Theorem ismhm0 18715

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	⊢ + = (+g‘𝑆)
ismhm0.q	⊢ ⨣ = (+g‘𝑇)
ismhm0.z	⊢ 0 = (0g‘𝑆)
ismhm0.y	⊢ 𝑌 = (0g‘𝑇)

Assertion

Ref	Expression
ismhm0	⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹‘ 0 ) = 𝑌)))

Proof of Theorem ismhm0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismhm0.b	. . 3 ⊢ 𝐵 = (Base‘𝑆)
2		ismhm0.c	. . 3 ⊢ 𝐶 = (Base‘𝑇)
3		ismhm0.p	. . 3 ⊢ + = (+g‘𝑆)
4		ismhm0.q	. . 3 ⊢ ⨣ = (+g‘𝑇)
5		ismhm0.z	. . 3 ⊢ 0 = (0g‘𝑆)
6		ismhm0.y	. . 3 ⊢ 𝑌 = (0g‘𝑇)
7	1, 2, 3, 4, 5, 6	ismhm 18710	. 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))
8		df-3an 1088	. . . 4 ⊢ ((𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌) ↔ ((𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) ∧ (𝐹‘ 0 ) = 𝑌))
9		mndmgm 18666	. . . . . . . 8 ⊢ (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
10		mndmgm 18666	. . . . . . . 8 ⊢ (𝑇 ∈ Mnd → 𝑇 ∈ Mgm)
11	9, 10	anim12i 613	. . . . . . 7 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
12	11	biantrurd 532	. . . . . 6 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))))
13	1, 2, 3, 4	ismgmhm 18621	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))))
14	12, 13	bitr4di 289	. . . . 5 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) ↔ 𝐹 ∈ (𝑆 MgmHom 𝑇)))
15	14	anbi1d 631	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (((𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) ∧ (𝐹‘ 0 ) = 𝑌) ↔ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹‘ 0 ) = 𝑌)))
16	8, 15	bitrid 283	. . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌) ↔ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹‘ 0 ) = 𝑌)))
17	16	pm5.32i 574	. 2 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹‘ 0 ) = 𝑌)))
18	7, 17	bitri 275	1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹‘ 0 ) = 𝑌)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  +_gcplusg 17177  0_gc0g 17359  Mgmcmgm 18563   MgmHom cmgmhm 18615  Mndcmnd 18659   MndHom cmhm 18706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8765  df-mgmhm 18617  df-sgrp 18644  df-mnd 18660  df-mhm 18708

This theorem is referenced by:  c0snmhm  20399