Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngohomadd Structured version   Visualization version   GIF version

Theorem rngohomadd 37499
Description: Ring homomorphisms preserve addition. (Contributed by Jeff Madsen, 3-Jan-2011.)
Hypotheses
Ref Expression
rnghomadd.1 𝐺 = (1st β€˜π‘…)
rnghomadd.2 𝑋 = ran 𝐺
rnghomadd.3 𝐽 = (1st β€˜π‘†)
Assertion
Ref Expression
rngohomadd (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜(𝐴𝐺𝐡)) = ((πΉβ€˜π΄)𝐽(πΉβ€˜π΅)))

Proof of Theorem rngohomadd
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnghomadd.1 . . . . . . 7 𝐺 = (1st β€˜π‘…)
2 eqid 2725 . . . . . . 7 (2nd β€˜π‘…) = (2nd β€˜π‘…)
3 rnghomadd.2 . . . . . . 7 𝑋 = ran 𝐺
4 eqid 2725 . . . . . . 7 (GIdβ€˜(2nd β€˜π‘…)) = (GIdβ€˜(2nd β€˜π‘…))
5 rnghomadd.3 . . . . . . 7 𝐽 = (1st β€˜π‘†)
6 eqid 2725 . . . . . . 7 (2nd β€˜π‘†) = (2nd β€˜π‘†)
7 eqid 2725 . . . . . . 7 ran 𝐽 = ran 𝐽
8 eqid 2725 . . . . . . 7 (GIdβ€˜(2nd β€˜π‘†)) = (GIdβ€˜(2nd β€˜π‘†))
91, 2, 3, 4, 5, 6, 7, 8isrngohom 37495 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ↔ (𝐹:π‘‹βŸΆran 𝐽 ∧ (πΉβ€˜(GIdβ€˜(2nd β€˜π‘…))) = (GIdβ€˜(2nd β€˜π‘†)) ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜(π‘₯𝐺𝑦)) = ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯)(2nd β€˜π‘†)(πΉβ€˜π‘¦))))))
109biimpa 475 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) β†’ (𝐹:π‘‹βŸΆran 𝐽 ∧ (πΉβ€˜(GIdβ€˜(2nd β€˜π‘…))) = (GIdβ€˜(2nd β€˜π‘†)) ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜(π‘₯𝐺𝑦)) = ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯)(2nd β€˜π‘†)(πΉβ€˜π‘¦)))))
1110simp3d 1141 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜(π‘₯𝐺𝑦)) = ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯)(2nd β€˜π‘†)(πΉβ€˜π‘¦))))
12113impa 1107 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜(π‘₯𝐺𝑦)) = ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯)(2nd β€˜π‘†)(πΉβ€˜π‘¦))))
13 simpl 481 . . . 4 (((πΉβ€˜(π‘₯𝐺𝑦)) = ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯)(2nd β€˜π‘†)(πΉβ€˜π‘¦))) β†’ (πΉβ€˜(π‘₯𝐺𝑦)) = ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)))
14132ralimi 3113 . . 3 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜(π‘₯𝐺𝑦)) = ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯)(2nd β€˜π‘†)(πΉβ€˜π‘¦))) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜(π‘₯𝐺𝑦)) = ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)))
1512, 14syl 17 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜(π‘₯𝐺𝑦)) = ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)))
16 fvoveq1 7439 . . . 4 (π‘₯ = 𝐴 β†’ (πΉβ€˜(π‘₯𝐺𝑦)) = (πΉβ€˜(𝐴𝐺𝑦)))
17 fveq2 6892 . . . . 5 (π‘₯ = 𝐴 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π΄))
1817oveq1d 7431 . . . 4 (π‘₯ = 𝐴 β†’ ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)) = ((πΉβ€˜π΄)𝐽(πΉβ€˜π‘¦)))
1916, 18eqeq12d 2741 . . 3 (π‘₯ = 𝐴 β†’ ((πΉβ€˜(π‘₯𝐺𝑦)) = ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)) ↔ (πΉβ€˜(𝐴𝐺𝑦)) = ((πΉβ€˜π΄)𝐽(πΉβ€˜π‘¦))))
20 oveq2 7424 . . . . 5 (𝑦 = 𝐡 β†’ (𝐴𝐺𝑦) = (𝐴𝐺𝐡))
2120fveq2d 6896 . . . 4 (𝑦 = 𝐡 β†’ (πΉβ€˜(𝐴𝐺𝑦)) = (πΉβ€˜(𝐴𝐺𝐡)))
22 fveq2 6892 . . . . 5 (𝑦 = 𝐡 β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π΅))
2322oveq2d 7432 . . . 4 (𝑦 = 𝐡 β†’ ((πΉβ€˜π΄)𝐽(πΉβ€˜π‘¦)) = ((πΉβ€˜π΄)𝐽(πΉβ€˜π΅)))
2421, 23eqeq12d 2741 . . 3 (𝑦 = 𝐡 β†’ ((πΉβ€˜(𝐴𝐺𝑦)) = ((πΉβ€˜π΄)𝐽(πΉβ€˜π‘¦)) ↔ (πΉβ€˜(𝐴𝐺𝐡)) = ((πΉβ€˜π΄)𝐽(πΉβ€˜π΅))))
2519, 24rspc2v 3612 . 2 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜(π‘₯𝐺𝑦)) = ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)) β†’ (πΉβ€˜(𝐴𝐺𝐡)) = ((πΉβ€˜π΄)𝐽(πΉβ€˜π΅))))
2615, 25mpan9 505 1 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜(𝐴𝐺𝐡)) = ((πΉβ€˜π΄)𝐽(πΉβ€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  ran crn 5673  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7416  1st c1st 7989  2nd c2nd 7990  GIdcgi 30344  RingOpscrngo 37424   RingOpsHom crngohom 37490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-map 8845  df-rngohom 37493
This theorem is referenced by:  rngogrphom  37501  rngohomco  37504  rngoisocnv  37511  keridl  37562
  Copyright terms: Public domain W3C validator