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Theorem c0mhm 41675
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 2620 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
2 c0mhm.0 . . . . . . . 8 0 = (0g𝑇)
31, 2mndidcl 17289 . . . . . . 7 (𝑇 ∈ Mnd → 0 ∈ (Base‘𝑇))
43adantl 482 . . . . . 6 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 0 ∈ (Base‘𝑇))
54adantr 481 . . . . 5 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ 𝑥𝐵) → 0 ∈ (Base‘𝑇))
6 c0mhm.h . . . . 5 𝐻 = (𝑥𝐵0 )
75, 6fmptd 6371 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻:𝐵⟶(Base‘𝑇))
83ancli 573 . . . . . . . . 9 (𝑇 ∈ Mnd → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
98adantl 482 . . . . . . . 8 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
10 eqid 2620 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
111, 10, 2mndlid 17292 . . . . . . . 8 ((𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)) → ( 0 (+g𝑇) 0 ) = 0 )
129, 11syl 17 . . . . . . 7 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ( 0 (+g𝑇) 0 ) = 0 )
1312adantr 481 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ( 0 (+g𝑇) 0 ) = 0 )
146a1i 11 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝐻 = (𝑥𝐵0 ))
15 eqidd 2621 . . . . . . . 8 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑎) → 0 = 0 )
16 simprl 793 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
174adantr 481 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 0 ∈ (Base‘𝑇))
1814, 15, 16, 17fvmptd 6275 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑎) = 0 )
19 eqidd 2621 . . . . . . . 8 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑏) → 0 = 0 )
20 simprr 795 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
2114, 19, 20, 17fvmptd 6275 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑏) = 0 )
2218, 21oveq12d 6653 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) = ( 0 (+g𝑇) 0 ))
23 eqidd 2621 . . . . . . 7 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = (𝑎(+g𝑆)𝑏)) → 0 = 0 )
24 c0mhm.b . . . . . . . . . 10 𝐵 = (Base‘𝑆)
25 eqid 2620 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
2624, 25mndcl 17282 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
27263expb 1264 . . . . . . . 8 ((𝑆 ∈ Mnd ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
2827adantlr 750 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
2914, 23, 28, 17fvmptd 6275 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = 0 )
3013, 22, 293eqtr4rd 2665 . . . . 5 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
3130ralrimivva 2968 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
326a1i 11 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 = (𝑥𝐵0 ))
33 eqidd 2621 . . . . 5 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ 𝑥 = (0g𝑆)) → 0 = 0 )
34 eqid 2620 . . . . . . 7 (0g𝑆) = (0g𝑆)
3524, 34mndidcl 17289 . . . . . 6 (𝑆 ∈ Mnd → (0g𝑆) ∈ 𝐵)
3635adantr 481 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (0g𝑆) ∈ 𝐵)
3732, 33, 36, 4fvmptd 6275 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐻‘(0g𝑆)) = 0 )
387, 31, 373jca 1240 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) ∧ (𝐻‘(0g𝑆)) = 0 ))
3938ancli 573 . 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 17318 . 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 224 1 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  cmpt 4720  wf 5872  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  0gc0g 16081  Mndcmnd 17275   MndHom cmhm 17314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-map 7844  df-0g 16083  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-mhm 17316
This theorem is referenced by:  c0ghm  41676  c0rhm  41677
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