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Theorem 1idpru 7419
Description: Lemma for 1idpr 7420. (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 3118 . . . . . 6 (2nd ‘1P) ⊆ (2nd ‘1P)
2 rexss 3165 . . . . . 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 7382 . . . . . . . . . . 11 1PP
5 prop 7303 . . . . . . . . . . . 12 (1PP → ⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P)
6 elprnqu 7310 . . . . . . . . . . . 12 ((⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P ∈ (2nd ‘1P)) → Q)
75, 6sylan 281 . . . . . . . . . . 11 ((1PP ∈ (2nd ‘1P)) → Q)
84, 7mpan 421 . . . . . . . . . 10 ( ∈ (2nd ‘1P) → Q)
9 prop 7303 . . . . . . . . . . . 12 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
10 elprnqu 7310 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
119, 10sylan 281 . . . . . . . . . . 11 ((𝐴P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
12 breq2 3937 . . . . . . . . . . . . 13 (𝑥 = (𝑓 ·Q ) → (𝑓 <Q 𝑥𝑓 <Q (𝑓 ·Q )))
13123ad2ant3 1005 . . . . . . . . . . . 12 ((𝑓QQ𝑥 = (𝑓 ·Q )) → (𝑓 <Q 𝑥𝑓 <Q (𝑓 ·Q )))
14 1pru 7384 . . . . . . . . . . . . . . 15 (2nd ‘1P) = { ∣ 1Q <Q }
1514abeq2i 2251 . . . . . . . . . . . . . 14 ( ∈ (2nd ‘1P) ↔ 1Q <Q )
16 1nq 7194 . . . . . . . . . . . . . . . . 17 1QQ
17 ltmnqg 7229 . . . . . . . . . . . . . . . . 17 ((1QQQ𝑓Q) → (1Q <Q ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q )))
1816, 17mp3an1 1303 . . . . . . . . . . . . . . . 16 ((Q𝑓Q) → (1Q <Q ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q )))
1918ancoms 266 . . . . . . . . . . . . . . 15 ((𝑓QQ) → (1Q <Q ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q )))
20 mulidnq 7217 . . . . . . . . . . . . . . . . 17 (𝑓Q → (𝑓 ·Q 1Q) = 𝑓)
2120breq1d 3943 . . . . . . . . . . . . . . . 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, 23syl5rbb 192 . . . . . . . . . . . . 13 ((𝑓QQ) → (𝑓 <Q (𝑓 ·Q ) ↔ ∈ (2nd ‘1P)))
25243adant3 1002 . . . . . . . . . . . 12 ((𝑓QQ𝑥 = (𝑓 ·Q )) → (𝑓 <Q (𝑓 ·Q ) ↔ ∈ (2nd ‘1P)))
2613, 25bitrd 187 . . . . . . . . . . 11 ((𝑓QQ𝑥 = (𝑓 ·Q )) → (𝑓 <Q 𝑥 ∈ (2nd ‘1P)))
2711, 26syl3an1 1250 . . . . . . . . . 10 (((𝐴P𝑓 ∈ (2nd𝐴)) ∧ Q𝑥 = (𝑓 ·Q )) → (𝑓 <Q 𝑥 ∈ (2nd ‘1P)))
288, 27syl3an2 1251 . . . . . . . . 9 (((𝐴P𝑓 ∈ (2nd𝐴)) ∧ ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q )) → (𝑓 <Q 𝑥 ∈ (2nd ‘1P)))
29283expia 1184 . . . . . . . 8 (((𝐴P𝑓 ∈ (2nd𝐴)) ∧ ∈ (2nd ‘1P)) → (𝑥 = (𝑓 ·Q ) → (𝑓 <Q 𝑥 ∈ (2nd ‘1P))))
3029pm5.32rd 447 . . . . . . 7 (((𝐴P𝑓 ∈ (2nd𝐴)) ∧ ∈ (2nd ‘1P)) → ((𝑓 <Q 𝑥𝑥 = (𝑓 ·Q )) ↔ ( ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ))))
3130rexbidva 2435 . . . . . 6 ((𝐴P𝑓 ∈ (2nd𝐴)) → (∃ ∈ (2nd ‘1P)(𝑓 <Q 𝑥𝑥 = (𝑓 ·Q )) ↔ ∃ ∈ (2nd ‘1P)( ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ))))
32 r19.42v 2589 . . . . . 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 2435 . . 3 (𝐴P → (∃𝑓 ∈ (2nd𝐴)∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ) ↔ ∃𝑓 ∈ (2nd𝐴)(𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))))
36 df-imp 7297 . . . . 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 7204 . . . . 5 ((𝑢Q𝑣Q) → (𝑢 ·Q 𝑣) ∈ Q)
3836, 37genpelvu 7341 . . . 4 ((𝐴P ∧ 1PP) → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (2nd𝐴)∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )))
394, 38mpan2 422 . . 3 (𝐴P → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (2nd𝐴)∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )))
40 prnminu 7317 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (2nd𝐴)) → ∃𝑓 ∈ (2nd𝐴)𝑓 <Q 𝑥)
419, 40sylan 281 . . . . . 6 ((𝐴P𝑥 ∈ (2nd𝐴)) → ∃𝑓 ∈ (2nd𝐴)𝑓 <Q 𝑥)
42 ltrelnq 7193 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
4342brel 4595 . . . . . . . . . . . . 13 (𝑓 <Q 𝑥 → (𝑓Q𝑥Q))
4443ancomd 265 . . . . . . . . . . . 12 (𝑓 <Q 𝑥 → (𝑥Q𝑓Q))
45 ltmnqg 7229 . . . . . . . . . . . . . . . 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 7220 . . . . . . . . . . . . . . . 16 (𝑓Q → (*Q𝑓) ∈ Q)
5049adantl 275 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → (*Q𝑓) ∈ Q)
51 mulcomnqg 7211 . . . . . . . . . . . . . . . 16 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5251adantl 275 . . . . . . . . . . . . . . 15 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5346, 47, 48, 50, 52caovord2d 5944 . . . . . . . . . . . . . 14 ((𝑥Q𝑓Q) → (𝑓 <Q 𝑥 ↔ (𝑓 ·Q (*Q𝑓)) <Q (𝑥 ·Q (*Q𝑓))))
54 recidnq 7221 . . . . . . . . . . . . . . . 16 (𝑓Q → (𝑓 ·Q (*Q𝑓)) = 1Q)
5554breq1d 3943 . . . . . . . . . . . . . . 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 7204 . . . . . . . . . . . . 13 ((𝑥Q ∧ (*Q𝑓) ∈ Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
6149, 60sylan2 284 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
62 breq2 3937 . . . . . . . . . . . . 13 ( = (𝑥 ·Q (*Q𝑓)) → (1Q <Q ↔ 1Q <Q (𝑥 ·Q (*Q𝑓))))
6362, 14elab2g 2832 . . . . . . . . . . . 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 7212 . . . . . . . . . . . . . 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 5956 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑓 ·Q (𝑥 ·Q (*Q𝑓))) = (𝑥 ·Q (𝑓 ·Q (*Q𝑓))))
6954oveq2d 5794 . . . . . . . . . . . . 13 (𝑓Q → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
7069adantl 275 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
71 mulidnq 7217 . . . . . . . . . . . . 13 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
7271adantr 274 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q 1Q) = 𝑥)
7368, 70, 723eqtrrd 2178 . . . . . . . . . . 11 ((𝑥Q𝑓Q) → 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
7444, 73syl 14 . . . . . . . . . 10 (𝑓 <Q 𝑥𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
75 oveq2 5786 . . . . . . . . . . . 12 ( = (𝑥 ·Q (*Q𝑓)) → (𝑓 ·Q ) = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
7675eqeq2d 2152 . . . . . . . . . . 11 ( = (𝑥 ·Q (*Q𝑓)) → (𝑥 = (𝑓 ·Q ) ↔ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓)))))
7776rspcev 2790 . . . . . . . . . 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 2528 . . . . . 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 7313 . . . . . . 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 2550 . . . 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 2138 1 (𝐴P → (2nd ‘(𝐴 ·P 1P)) = (2nd𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 963   = wceq 1332  wcel 1481  wrex 2418  wss 3072  cop 3531   class class class wbr 3933  cfv 5127  (class class class)co 5778  1st c1st 6040  2nd c2nd 6041  Qcnq 7108  1Qc1q 7109   ·Q cmq 7111  *Qcrq 7112   <Q cltq 7113  Pcnp 7119  1Pc1p 7120   ·P cmp 7122
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-eprel 4215  df-id 4219  df-po 4222  df-iso 4223  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-irdg 6271  df-1o 6317  df-oadd 6321  df-omul 6322  df-er 6433  df-ec 6435  df-qs 6439  df-ni 7132  df-pli 7133  df-mi 7134  df-lti 7135  df-plpq 7172  df-mpq 7173  df-enq 7175  df-nqqs 7176  df-plqqs 7177  df-mqqs 7178  df-1nqqs 7179  df-rq 7180  df-ltnqqs 7181  df-inp 7294  df-i1p 7295  df-imp 7297
This theorem is referenced by:  1idpr  7420
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