 13 ((𝑥Q ∧ (*Q𝑓) ∈ Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
6149, 60sylan2 284 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
62 breq2 3991 . . . . . . . . . . . . 13 ( = (𝑥 ·Q (*Q𝑓)) → (1Q <Q ↔ 1Q <Q (𝑥 ·Q (*Q𝑓))))
6362, 14elab2g 2877 . . . . . . . . . . . 12 ((𝑥 ·Q (*Q𝑓)) ∈ Q → ((𝑥 ·Q (*Q𝑓)) ∈ (2nd ‘1P) ↔ 1Q <Q (𝑥 ·Q (*Q𝑓))))
6444, 61, 633syl 17 . . . . . . . . . . 11 (𝑓 <Q 𝑥 → ((𝑥 ·Q (*Q𝑓)) ∈ (2nd ‘1P) ↔ 1Q <Q (𝑥 ·Q (*Q𝑓))))
6559, 64mpbird 166 . . . . . . . . . 10 (𝑓 <Q 𝑥 → (𝑥 ·Q (*Q𝑓)) ∈ (2nd ‘1P))
66 mulassnqg 7333 . . . . . . . . . . . . . 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, 48, 50, 52, 67caov12d 6031 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑓 ·Q (𝑥 ·Q (*Q𝑓))) = (𝑥 ·Q (𝑓 ·Q (*Q𝑓))))
6954oveq2d 5866 . . . . . . . . . . . . 13 (𝑓Q → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
7069adantl 275 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
71 mulidnq 7338 . . . . . . . . . . . . 13 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
7271adantr 274 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q 1Q) = 𝑥)
7368, 70, 723eqtrrd 2208 . . . . . . . . . . 11 ((𝑥Q𝑓Q) → 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
7444, 73syl 14 . . . . . . . . . 10 (𝑓 <Q 𝑥𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
75 oveq2 5858 . . . . . . . . . . . 12 ( = (𝑥 ·Q (*Q𝑓)) → (𝑓 ·Q ) = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
7675eqeq2d 2182 . . . . . . . . . . 11 ( = (𝑥 ·Q (*Q𝑓)) → (𝑥 = (𝑓 ·Q ) ↔ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓)))))
7776rspcev 2834 . . . . . . . . . 10 (((𝑥 ·Q (*Q𝑓)) ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓)))) → ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))
7865, 74, 77syl2anc 409 . . . . . . . . 9 (𝑓 <Q 𝑥 → ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))
7978a1i 9 . . . . . . . 8 (𝑓 ∈ (2nd𝐴) → (𝑓 <Q 𝑥 → ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )))
8079ancld 323 . . . . . . 7 (𝑓 ∈ (2nd𝐴) → (𝑓 <Q 𝑥 → (𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))))
8180reximia 2565 . . . . . 6 (∃𝑓 ∈ (2nd𝐴)𝑓 <Q 𝑥 → ∃𝑓 ∈ (2nd𝐴)(𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )))
8241, 81syl 14 . . . . 5 ((𝐴P𝑥 ∈ (2nd𝐴)) → ∃𝑓 ∈ (2nd𝐴)(𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )))
8382ex 114 . . . 4 (𝐴P → (𝑥 ∈ (2nd𝐴) → ∃𝑓 ∈ (2nd𝐴)(𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))))
84 prcunqu 7434 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (2nd𝐴)) → (𝑓 <Q 𝑥𝑥 ∈ (2nd𝐴)))
859, 84sylan 281 . . . . . 6 ((𝐴P𝑓 ∈ (2nd𝐴)) → (𝑓 <Q 𝑥𝑥 ∈ (2nd𝐴)))
8685adantrd 277 . . . . 5 ((𝐴P𝑓 ∈ (2nd𝐴)) → ((𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )) → 𝑥 ∈ (2nd𝐴)))
8786rexlimdva 2587 . . . 4 (𝐴P → (∃𝑓 ∈ (2nd𝐴)(𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )) → 𝑥 ∈ (2nd𝐴)))
8883, 87impbid 128 . . 3 (𝐴P → (𝑥 ∈ (2nd𝐴) ↔ ∃𝑓 ∈ (2nd𝐴)(𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))))
8935, 39, 883bitr4d 219 . 2 (𝐴P → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ 𝑥 ∈ (2nd𝐴)))
9089eqrdv 2168 1 (𝐴P → (2nd ‘(𝐴 ·P 1P)) = (2nd𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 973   = wceq 1348  wcel 2141  wrex 2449  wss 3121  cop 3584   class class class wbr 3987  cfv 5196  (class class class)co 5850  1st c1st 6114  2nd c2nd 6115  Qcnq 7229  1Qc1q 7230   ·Q cmq 7232  *Qcrq 7233   <Q cltq 7234  Pcnp 7240  1Pc1p 7241   ·P cmp 7243
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-1o 6392  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-pli 7254  df-mi 7255  df-lti 7256  df-plpq 7293  df-mpq 7294  df-enq 7296  df-nqqs 7297  df-plqqs 7298  df-mqqs 7299  df-1nqqs 7300  df-rq 7301  df-ltnqqs 7302  df-inp 7415  df-i1p 7416  df-imp 7418
This theorem is referenced by:  1idpr  7541
  Copyright terms: Public domain W3C validator