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Theorem 1idpru 6746
Description: Lemma for 1idpr 6747. (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 2991 . . . . . 6 (2nd ‘1P) ⊆ (2nd ‘1P)
2 rexss 3034 . . . . . 6 ((2nd ‘1P) ⊆ (2nd ‘1P) → (∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ) ↔ ∃ ∈ (2nd ‘1P)( ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ))))
31, 2ax-mp 7 . . . . 5 (∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ) ↔ ∃ ∈ (2nd ‘1P)( ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q )))
4 r19.42v 2484 . . . . . 6 (∃ ∈ (2nd ‘1P)(𝑓 <Q 𝑥𝑥 = (𝑓 ·Q )) ↔ (𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )))
5 1pr 6709 . . . . . . . . . . 11 1PP
6 prop 6630 . . . . . . . . . . . 12 (1PP → ⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P)
7 elprnqu 6637 . . . . . . . . . . . 12 ((⟨(1st ‘1P), (2nd ‘1P)⟩ ∈ P ∈ (2nd ‘1P)) → Q)
86, 7sylan 271 . . . . . . . . . . 11 ((1PP ∈ (2nd ‘1P)) → Q)
95, 8mpan 408 . . . . . . . . . 10 ( ∈ (2nd ‘1P) → Q)
10 prop 6630 . . . . . . . . . . . 12 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
11 elprnqu 6637 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
1210, 11sylan 271 . . . . . . . . . . 11 ((𝐴P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
13 breq2 3795 . . . . . . . . . . . . 13 (𝑥 = (𝑓 ·Q ) → (𝑓 <Q 𝑥𝑓 <Q (𝑓 ·Q )))
14133ad2ant3 938 . . . . . . . . . . . 12 ((𝑓QQ𝑥 = (𝑓 ·Q )) → (𝑓 <Q 𝑥𝑓 <Q (𝑓 ·Q )))
15 1pru 6711 . . . . . . . . . . . . . . 15 (2nd ‘1P) = { ∣ 1Q <Q }
1615abeq2i 2164 . . . . . . . . . . . . . 14 ( ∈ (2nd ‘1P) ↔ 1Q <Q )
17 1nq 6521 . . . . . . . . . . . . . . . . 17 1QQ
18 ltmnqg 6556 . . . . . . . . . . . . . . . . 17 ((1QQQ𝑓Q) → (1Q <Q ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q )))
1917, 18mp3an1 1230 . . . . . . . . . . . . . . . 16 ((Q𝑓Q) → (1Q <Q ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q )))
2019ancoms 259 . . . . . . . . . . . . . . 15 ((𝑓QQ) → (1Q <Q ↔ (𝑓 ·Q 1Q) <Q (𝑓 ·Q )))
21 mulidnq 6544 . . . . . . . . . . . . . . . . 17 (𝑓Q → (𝑓 ·Q 1Q) = 𝑓)
2221breq1d 3801 . . . . . . . . . . . . . . . 16 (𝑓Q → ((𝑓 ·Q 1Q) <Q (𝑓 ·Q ) ↔ 𝑓 <Q (𝑓 ·Q )))
2322adantr 265 . . . . . . . . . . . . . . 15 ((𝑓QQ) → ((𝑓 ·Q 1Q) <Q (𝑓 ·Q ) ↔ 𝑓 <Q (𝑓 ·Q )))
2420, 23bitrd 181 . . . . . . . . . . . . . 14 ((𝑓QQ) → (1Q <Q 𝑓 <Q (𝑓 ·Q )))
2516, 24syl5rbb 186 . . . . . . . . . . . . 13 ((𝑓QQ) → (𝑓 <Q (𝑓 ·Q ) ↔ ∈ (2nd ‘1P)))
26253adant3 935 . . . . . . . . . . . 12 ((𝑓QQ𝑥 = (𝑓 ·Q )) → (𝑓 <Q (𝑓 ·Q ) ↔ ∈ (2nd ‘1P)))
2714, 26bitrd 181 . . . . . . . . . . 11 ((𝑓QQ𝑥 = (𝑓 ·Q )) → (𝑓 <Q 𝑥 ∈ (2nd ‘1P)))
2812, 27syl3an1 1179 . . . . . . . . . 10 (((𝐴P𝑓 ∈ (2nd𝐴)) ∧ Q𝑥 = (𝑓 ·Q )) → (𝑓 <Q 𝑥 ∈ (2nd ‘1P)))
299, 28syl3an2 1180 . . . . . . . . 9 (((𝐴P𝑓 ∈ (2nd𝐴)) ∧ ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q )) → (𝑓 <Q 𝑥 ∈ (2nd ‘1P)))
30293expia 1117 . . . . . . . 8 (((𝐴P𝑓 ∈ (2nd𝐴)) ∧ ∈ (2nd ‘1P)) → (𝑥 = (𝑓 ·Q ) → (𝑓 <Q 𝑥 ∈ (2nd ‘1P))))
3130pm5.32rd 432 . . . . . . 7 (((𝐴P𝑓 ∈ (2nd𝐴)) ∧ ∈ (2nd ‘1P)) → ((𝑓 <Q 𝑥𝑥 = (𝑓 ·Q )) ↔ ( ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ))))
3231rexbidva 2340 . . . . . 6 ((𝐴P𝑓 ∈ (2nd𝐴)) → (∃ ∈ (2nd ‘1P)(𝑓 <Q 𝑥𝑥 = (𝑓 ·Q )) ↔ ∃ ∈ (2nd ‘1P)( ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q ))))
334, 32syl5rbbr 188 . . . . 5 ((𝐴P𝑓 ∈ (2nd𝐴)) → (∃ ∈ (2nd ‘1P)( ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q )) ↔ (𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))))
343, 33syl5bb 185 . . . 4 ((𝐴P𝑓 ∈ (2nd𝐴)) → (∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ) ↔ (𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))))
3534rexbidva 2340 . . 3 (𝐴P → (∃𝑓 ∈ (2nd𝐴)∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ) ↔ ∃𝑓 ∈ (2nd𝐴)(𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))))
36 df-imp 6624 . . . . 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 6531 . . . . 5 ((𝑢Q𝑣Q) → (𝑢 ·Q 𝑣) ∈ Q)
3836, 37genpelvu 6668 . . . 4 ((𝐴P ∧ 1PP) → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (2nd𝐴)∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )))
395, 38mpan2 409 . . 3 (𝐴P → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ ∃𝑓 ∈ (2nd𝐴)∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )))
40 prnminu 6644 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (2nd𝐴)) → ∃𝑓 ∈ (2nd𝐴)𝑓 <Q 𝑥)
4110, 40sylan 271 . . . . . 6 ((𝐴P𝑥 ∈ (2nd𝐴)) → ∃𝑓 ∈ (2nd𝐴)𝑓 <Q 𝑥)
42 ltrelnq 6520 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
4342brel 4419 . . . . . . . . . . . . 13 (𝑓 <Q 𝑥 → (𝑓Q𝑥Q))
4443ancomd 258 . . . . . . . . . . . 12 (𝑓 <Q 𝑥 → (𝑥Q𝑓Q))
45 ltmnqg 6556 . . . . . . . . . . . . . . . 16 ((𝑦Q𝑧Q𝑤Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
4645adantl 266 . . . . . . . . . . . . . . 15 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q𝑤Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
47 simpr 107 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → 𝑓Q)
48 simpl 106 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → 𝑥Q)
49 recclnq 6547 . . . . . . . . . . . . . . . 16 (𝑓Q → (*Q𝑓) ∈ Q)
5049adantl 266 . . . . . . . . . . . . . . 15 ((𝑥Q𝑓Q) → (*Q𝑓) ∈ Q)
51 mulcomnqg 6538 . . . . . . . . . . . . . . . 16 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5251adantl 266 . . . . . . . . . . . . . . 15 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
5346, 47, 48, 50, 52caovord2d 5697 . . . . . . . . . . . . . 14 ((𝑥Q𝑓Q) → (𝑓 <Q 𝑥 ↔ (𝑓 ·Q (*Q𝑓)) <Q (𝑥 ·Q (*Q𝑓))))
54 recidnq 6548 . . . . . . . . . . . . . . . 16 (𝑓Q → (𝑓 ·Q (*Q𝑓)) = 1Q)
5554breq1d 3801 . . . . . . . . . . . . . . 15 (𝑓Q → ((𝑓 ·Q (*Q𝑓)) <Q (𝑥 ·Q (*Q𝑓)) ↔ 1Q <Q (𝑥 ·Q (*Q𝑓))))
5655adantl 266 . . . . . . . . . . . . . 14 ((𝑥Q𝑓Q) → ((𝑓 ·Q (*Q𝑓)) <Q (𝑥 ·Q (*Q𝑓)) ↔ 1Q <Q (𝑥 ·Q (*Q𝑓))))
5753, 56bitrd 181 . . . . . . . . . . . . 13 ((𝑥Q𝑓Q) → (𝑓 <Q 𝑥 ↔ 1Q <Q (𝑥 ·Q (*Q𝑓))))
5857biimpd 136 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑓 <Q 𝑥 → 1Q <Q (𝑥 ·Q (*Q𝑓))))
5944, 58mpcom 36 . . . . . . . . . . 11 (𝑓 <Q 𝑥 → 1Q <Q (𝑥 ·Q (*Q𝑓)))
60 mulclnq 6531 . . . . . . . . . . . . 13 ((𝑥Q ∧ (*Q𝑓) ∈ Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
6149, 60sylan2 274 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q (*Q𝑓)) ∈ Q)
62 breq2 3795 . . . . . . . . . . . . 13 ( = (𝑥 ·Q (*Q𝑓)) → (1Q <Q ↔ 1Q <Q (𝑥 ·Q (*Q𝑓))))
6362, 15elab2g 2711 . . . . . . . . . . . 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 160 . . . . . . . . . 10 (𝑓 <Q 𝑥 → (𝑥 ·Q (*Q𝑓)) ∈ (2nd ‘1P))
66 mulassnqg 6539 . . . . . . . . . . . . . 14 ((𝑦Q𝑧Q𝑤Q) → ((𝑦 ·Q 𝑧) ·Q 𝑤) = (𝑦 ·Q (𝑧 ·Q 𝑤)))
6766adantl 266 . . . . . . . . . . . . 13 (((𝑥Q𝑓Q) ∧ (𝑦Q𝑧Q𝑤Q)) → ((𝑦 ·Q 𝑧) ·Q 𝑤) = (𝑦 ·Q (𝑧 ·Q 𝑤)))
6847, 48, 50, 52, 67caov12d 5709 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑓 ·Q (𝑥 ·Q (*Q𝑓))) = (𝑥 ·Q (𝑓 ·Q (*Q𝑓))))
6954oveq2d 5555 . . . . . . . . . . . . 13 (𝑓Q → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
7069adantl 266 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
71 mulidnq 6544 . . . . . . . . . . . . 13 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
7271adantr 265 . . . . . . . . . . . 12 ((𝑥Q𝑓Q) → (𝑥 ·Q 1Q) = 𝑥)
7368, 70, 723eqtrrd 2093 . . . . . . . . . . 11 ((𝑥Q𝑓Q) → 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
7444, 73syl 14 . . . . . . . . . 10 (𝑓 <Q 𝑥𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
75 oveq2 5547 . . . . . . . . . . . 12 ( = (𝑥 ·Q (*Q𝑓)) → (𝑓 ·Q ) = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
7675eqeq2d 2067 . . . . . . . . . . 11 ( = (𝑥 ·Q (*Q𝑓)) → (𝑥 = (𝑓 ·Q ) ↔ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓)))))
7776rspcev 2673 . . . . . . . . . 10 (((𝑥 ·Q (*Q𝑓)) ∈ (2nd ‘1P) ∧ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓)))) → ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))
7865, 74, 77syl2anc 397 . . . . . . . . 9 (𝑓 <Q 𝑥 → ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))
7978a1i 9 . . . . . . . 8 (𝑓 ∈ (2nd𝐴) → (𝑓 <Q 𝑥 → ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )))
8079ancld 312 . . . . . . 7 (𝑓 ∈ (2nd𝐴) → (𝑓 <Q 𝑥 → (𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))))
8180reximia 2431 . . . . . 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 112 . . . 4 (𝐴P → (𝑥 ∈ (2nd𝐴) → ∃𝑓 ∈ (2nd𝐴)(𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))))
84 prcunqu 6640 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (2nd𝐴)) → (𝑓 <Q 𝑥𝑥 ∈ (2nd𝐴)))
8510, 84sylan 271 . . . . . 6 ((𝐴P𝑓 ∈ (2nd𝐴)) → (𝑓 <Q 𝑥𝑥 ∈ (2nd𝐴)))
8685adantrd 268 . . . . 5 ((𝐴P𝑓 ∈ (2nd𝐴)) → ((𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )) → 𝑥 ∈ (2nd𝐴)))
8786rexlimdva 2450 . . . 4 (𝐴P → (∃𝑓 ∈ (2nd𝐴)(𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q )) → 𝑥 ∈ (2nd𝐴)))
8883, 87impbid 124 . . 3 (𝐴P → (𝑥 ∈ (2nd𝐴) ↔ ∃𝑓 ∈ (2nd𝐴)(𝑓 <Q 𝑥 ∧ ∃ ∈ (2nd ‘1P)𝑥 = (𝑓 ·Q ))))
8935, 39, 883bitr4d 213 . 2 (𝐴P → (𝑥 ∈ (2nd ‘(𝐴 ·P 1P)) ↔ 𝑥 ∈ (2nd𝐴)))
9089eqrdv 2054 1 (𝐴P → (2nd ‘(𝐴 ·P 1P)) = (2nd𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  wrex 2324  wss 2944  cop 3405   class class class wbr 3791  cfv 4929  (class class class)co 5539  1st c1st 5792  2nd c2nd 5793  Qcnq 6435  1Qc1q 6436   ·Q cmq 6438  *Qcrq 6439   <Q cltq 6440  Pcnp 6446  1Pc1p 6447   ·P cmp 6449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-inp 6621  df-i1p 6622  df-imp 6624
This theorem is referenced by:  1idpr  6747
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