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Theorem 1idpru 7553
Description: Lemma for 1idpr 7554. (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 7516 . . . . . . . . . . 11 1PP
5 prop 7437 . . . . . . . . . . . 12 (1PP → ⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P)
6 elprnqu 7444 . . . . . . . . . . . 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 7437 . . . . . . . . . . . 12 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
10 elprnqu 7444 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
119, 10sylan 281 . . . . . . . . . . 11 ((𝐴P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
12 breq2 3993 . . . . . . . . . . . . 13 (𝑥 = (𝑓 ·Q ) → (𝑓 <Q 𝑥𝑓 <Q (𝑓 ·Q )))
13123ad2ant3 1015 . . . . . . . . . . . 12 ((𝑓QQ𝑥 = (𝑓 ·Q )) → (𝑓 <Q 𝑥𝑓 <Q (𝑓 ·Q )))
14 1pru 7518 . . . . . . . . . . . . . . 15 (2nd ‘1P) = { ∣ 1Q <Q }
1514abeq2i 2281 . . . . . . . . . . . . . 14 ( ∈ (2nd ‘1P) ↔ 1Q <Q )
16 1nq 7328 . . . . . . . . . . . . . . . . 17 1QQ
17 ltmnqg 7363 . . . . . . . . . . . . . . . . 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 7351 . . . . . . . . . . . . . . . . 17 (𝑓Q → (𝑓 ·Q 1Q) = 𝑓)
2120breq1d 3999 . . . . . . . . . . . . . . . 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 7431 . . . . 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 7338 . . . . 5 ((𝑢Q𝑣Q) → (𝑢 ·Q 𝑣) ∈ Q)
3836, 37genpelvu 7475 . . . 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 7451 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (2nd𝐴)) → ∃𝑓 ∈ (2nd𝐴)𝑓 <Q 𝑥)
419, 40sylan 281 . . . . . 6 ((𝐴P𝑥 ∈ (2nd𝐴)) → ∃𝑓 ∈ (2nd𝐴)𝑓 <Q 𝑥)
42 ltrelnq 7327 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
4342brel 4663 . . . . . . . . . . . . 13 (𝑓 <Q 𝑥 → (𝑓Q𝑥Q))
4443ancomd 265 . . . . . . . . . . . 12 (𝑓 <Q 𝑥 → (𝑥Q𝑓Q))
45 ltmnqg 7363 . . . . . . . . . . . . . . . 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 7354 . . . . . . . . . . . . . . . 16 (𝑓Q → (*Q𝑓) ∈ Q)
5049adantl 275 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → (*Q𝑓) ∈ Q)
51 mulcomnqg 7345 . . . . . . . . . . . . . . . 16 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5251adantl 275 . . . . . . . . . . . . . . 15 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5346, 47, 48, 50, 52caovord2d 6022 . . . . . . . . . . . . . 14 ((𝑥Q𝑓Q) → (𝑓 <Q 𝑥 ↔ (𝑓 ·Q (*Q𝑓)) <Q (𝑥 ·Q (*Q𝑓))))
54 recidnq 7355 . . . . . . . . . . . . . . . 16 (𝑓Q → (𝑓 ·Q (*Q𝑓)) = 1Q)
5554breq1d 3999 . . . . . . . . . . . . . . 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 7338 . . . . . . . . . . . . 13 ((𝑥Q ∧ (*Q𝑓) ∈ Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
6149, 60sylan2 284 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
62 breq2 3993 . . . . . . . . . . . . 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 7346 . . . . . . . . . . . . . 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 6034 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑓 ·Q (𝑥 ·Q (*Q𝑓))) = (𝑥 ·Q (𝑓 ·Q (*Q𝑓))))
6954oveq2d 5869 . . . . . . . . . . . . 13 (𝑓Q → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
7069adantl 275 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
71 mulidnq 7351 . . . . . . . . . . . . 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 5861 . . . . . . . . . . . 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 7447 . . . . . . 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 3586   class class class wbr 3989  cfv 5198  (class class class)co 5853  1st c1st 6117  2nd c2nd 6118  Qcnq 7242  1Qc1q 7243   ·Q cmq 7245  *Qcrq 7246   <Q cltq 7247  Pcnp 7253  1Pc1p 7254   ·P cmp 7256
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 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
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 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-inp 7428  df-i1p 7429  df-imp 7431
This theorem is referenced by:  1idpr  7554
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