MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ismhm0 Structured version   Visualization version   GIF version

Theorem ismhm0 18718
Description: Property of a monoid homomorphism, expressed by a magma homomorphism. (Contributed by AV, 17-Apr-2020.)
Hypotheses
Ref Expression
ismhm0.b 𝐵 = (Base‘𝑆)
ismhm0.c 𝐶 = (Base‘𝑇)
ismhm0.p + = (+g𝑆)
ismhm0.q = (+g𝑇)
ismhm0.z 0 = (0g𝑆)
ismhm0.y 𝑌 = (0g𝑇)
Assertion
Ref Expression
ismhm0 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹0 ) = 𝑌)))

Proof of Theorem ismhm0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismhm0.b . . 3 𝐵 = (Base‘𝑆)
2 ismhm0.c . . 3 𝐶 = (Base‘𝑇)
3 ismhm0.p . . 3 + = (+g𝑆)
4 ismhm0.q . . 3 = (+g𝑇)
5 ismhm0.z . . 3 0 = (0g𝑆)
6 ismhm0.y . . 3 𝑌 = (0g𝑇)
71, 2, 3, 4, 5, 6ismhm 18713 . 2 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
8 df-3an 1086 . . . 4 ((𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌) ↔ ((𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ∧ (𝐹0 ) = 𝑌))
9 mndmgm 18672 . . . . . . . 8 (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
10 mndmgm 18672 . . . . . . . 8 (𝑇 ∈ Mnd → 𝑇 ∈ Mgm)
119, 10anim12i 612 . . . . . . 7 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
1211biantrurd 532 . . . . . 6 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))))
131, 2, 3, 4ismgmhm 18627 . . . . . 6 (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
1412, 13bitr4di 289 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ↔ 𝐹 ∈ (𝑆 MgmHom 𝑇)))
1514anbi1d 629 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (((𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ∧ (𝐹0 ) = 𝑌) ↔ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹0 ) = 𝑌)))
168, 15bitrid 283 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌) ↔ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹0 ) = 𝑌)))
1716pm5.32i 574 . 2 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹0 ) = 𝑌)))
187, 17bitri 275 1 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹0 ) = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3055  wf 6532  cfv 6536  (class class class)co 7404  Basecbs 17151  +gcplusg 17204  0gc0g 17392  Mgmcmgm 18569   MgmHom cmgmhm 18621  Mndcmnd 18665   MndHom cmhm 18709
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-mgmhm 18623  df-sgrp 18650  df-mnd 18666  df-mhm 18711
This theorem is referenced by:  c0snmhm  20363
  Copyright terms: Public domain W3C validator