Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  c0mhm Structured version   Visualization version   GIF version

Theorem c0mhm 46709
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 2733 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
2 c0mhm.0 . . . . . . . 8 0 = (0g𝑇)
31, 2mndidcl 18640 . . . . . . 7 (𝑇 ∈ Mnd → 0 ∈ (Base‘𝑇))
43adantl 483 . . . . . 6 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 0 ∈ (Base‘𝑇))
54adantr 482 . . . . 5 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ 𝑥𝐵) → 0 ∈ (Base‘𝑇))
6 c0mhm.h . . . . 5 𝐻 = (𝑥𝐵0 )
75, 6fmptd 7114 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻:𝐵⟶(Base‘𝑇))
83ancli 550 . . . . . . . . 9 (𝑇 ∈ Mnd → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
98adantl 483 . . . . . . . 8 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
10 eqid 2733 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
111, 10, 2mndlid 18645 . . . . . . . 8 ((𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)) → ( 0 (+g𝑇) 0 ) = 0 )
129, 11syl 17 . . . . . . 7 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ( 0 (+g𝑇) 0 ) = 0 )
1312adantr 482 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ( 0 (+g𝑇) 0 ) = 0 )
146a1i 11 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝐻 = (𝑥𝐵0 ))
15 eqidd 2734 . . . . . . . 8 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑎) → 0 = 0 )
16 simprl 770 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
174adantr 482 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 0 ∈ (Base‘𝑇))
1814, 15, 16, 17fvmptd 7006 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑎) = 0 )
19 eqidd 2734 . . . . . . . 8 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑏) → 0 = 0 )
20 simprr 772 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
2114, 19, 20, 17fvmptd 7006 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑏) = 0 )
2218, 21oveq12d 7427 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) = ( 0 (+g𝑇) 0 ))
23 eqidd 2734 . . . . . . 7 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = (𝑎(+g𝑆)𝑏)) → 0 = 0 )
24 c0mhm.b . . . . . . . . . 10 𝐵 = (Base‘𝑆)
25 eqid 2733 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
2624, 25mndcl 18633 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
27263expb 1121 . . . . . . . 8 ((𝑆 ∈ Mnd ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
2827adantlr 714 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
2914, 23, 28, 17fvmptd 7006 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = 0 )
3013, 22, 293eqtr4rd 2784 . . . . 5 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
3130ralrimivva 3201 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
326a1i 11 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 = (𝑥𝐵0 ))
33 eqidd 2734 . . . . 5 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ 𝑥 = (0g𝑆)) → 0 = 0 )
34 eqid 2733 . . . . . . 7 (0g𝑆) = (0g𝑆)
3524, 34mndidcl 18640 . . . . . 6 (𝑆 ∈ Mnd → (0g𝑆) ∈ 𝐵)
3635adantr 482 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (0g𝑆) ∈ 𝐵)
3732, 33, 36, 4fvmptd 7006 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐻‘(0g𝑆)) = 0 )
387, 31, 373jca 1129 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) ∧ (𝐻‘(0g𝑆)) = 0 ))
3938ancli 550 . 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 18673 . 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 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  cmpt 5232  wf 6540  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  0gc0g 17385  Mndcmnd 18625   MndHom cmhm 18669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671
This theorem is referenced by:  c0ghm  46710  c0rhm  46711
  Copyright terms: Public domain W3C validator