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Theorem 1idpru 6565
Description: Lemma for 1idpr 6566. (Contributed by Jim Kingdon, 13-Dec-2019.)
Assertion
Ref Expression
1idpru (A P → (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 2958 . . . . . 6 (2nd ‘1P) ⊆ (2nd ‘1P)
2 rexss 3001 . . . . . 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 2461 . . . . . 6 ( (2nd ‘1P)(f <Q x x = (f ·Q )) ↔ (f <Q x (2nd ‘1P)x = (f ·Q )))
5 1pr 6534 . . . . . . . . . . 11 1P P
6 prop 6457 . . . . . . . . . . . 12 (1P P → ⟨(1st ‘1P), (2nd ‘1P)⟩ P)
7 elprnqu 6464 . . . . . . . . . . . 12 ((⟨(1st ‘1P), (2nd ‘1P)⟩ P (2nd ‘1P)) → Q)
86, 7sylan 267 . . . . . . . . . . 11 ((1P P (2nd ‘1P)) → Q)
95, 8mpan 400 . . . . . . . . . 10 ( (2nd ‘1P) → Q)
10 prop 6457 . . . . . . . . . . . 12 (A P → ⟨(1stA), (2ndA)⟩ P)
11 elprnqu 6464 . . . . . . . . . . . 12 ((⟨(1stA), (2ndA)⟩ P f (2ndA)) → f Q)
1210, 11sylan 267 . . . . . . . . . . 11 ((A P f (2ndA)) → f Q)
13 breq2 3759 . . . . . . . . . . . . 13 (x = (f ·Q ) → (f <Q xf <Q (f ·Q )))
14133ad2ant3 926 . . . . . . . . . . . 12 ((f Q Q x = (f ·Q )) → (f <Q xf <Q (f ·Q )))
15 1pru 6536 . . . . . . . . . . . . . . 15 (2nd ‘1P) = { ∣ 1Q <Q }
1615abeq2i 2145 . . . . . . . . . . . . . 14 ( (2nd ‘1P) ↔ 1Q <Q )
17 1nq 6350 . . . . . . . . . . . . . . . . 17 1Q Q
18 ltmnqg 6385 . . . . . . . . . . . . . . . . 17 ((1Q Q Q f Q) → (1Q <Q ↔ (f ·Q 1Q) <Q (f ·Q )))
1917, 18mp3an1 1218 . . . . . . . . . . . . . . . 16 (( Q f Q) → (1Q <Q ↔ (f ·Q 1Q) <Q (f ·Q )))
2019ancoms 255 . . . . . . . . . . . . . . 15 ((f Q Q) → (1Q <Q ↔ (f ·Q 1Q) <Q (f ·Q )))
21 mulidnq 6373 . . . . . . . . . . . . . . . . 17 (f Q → (f ·Q 1Q) = f)
2221breq1d 3765 . . . . . . . . . . . . . . . 16 (f Q → ((f ·Q 1Q) <Q (f ·Q ) ↔ f <Q (f ·Q )))
2322adantr 261 . . . . . . . . . . . . . . 15 ((f Q Q) → ((f ·Q 1Q) <Q (f ·Q ) ↔ f <Q (f ·Q )))
2420, 23bitrd 177 . . . . . . . . . . . . . 14 ((f Q Q) → (1Q <Q f <Q (f ·Q )))
2516, 24syl5rbb 182 . . . . . . . . . . . . 13 ((f Q Q) → (f <Q (f ·Q ) ↔ (2nd ‘1P)))
26253adant3 923 . . . . . . . . . . . 12 ((f Q Q x = (f ·Q )) → (f <Q (f ·Q ) ↔ (2nd ‘1P)))
2714, 26bitrd 177 . . . . . . . . . . 11 ((f Q Q x = (f ·Q )) → (f <Q x (2nd ‘1P)))
2812, 27syl3an1 1167 . . . . . . . . . 10 (((A P f (2ndA)) Q x = (f ·Q )) → (f <Q x (2nd ‘1P)))
299, 28syl3an2 1168 . . . . . . . . 9 (((A P f (2ndA)) (2nd ‘1P) x = (f ·Q )) → (f <Q x (2nd ‘1P)))
30293expia 1105 . . . . . . . 8 (((A P f (2ndA)) (2nd ‘1P)) → (x = (f ·Q ) → (f <Q x (2nd ‘1P))))
3130pm5.32rd 424 . . . . . . 7 (((A P f (2ndA)) (2nd ‘1P)) → ((f <Q x x = (f ·Q )) ↔ ( (2nd ‘1P) x = (f ·Q ))))
3231rexbidva 2317 . . . . . 6 ((A P f (2ndA)) → ( (2nd ‘1P)(f <Q x x = (f ·Q )) ↔ (2nd ‘1P)( (2nd ‘1P) x = (f ·Q ))))
334, 32syl5rbbr 184 . . . . 5 ((A P f (2ndA)) → ( (2nd ‘1P)( (2nd ‘1P) x = (f ·Q )) ↔ (f <Q x (2nd ‘1P)x = (f ·Q ))))
343, 33syl5bb 181 . . . 4 ((A P f (2ndA)) → ( (2nd ‘1P)x = (f ·Q ) ↔ (f <Q x (2nd ‘1P)x = (f ·Q ))))
3534rexbidva 2317 . . 3 (A P → (f (2ndA) (2nd ‘1P)x = (f ·Q ) ↔ f (2ndA)(f <Q x (2nd ‘1P)x = (f ·Q ))))
36 df-imp 6451 . . . . 5 ·P = (y P, z P ↦ ⟨{w Qu Q v Q (u (1sty) v (1stz) w = (u ·Q v))}, {w Qu Q v Q (u (2ndy) v (2ndz) w = (u ·Q v))}⟩)
37 mulclnq 6360 . . . . 5 ((u Q v Q) → (u ·Q v) Q)
3836, 37genpelvu 6495 . . . 4 ((A P 1P P) → (x (2nd ‘(A ·P 1P)) ↔ f (2ndA) (2nd ‘1P)x = (f ·Q )))
395, 38mpan2 401 . . 3 (A P → (x (2nd ‘(A ·P 1P)) ↔ f (2ndA) (2nd ‘1P)x = (f ·Q )))
40 prnminu 6471 . . . . . . 7 ((⟨(1stA), (2ndA)⟩ P x (2ndA)) → f (2ndA)f <Q x)
4110, 40sylan 267 . . . . . 6 ((A P x (2ndA)) → f (2ndA)f <Q x)
42 ltrelnq 6349 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
4342brel 4335 . . . . . . . . . . . . 13 (f <Q x → (f Q x Q))
4443ancomd 254 . . . . . . . . . . . 12 (f <Q x → (x Q f Q))
45 ltmnqg 6385 . . . . . . . . . . . . . . . 16 ((y Q z Q w Q) → (y <Q z ↔ (w ·Q y) <Q (w ·Q z)))
4645adantl 262 . . . . . . . . . . . . . . 15 (((x Q f Q) (y Q z Q w Q)) → (y <Q z ↔ (w ·Q y) <Q (w ·Q z)))
47 simpr 103 . . . . . . . . . . . . . . 15 ((x Q f Q) → f Q)
48 simpl 102 . . . . . . . . . . . . . . 15 ((x Q f Q) → x Q)
49 recclnq 6376 . . . . . . . . . . . . . . . 16 (f Q → (*Qf) Q)
5049adantl 262 . . . . . . . . . . . . . . 15 ((x Q f Q) → (*Qf) Q)
51 mulcomnqg 6367 . . . . . . . . . . . . . . . 16 ((y Q z Q) → (y ·Q z) = (z ·Q y))
5251adantl 262 . . . . . . . . . . . . . . 15 (((x Q f Q) (y Q z Q)) → (y ·Q z) = (z ·Q y))
5346, 47, 48, 50, 52caovord2d 5612 . . . . . . . . . . . . . 14 ((x Q f Q) → (f <Q x ↔ (f ·Q (*Qf)) <Q (x ·Q (*Qf))))
54 recidnq 6377 . . . . . . . . . . . . . . . 16 (f Q → (f ·Q (*Qf)) = 1Q)
5554breq1d 3765 . . . . . . . . . . . . . . 15 (f Q → ((f ·Q (*Qf)) <Q (x ·Q (*Qf)) ↔ 1Q <Q (x ·Q (*Qf))))
5655adantl 262 . . . . . . . . . . . . . 14 ((x Q f Q) → ((f ·Q (*Qf)) <Q (x ·Q (*Qf)) ↔ 1Q <Q (x ·Q (*Qf))))
5753, 56bitrd 177 . . . . . . . . . . . . 13 ((x Q f Q) → (f <Q x ↔ 1Q <Q (x ·Q (*Qf))))
5857biimpd 132 . . . . . . . . . . . 12 ((x Q f Q) → (f <Q x → 1Q <Q (x ·Q (*Qf))))
5944, 58mpcom 32 . . . . . . . . . . 11 (f <Q x → 1Q <Q (x ·Q (*Qf)))
60 mulclnq 6360 . . . . . . . . . . . . 13 ((x Q (*Qf) Q) → (x ·Q (*Qf)) Q)
6149, 60sylan2 270 . . . . . . . . . . . 12 ((x Q f Q) → (x ·Q (*Qf)) Q)
62 breq2 3759 . . . . . . . . . . . . 13 ( = (x ·Q (*Qf)) → (1Q <Q ↔ 1Q <Q (x ·Q (*Qf))))
6362, 15elab2g 2683 . . . . . . . . . . . 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 156 . . . . . . . . . 10 (f <Q x → (x ·Q (*Qf)) (2nd ‘1P))
66 mulassnqg 6368 . . . . . . . . . . . . . 14 ((y Q z Q w Q) → ((y ·Q z) ·Q w) = (y ·Q (z ·Q w)))
6766adantl 262 . . . . . . . . . . . . 13 (((x Q f Q) (y Q z Q w Q)) → ((y ·Q z) ·Q w) = (y ·Q (z ·Q w)))
6847, 48, 50, 52, 67caov12d 5624 . . . . . . . . . . . 12 ((x Q f Q) → (f ·Q (x ·Q (*Qf))) = (x ·Q (f ·Q (*Qf))))
6954oveq2d 5471 . . . . . . . . . . . . 13 (f Q → (x ·Q (f ·Q (*Qf))) = (x ·Q 1Q))
7069adantl 262 . . . . . . . . . . . 12 ((x Q f Q) → (x ·Q (f ·Q (*Qf))) = (x ·Q 1Q))
71 mulidnq 6373 . . . . . . . . . . . . 13 (x Q → (x ·Q 1Q) = x)
7271adantr 261 . . . . . . . . . . . 12 ((x Q f Q) → (x ·Q 1Q) = x)
7368, 70, 723eqtrrd 2074 . . . . . . . . . . 11 ((x Q f Q) → x = (f ·Q (x ·Q (*Qf))))
7444, 73syl 14 . . . . . . . . . 10 (f <Q xx = (f ·Q (x ·Q (*Qf))))
75 oveq2 5463 . . . . . . . . . . . 12 ( = (x ·Q (*Qf)) → (f ·Q ) = (f ·Q (x ·Q (*Qf))))
7675eqeq2d 2048 . . . . . . . . . . 11 ( = (x ·Q (*Qf)) → (x = (f ·Q ) ↔ x = (f ·Q (x ·Q (*Qf)))))
7776rspcev 2650 . . . . . . . . . 10 (((x ·Q (*Qf)) (2nd ‘1P) x = (f ·Q (x ·Q (*Qf)))) → (2nd ‘1P)x = (f ·Q ))
7865, 74, 77syl2anc 391 . . . . . . . . 9 (f <Q x (2nd ‘1P)x = (f ·Q ))
7978a1i 9 . . . . . . . 8 (f (2ndA) → (f <Q x (2nd ‘1P)x = (f ·Q )))
8079ancld 308 . . . . . . 7 (f (2ndA) → (f <Q x → (f <Q x (2nd ‘1P)x = (f ·Q ))))
8180reximia 2408 . . . . . 6 (f (2ndA)f <Q xf (2ndA)(f <Q x (2nd ‘1P)x = (f ·Q )))
8241, 81syl 14 . . . . 5 ((A P x (2ndA)) → f (2ndA)(f <Q x (2nd ‘1P)x = (f ·Q )))
8382ex 108 . . . 4 (A P → (x (2ndA) → f (2ndA)(f <Q x (2nd ‘1P)x = (f ·Q ))))
84 prcunqu 6467 . . . . . . 7 ((⟨(1stA), (2ndA)⟩ P f (2ndA)) → (f <Q xx (2ndA)))
8510, 84sylan 267 . . . . . 6 ((A P f (2ndA)) → (f <Q xx (2ndA)))
8685adantrd 264 . . . . 5 ((A P f (2ndA)) → ((f <Q x (2nd ‘1P)x = (f ·Q )) → x (2ndA)))
8786rexlimdva 2427 . . . 4 (A P → (f (2ndA)(f <Q x (2nd ‘1P)x = (f ·Q )) → x (2ndA)))
8883, 87impbid 120 . . 3 (A P → (x (2ndA) ↔ f (2ndA)(f <Q x (2nd ‘1P)x = (f ·Q ))))
8935, 39, 883bitr4d 209 . 2 (A P → (x (2nd ‘(A ·P 1P)) ↔ x (2ndA)))
9089eqrdv 2035 1 (A P → (2nd ‘(A ·P 1P)) = (2ndA))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  wrex 2301  wss 2911  cop 3370   class class class wbr 3755  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  1Qc1q 6265   ·Q cmq 6267  *Qcrq 6268   <Q cltq 6269  Pcnp 6275  1Pc1p 6276   ·P cmp 6278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6448  df-i1p 6449  df-imp 6451
This theorem is referenced by:  1idpr  6566
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