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Theorem rngohomadd 36431
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 RngHom 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 2737 . . . . . . 7 (2nd β€˜π‘…) = (2nd β€˜π‘…)
3 rnghomadd.2 . . . . . . 7 𝑋 = ran 𝐺
4 eqid 2737 . . . . . . 7 (GIdβ€˜(2nd β€˜π‘…)) = (GIdβ€˜(2nd β€˜π‘…))
5 rnghomadd.3 . . . . . . 7 𝐽 = (1st β€˜π‘†)
6 eqid 2737 . . . . . . 7 (2nd β€˜π‘†) = (2nd β€˜π‘†)
7 eqid 2737 . . . . . . 7 ran 𝐽 = ran 𝐽
8 eqid 2737 . . . . . . 7 (GIdβ€˜(2nd β€˜π‘†)) = (GIdβ€˜(2nd β€˜π‘†))
91, 2, 3, 4, 5, 6, 7, 8isrngohom 36427 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (𝐹 ∈ (𝑅 RngHom 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 478 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 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 1145 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 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 1111 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 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 484 . . . 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 3127 . . 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 RngHom 𝑆)) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜(π‘₯𝐺𝑦)) = ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)))
16 fvoveq1 7381 . . . 4 (π‘₯ = 𝐴 β†’ (πΉβ€˜(π‘₯𝐺𝑦)) = (πΉβ€˜(𝐴𝐺𝑦)))
17 fveq2 6843 . . . . 5 (π‘₯ = 𝐴 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π΄))
1817oveq1d 7373 . . . 4 (π‘₯ = 𝐴 β†’ ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)) = ((πΉβ€˜π΄)𝐽(πΉβ€˜π‘¦)))
1916, 18eqeq12d 2753 . . 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 7366 . . . . 5 (𝑦 = 𝐡 β†’ (𝐴𝐺𝑦) = (𝐴𝐺𝐡))
2120fveq2d 6847 . . . 4 (𝑦 = 𝐡 β†’ (πΉβ€˜(𝐴𝐺𝑦)) = (πΉβ€˜(𝐴𝐺𝐡)))
22 fveq2 6843 . . . . 5 (𝑦 = 𝐡 β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π΅))
2322oveq2d 7374 . . . 4 (𝑦 = 𝐡 β†’ ((πΉβ€˜π΄)𝐽(πΉβ€˜π‘¦)) = ((πΉβ€˜π΄)𝐽(πΉβ€˜π΅)))
2421, 23eqeq12d 2753 . . 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 3591 . 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 508 1 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  ran crn 5635  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  GIdcgi 29435  RingOpscrngo 36356   RngHom crnghom 36422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8768  df-rngohom 36425
This theorem is referenced by:  rngogrphom  36433  rngohomco  36436  rngoisocnv  36443  keridl  36494
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