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Theorem ismhm0 46185
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 18608 . 2 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
8 df-3an 1090 . . . 4 ((𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌) ↔ ((𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ∧ (𝐹0 ) = 𝑌))
9 mndmgm 18568 . . . . . . . 8 (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
10 mndmgm 18568 . . . . . . . 8 (𝑇 ∈ Mnd → 𝑇 ∈ Mgm)
119, 10anim12i 614 . . . . . . 7 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
1211biantrurd 534 . . . . . 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 46163 . . . . . 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 631 . . . 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 576 . 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 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wf 6493  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  Mgmcmgm 18500  Mndcmnd 18561   MndHom cmhm 18604   MgmHom cmgmhm 46157
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-sgrp 18551  df-mnd 18562  df-mhm 18606  df-mgmhm 46159
This theorem is referenced by:  c0snmhm  46299
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