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Theorem rngohom1 37474
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 RingOpsHom 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 2728 . . . . 5 (1st β€˜π‘…) = (1st β€˜π‘…)
2 rnghom1.1 . . . . 5 𝐻 = (2nd β€˜π‘…)
3 eqid 2728 . . . . 5 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
4 rnghom1.2 . . . . 5 π‘ˆ = (GIdβ€˜π»)
5 eqid 2728 . . . . 5 (1st β€˜π‘†) = (1st β€˜π‘†)
6 rnghom1.3 . . . . 5 𝐾 = (2nd β€˜π‘†)
7 eqid 2728 . . . . 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 37471 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (𝐹 ∈ (𝑅 RingOpsHom 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 475 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 1140 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) β†’ (πΉβ€˜π‘ˆ) = 𝑉)
12113impa 1107 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) β†’ (πΉβ€˜π‘ˆ) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  ran crn 5683  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426  1st c1st 7997  2nd c2nd 7998  GIdcgi 30320  RingOpscrngo 37400   RingOpsHom crngohom 37466
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 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-map 8853  df-rngohom 37469
This theorem is referenced by:  rngohomco  37480  rngoisocnv  37487
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