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Theorem genpassg 6509
Description: Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.)
Hypotheses
Ref Expression
genpelvl.1 𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)
genpelvl.2 ((y Q z Q) → (y𝐺z) Q)
genpassg.4 dom 𝐹 = (P × P)
genpassg.5 ((f P g P) → (f𝐹g) P)
genpassg.6 ((f Q g Q Q) → ((f𝐺g)𝐺) = (f𝐺(g𝐺)))
Assertion
Ref Expression
genpassg ((A P B P 𝐶 P) → ((A𝐹B)𝐹𝐶) = (A𝐹(B𝐹𝐶)))
Distinct variable groups:   x,y,z,f,g,,w,v,A   x,B,y,z,f,g,,w,v   x,𝐺,y,z,f,g,,w,v   f,𝐹,g   𝐶,f,g,,v,w,x,y,z   ,𝐹,v,w,x,y,z

Proof of Theorem genpassg
StepHypRef Expression
1 genpelvl.1 . . 3 𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)
2 genpelvl.2 . . 3 ((y Q z Q) → (y𝐺z) Q)
3 genpassg.4 . . 3 dom 𝐹 = (P × P)
4 genpassg.5 . . 3 ((f P g P) → (f𝐹g) P)
5 genpassg.6 . . 3 ((f Q g Q Q) → ((f𝐺g)𝐺) = (f𝐺(g𝐺)))
61, 2, 3, 4, 5genpassl 6507 . 2 ((A P B P 𝐶 P) → (1st ‘((A𝐹B)𝐹𝐶)) = (1st ‘(A𝐹(B𝐹𝐶))))
71, 2, 3, 4, 5genpassu 6508 . 2 ((A P B P 𝐶 P) → (2nd ‘((A𝐹B)𝐹𝐶)) = (2nd ‘(A𝐹(B𝐹𝐶))))
84caovcl 5597 . . . . 5 ((A P B P) → (A𝐹B) P)
94caovcl 5597 . . . . 5 (((A𝐹B) P 𝐶 P) → ((A𝐹B)𝐹𝐶) P)
108, 9sylan 267 . . . 4 (((A P B P) 𝐶 P) → ((A𝐹B)𝐹𝐶) P)
11103impa 1098 . . 3 ((A P B P 𝐶 P) → ((A𝐹B)𝐹𝐶) P)
124caovcl 5597 . . . . 5 ((B P 𝐶 P) → (B𝐹𝐶) P)
134caovcl 5597 . . . . 5 ((A P (B𝐹𝐶) P) → (A𝐹(B𝐹𝐶)) P)
1412, 13sylan2 270 . . . 4 ((A P (B P 𝐶 P)) → (A𝐹(B𝐹𝐶)) P)
15143impb 1099 . . 3 ((A P B P 𝐶 P) → (A𝐹(B𝐹𝐶)) P)
16 preqlu 6454 . . 3 ((((A𝐹B)𝐹𝐶) P (A𝐹(B𝐹𝐶)) P) → (((A𝐹B)𝐹𝐶) = (A𝐹(B𝐹𝐶)) ↔ ((1st ‘((A𝐹B)𝐹𝐶)) = (1st ‘(A𝐹(B𝐹𝐶))) (2nd ‘((A𝐹B)𝐹𝐶)) = (2nd ‘(A𝐹(B𝐹𝐶))))))
1711, 15, 16syl2anc 391 . 2 ((A P B P 𝐶 P) → (((A𝐹B)𝐹𝐶) = (A𝐹(B𝐹𝐶)) ↔ ((1st ‘((A𝐹B)𝐹𝐶)) = (1st ‘(A𝐹(B𝐹𝐶))) (2nd ‘((A𝐹B)𝐹𝐶)) = (2nd ‘(A𝐹(B𝐹𝐶))))))
186, 7, 17mpbir2and 850 1 ((A P B P 𝐶 P) → ((A𝐹B)𝐹𝐶) = (A𝐹(B𝐹𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  wrex 2301  {crab 2304  cop 3370   × cxp 4286  dom cdm 4288  cfv 4845  (class class class)co 5455  cmpt2 5457  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  Pcnp 6275
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-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  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-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-id 4021  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-qs 6048  df-ni 6288  df-nqqs 6332  df-inp 6448
This theorem is referenced by:  addassprg  6553  mulassprg  6555
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