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Theorem c0mhm 20354
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 2724 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
2 c0mhm.0 . . . . . . . 8 0 = (0g𝑇)
31, 2mndidcl 18674 . . . . . . 7 (𝑇 ∈ Mnd → 0 ∈ (Base‘𝑇))
43adantl 481 . . . . . 6 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 0 ∈ (Base‘𝑇))
54adantr 480 . . . . 5 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ 𝑥𝐵) → 0 ∈ (Base‘𝑇))
6 c0mhm.h . . . . 5 𝐻 = (𝑥𝐵0 )
75, 6fmptd 7106 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻:𝐵⟶(Base‘𝑇))
83ancli 548 . . . . . . . . 9 (𝑇 ∈ Mnd → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
98adantl 481 . . . . . . . 8 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
10 eqid 2724 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
111, 10, 2mndlid 18679 . . . . . . . 8 ((𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)) → ( 0 (+g𝑇) 0 ) = 0 )
129, 11syl 17 . . . . . . 7 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ( 0 (+g𝑇) 0 ) = 0 )
1312adantr 480 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ( 0 (+g𝑇) 0 ) = 0 )
146a1i 11 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝐻 = (𝑥𝐵0 ))
15 eqidd 2725 . . . . . . . 8 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑎) → 0 = 0 )
16 simprl 768 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
174adantr 480 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 0 ∈ (Base‘𝑇))
1814, 15, 16, 17fvmptd 6996 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑎) = 0 )
19 eqidd 2725 . . . . . . . 8 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑏) → 0 = 0 )
20 simprr 770 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
2114, 19, 20, 17fvmptd 6996 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑏) = 0 )
2218, 21oveq12d 7420 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) = ( 0 (+g𝑇) 0 ))
23 eqidd 2725 . . . . . . 7 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = (𝑎(+g𝑆)𝑏)) → 0 = 0 )
24 c0mhm.b . . . . . . . . . 10 𝐵 = (Base‘𝑆)
25 eqid 2724 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
2624, 25mndcl 18667 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
27263expb 1117 . . . . . . . 8 ((𝑆 ∈ Mnd ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
2827adantlr 712 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
2914, 23, 28, 17fvmptd 6996 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = 0 )
3013, 22, 293eqtr4rd 2775 . . . . 5 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
3130ralrimivva 3192 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
326a1i 11 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 = (𝑥𝐵0 ))
33 eqidd 2725 . . . . 5 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ 𝑥 = (0g𝑆)) → 0 = 0 )
34 eqid 2724 . . . . . . 7 (0g𝑆) = (0g𝑆)
3524, 34mndidcl 18674 . . . . . 6 (𝑆 ∈ Mnd → (0g𝑆) ∈ 𝐵)
3635adantr 480 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (0g𝑆) ∈ 𝐵)
3732, 33, 36, 4fvmptd 6996 . . . 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 548 . 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 18707 . 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 233 1 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053  cmpt 5222  wf 6530  cfv 6534  (class class class)co 7402  Basecbs 17145  +gcplusg 17198  0gc0g 17386  Mndcmnd 18659   MndHom cmhm 18703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-0g 17388  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18705
This theorem is referenced by:  c0ghm  20355  c0rhm  20426
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