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Theorem rngohom1 36430
Description: A ring homomorphism preserves 1. (Contributed by Jeff Madsen, 24-Jun-2011.)
Hypotheses
Ref Expression
rnghom1.1 𝐻 = (2nd β€˜π‘…)
rnghom1.2 π‘ˆ = (GIdβ€˜π»)
rnghom1.3 𝐾 = (2nd β€˜π‘†)
rnghom1.4 𝑉 = (GIdβ€˜πΎ)
Assertion
Ref Expression
rngohom1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (πΉβ€˜π‘ˆ) = 𝑉)

Proof of Theorem rngohom1
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . 5 (1st β€˜π‘…) = (1st β€˜π‘…)
2 rnghom1.1 . . . . 5 𝐻 = (2nd β€˜π‘…)
3 eqid 2737 . . . . 5 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
4 rnghom1.2 . . . . 5 π‘ˆ = (GIdβ€˜π»)
5 eqid 2737 . . . . 5 (1st β€˜π‘†) = (1st β€˜π‘†)
6 rnghom1.3 . . . . 5 𝐾 = (2nd β€˜π‘†)
7 eqid 2737 . . . . 5 ran (1st β€˜π‘†) = ran (1st β€˜π‘†)
8 rnghom1.4 . . . . 5 𝑉 = (GIdβ€˜πΎ)
91, 2, 3, 4, 5, 6, 7, 8isrngohom 36427 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹:ran (1st β€˜π‘…)⟢ran (1st β€˜π‘†) ∧ (πΉβ€˜π‘ˆ) = 𝑉 ∧ βˆ€π‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)((πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯)(1st β€˜π‘†)(πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯𝐻𝑦)) = ((πΉβ€˜π‘₯)𝐾(πΉβ€˜π‘¦))))))
109biimpa 478 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (𝐹:ran (1st β€˜π‘…)⟢ran (1st β€˜π‘†) ∧ (πΉβ€˜π‘ˆ) = 𝑉 ∧ βˆ€π‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)((πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯)(1st β€˜π‘†)(πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯𝐻𝑦)) = ((πΉβ€˜π‘₯)𝐾(πΉβ€˜π‘¦)))))
1110simp2d 1144 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (πΉβ€˜π‘ˆ) = 𝑉)
12113impa 1111 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (πΉβ€˜π‘ˆ) = 𝑉)
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:  rngohomco  36436  rngoisocnv  36443
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