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Theorem c0mhm 41700
 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 2610 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
2 c0mhm.0 . . . . . . . 8 0 = (0g𝑇)
31, 2mndidcl 17131 . . . . . . 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 6292 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻:𝐵⟶(Base‘𝑇))
83ancli 572 . . . . . . . . 9 (𝑇 ∈ Mnd → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
98adantl 481 . . . . . . . 8 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
10 eqid 2610 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
111, 10, 2mndlid 17134 . . . . . . . 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 2611 . . . . . . . 8 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑎) → 0 = 0 )
16 simprl 790 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
174adantr 480 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 0 ∈ (Base‘𝑇))
1814, 15, 16, 17fvmptd 6197 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑎) = 0 )
19 eqidd 2611 . . . . . . . 8 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑏) → 0 = 0 )
20 simprr 792 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
2114, 19, 20, 17fvmptd 6197 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑏) = 0 )
2218, 21oveq12d 6567 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) = ( 0 (+g𝑇) 0 ))
23 eqidd 2611 . . . . . . 7 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = (𝑎(+g𝑆)𝑏)) → 0 = 0 )
24 c0mhm.b . . . . . . . . . 10 𝐵 = (Base‘𝑆)
25 eqid 2610 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
2624, 25mndcl 17124 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
27263expb 1258 . . . . . . . 8 ((𝑆 ∈ Mnd ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
2827adantlr 747 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
2914, 23, 28, 17fvmptd 6197 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = 0 )
3013, 22, 293eqtr4rd 2655 . . . . 5 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
3130ralrimivva 2954 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
326a1i 11 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 = (𝑥𝐵0 ))
33 eqidd 2611 . . . . 5 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ 𝑥 = (0g𝑆)) → 0 = 0 )
34 eqid 2610 . . . . . . 7 (0g𝑆) = (0g𝑆)
3524, 34mndidcl 17131 . . . . . 6 (𝑆 ∈ Mnd → (0g𝑆) ∈ 𝐵)
3635adantr 480 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (0g𝑆) ∈ 𝐵)
3732, 33, 36, 4fvmptd 6197 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐻‘(0g𝑆)) = 0 )
387, 31, 373jca 1235 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) ∧ (𝐻‘(0g𝑆)) = 0 ))
3938ancli 572 . 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 17160 . 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 223 1 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MndHom 𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Mndcmnd 17117   MndHom cmhm 17156 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158 This theorem is referenced by:  c0ghm  41701  c0rhm  41702
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