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Theorem ismhm0 45359
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 18432 . 2 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
8 df-3an 1088 . . . 4 ((𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌) ↔ ((𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ∧ (𝐹0 ) = 𝑌))
9 mndmgm 18392 . . . . . . . 8 (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
10 mndmgm 18392 . . . . . . . 8 (𝑇 ∈ Mnd → 𝑇 ∈ Mgm)
119, 10anim12i 613 . . . . . . 7 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
1211biantrurd 533 . . . . . 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 45337 . . . . . 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 630 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (((𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ∧ (𝐹0 ) = 𝑌) ↔ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹0 ) = 𝑌)))
168, 15syl5bb 283 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌) ↔ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹0 ) = 𝑌)))
1716pm5.32i 575 . 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 274 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 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wf 6429  cfv 6433  (class class class)co 7275  Basecbs 16912  +gcplusg 16962  0gc0g 17150  Mgmcmgm 18324  Mndcmnd 18385   MndHom cmhm 18428   MgmHom cmgmhm 45331
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-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-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  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-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-mgmhm 45333
This theorem is referenced by:  c0snmhm  45473
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