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Theorem c0mhm 44473
Description: The constant mapping to zero is a monoid homomorphism. (Contributed by AV, 16-Apr-2020.)
Hypotheses
Ref Expression
c0mhm.b 𝐵 = (Base‘𝑆)
c0mhm.0 0 = (0g𝑇)
c0mhm.h 𝐻 = (𝑥𝐵0 )
Assertion
Ref Expression
c0mhm ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MndHom 𝑇))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑆   𝑥,𝑇   𝑥, 0
Allowed substitution hint:   𝐻(𝑥)

Proof of Theorem c0mhm
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2822 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
2 c0mhm.0 . . . . . . . 8 0 = (0g𝑇)
31, 2mndidcl 17917 . . . . . . 7 (𝑇 ∈ Mnd → 0 ∈ (Base‘𝑇))
43adantl 485 . . . . . 6 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 0 ∈ (Base‘𝑇))
54adantr 484 . . . . 5 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ 𝑥𝐵) → 0 ∈ (Base‘𝑇))
6 c0mhm.h . . . . 5 𝐻 = (𝑥𝐵0 )
75, 6fmptd 6860 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻:𝐵⟶(Base‘𝑇))
83ancli 552 . . . . . . . . 9 (𝑇 ∈ Mnd → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
98adantl 485 . . . . . . . 8 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
10 eqid 2822 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
111, 10, 2mndlid 17922 . . . . . . . 8 ((𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)) → ( 0 (+g𝑇) 0 ) = 0 )
129, 11syl 17 . . . . . . 7 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ( 0 (+g𝑇) 0 ) = 0 )
1312adantr 484 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ( 0 (+g𝑇) 0 ) = 0 )
146a1i 11 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝐻 = (𝑥𝐵0 ))
15 eqidd 2823 . . . . . . . 8 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑎) → 0 = 0 )
16 simprl 770 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
174adantr 484 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 0 ∈ (Base‘𝑇))
1814, 15, 16, 17fvmptd 6757 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑎) = 0 )
19 eqidd 2823 . . . . . . . 8 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑏) → 0 = 0 )
20 simprr 772 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
2114, 19, 20, 17fvmptd 6757 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑏) = 0 )
2218, 21oveq12d 7158 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) = ( 0 (+g𝑇) 0 ))
23 eqidd 2823 . . . . . . 7 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = (𝑎(+g𝑆)𝑏)) → 0 = 0 )
24 c0mhm.b . . . . . . . . . 10 𝐵 = (Base‘𝑆)
25 eqid 2822 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
2624, 25mndcl 17910 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
27263expb 1117 . . . . . . . 8 ((𝑆 ∈ Mnd ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
2827adantlr 714 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
2914, 23, 28, 17fvmptd 6757 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = 0 )
3013, 22, 293eqtr4rd 2868 . . . . 5 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
3130ralrimivva 3181 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
326a1i 11 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 = (𝑥𝐵0 ))
33 eqidd 2823 . . . . 5 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ 𝑥 = (0g𝑆)) → 0 = 0 )
34 eqid 2822 . . . . . . 7 (0g𝑆) = (0g𝑆)
3524, 34mndidcl 17917 . . . . . 6 (𝑆 ∈ Mnd → (0g𝑆) ∈ 𝐵)
3635adantr 484 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (0g𝑆) ∈ 𝐵)
3732, 33, 36, 4fvmptd 6757 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐻‘(0g𝑆)) = 0 )
387, 31, 373jca 1125 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) ∧ (𝐻‘(0g𝑆)) = 0 ))
3938ancli 552 . 2 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) ∧ (𝐻‘(0g𝑆)) = 0 )))
4024, 1, 25, 10, 34, 2ismhm 17949 . 2 (𝐻 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) ∧ (𝐻‘(0g𝑆)) = 0 )))
4139, 40sylibr 237 1 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114  wral 3130  cmpt 5122  wf 6330  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  0gc0g 16704  Mndcmnd 17902   MndHom cmhm 17945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-map 8395  df-0g 16706  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-mhm 17947
This theorem is referenced by:  c0ghm  44474  c0rhm  44475
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