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 42420
 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 42395 . . . 4 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
11 ablgrp 18404 . . . 4 (𝑅 ∈ Abel → 𝑅 ∈ Grp)
124, 10, 113syl 18 . . 3 (𝜑𝑅 ∈ Grp)
13 rngabl 42395 . . . 4 (𝑆 ∈ Rng → 𝑆 ∈ Abel)
14 ablgrp 18404 . . . 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 17876 . 2 (𝜑𝐹 ∈ (𝑅 GrpHom 𝑆))
191, 2, 3, 4, 5, 6, 18isrnghm2d 42419 1 (𝜑𝐹 ∈ (𝑅 RngHomo 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1630   ∈ wcel 2144  ⟶wf 6027  ‘cfv 6031  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  .rcmulr 16149  Grpcgrp 17629  Abelcabl 18400  Rngcrng 42392   RngHomo crngh 42403 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8010  df-ghm 17865  df-abl 18402  df-rng0 42393  df-rnghomo 42405 This theorem is referenced by: (None)
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