Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isrnghmd Structured version   Visualization version   GIF version

Theorem isrnghmd 45460
Description: Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.)
Hypotheses
Ref Expression
isrnghmd.b 𝐵 = (Base‘𝑅)
isrnghmd.t · = (.r𝑅)
isrnghmd.u × = (.r𝑆)
isrnghmd.r (𝜑𝑅 ∈ Rng)
isrnghmd.s (𝜑𝑆 ∈ Rng)
isrnghmd.ht ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))
isrnghmd.c 𝐶 = (Base‘𝑆)
isrnghmd.p + = (+g𝑅)
isrnghmd.q = (+g𝑆)
isrnghmd.f (𝜑𝐹:𝐵𝐶)
isrnghmd.hp ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
Assertion
Ref Expression
isrnghmd (𝜑𝐹 ∈ (𝑅 RngHomo 𝑆))
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝑥, + ,𝑦   𝑥, ,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   · (𝑥,𝑦)   × (𝑥,𝑦)

Proof of Theorem isrnghmd
StepHypRef Expression
1 isrnghmd.b . 2 𝐵 = (Base‘𝑅)
2 isrnghmd.t . 2 · = (.r𝑅)
3 isrnghmd.u . 2 × = (.r𝑆)
4 isrnghmd.r . 2 (𝜑𝑅 ∈ Rng)
5 isrnghmd.s . 2 (𝜑𝑆 ∈ Rng)
6 isrnghmd.ht . 2 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))
7 isrnghmd.c . . 3 𝐶 = (Base‘𝑆)
8 isrnghmd.p . . 3 + = (+g𝑅)
9 isrnghmd.q . . 3 = (+g𝑆)
10 rngabl 45435 . . . 4 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
11 ablgrp 19391 . . . 4 (𝑅 ∈ Abel → 𝑅 ∈ Grp)
124, 10, 113syl 18 . . 3 (𝜑𝑅 ∈ Grp)
13 rngabl 45435 . . . 4 (𝑆 ∈ Rng → 𝑆 ∈ Abel)
14 ablgrp 19391 . . . 4 (𝑆 ∈ Abel → 𝑆 ∈ Grp)
155, 13, 143syl 18 . . 3 (𝜑𝑆 ∈ Grp)
16 isrnghmd.f . . 3 (𝜑𝐹:𝐵𝐶)
17 isrnghmd.hp . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
181, 7, 8, 9, 12, 15, 16, 17isghmd 18843 . 2 (𝜑𝐹 ∈ (𝑅 GrpHom 𝑆))
191, 2, 3, 4, 5, 6, 18isrnghm2d 45459 1 (𝜑𝐹 ∈ (𝑅 RngHomo 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wf 6429  cfv 6433  (class class class)co 7275  Basecbs 16912  +gcplusg 16962  .rcmulr 16963  Grpcgrp 18577  Abelcabl 19387  Rngcrng 45432   RngHomo crngh 45443
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-ghm 18832  df-abl 19389  df-rng0 45433  df-rnghomo 45445
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator