ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ecovicom Structured version   GIF version

Theorem ecovicom 6150
Description: Lemma used to transfer a commutative law via an equivalence relation. (Contributed by Jim Kingdon, 15-Sep-2019.)
Hypotheses
Ref Expression
ecovicom.1 𝐶 = ((𝑆 × 𝑆) / )
ecovicom.2 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → ([⟨x, y⟩] + [⟨z, w⟩] ) = [⟨𝐷, 𝐺⟩] )
ecovicom.3 (((z 𝑆 w 𝑆) (x 𝑆 y 𝑆)) → ([⟨z, w⟩] + [⟨x, y⟩] ) = [⟨𝐻, 𝐽⟩] )
ecovicom.4 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → 𝐷 = 𝐻)
ecovicom.5 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → 𝐺 = 𝐽)
Assertion
Ref Expression
ecovicom ((A 𝐶 B 𝐶) → (A + B) = (B + A))
Distinct variable groups:   x,y,z,w,A   z,B,w   x, + ,y,z,w   x, ,y,z,w   x,𝑆,y,z,w   z,𝐶,w
Allowed substitution hints:   B(x,y)   𝐶(x,y)   𝐷(x,y,z,w)   𝐺(x,y,z,w)   𝐻(x,y,z,w)   𝐽(x,y,z,w)

Proof of Theorem ecovicom
StepHypRef Expression
1 ecovicom.1 . 2 𝐶 = ((𝑆 × 𝑆) / )
2 oveq1 5462 . . 3 ([⟨x, y⟩] = A → ([⟨x, y⟩] + [⟨z, w⟩] ) = (A + [⟨z, w⟩] ))
3 oveq2 5463 . . 3 ([⟨x, y⟩] = A → ([⟨z, w⟩] + [⟨x, y⟩] ) = ([⟨z, w⟩] + A))
42, 3eqeq12d 2051 . 2 ([⟨x, y⟩] = A → (([⟨x, y⟩] + [⟨z, w⟩] ) = ([⟨z, w⟩] + [⟨x, y⟩] ) ↔ (A + [⟨z, w⟩] ) = ([⟨z, w⟩] + A)))
5 oveq2 5463 . . 3 ([⟨z, w⟩] = B → (A + [⟨z, w⟩] ) = (A + B))
6 oveq1 5462 . . 3 ([⟨z, w⟩] = B → ([⟨z, w⟩] + A) = (B + A))
75, 6eqeq12d 2051 . 2 ([⟨z, w⟩] = B → ((A + [⟨z, w⟩] ) = ([⟨z, w⟩] + A) ↔ (A + B) = (B + A)))
8 ecovicom.4 . . . 4 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → 𝐷 = 𝐻)
9 ecovicom.5 . . . 4 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → 𝐺 = 𝐽)
10 opeq12 3542 . . . . 5 ((𝐷 = 𝐻 𝐺 = 𝐽) → ⟨𝐷, 𝐺⟩ = ⟨𝐻, 𝐽⟩)
1110eceq1d 6078 . . . 4 ((𝐷 = 𝐻 𝐺 = 𝐽) → [⟨𝐷, 𝐺⟩] = [⟨𝐻, 𝐽⟩] )
128, 9, 11syl2anc 391 . . 3 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → [⟨𝐷, 𝐺⟩] = [⟨𝐻, 𝐽⟩] )
13 ecovicom.2 . . 3 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → ([⟨x, y⟩] + [⟨z, w⟩] ) = [⟨𝐷, 𝐺⟩] )
14 ecovicom.3 . . . 4 (((z 𝑆 w 𝑆) (x 𝑆 y 𝑆)) → ([⟨z, w⟩] + [⟨x, y⟩] ) = [⟨𝐻, 𝐽⟩] )
1514ancoms 255 . . 3 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → ([⟨z, w⟩] + [⟨x, y⟩] ) = [⟨𝐻, 𝐽⟩] )
1612, 13, 153eqtr4d 2079 . 2 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → ([⟨x, y⟩] + [⟨z, w⟩] ) = ([⟨z, w⟩] + [⟨x, y⟩] ))
171, 4, 7, 162ecoptocl 6130 1 ((A 𝐶 B 𝐶) → (A + B) = (B + A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  cop 3370   × cxp 4286  (class class class)co 5455  [cec 6040   / cqs 6041
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-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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  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-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fv 4853  df-ov 5458  df-ec 6044  df-qs 6048
This theorem is referenced by:  addcomnqg  6365  mulcomnqg  6367  addcomsrg  6643  mulcomsrg  6645  axmulcom  6715
  Copyright terms: Public domain W3C validator