Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  c0snmgmhm Structured version   Visualization version   GIF version

Theorem c0snmgmhm 44706
 Description: The constant mapping to zero is a magma homomorphism from a magma with one element to any monoid. (Contributed by AV, 17-Apr-2020.)
Hypotheses
Ref Expression
zrrhm.b 𝐵 = (Base‘𝑇)
zrrhm.0 0 = (0g𝑆)
zrrhm.h 𝐻 = (𝑥𝐵0 )
Assertion
Ref Expression
c0snmgmhm ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑆   𝑥,𝑇   𝑥, 0
Allowed substitution hint:   𝐻(𝑥)

Proof of Theorem c0snmgmhm
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mndmgm 17930 . . . . 5 (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
21anim1i 617 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
323adant3 1129 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
43ancomd 465 . 2 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝑇 ∈ Mgm ∧ 𝑆 ∈ Mgm))
5 zrrhm.b . . . . . 6 𝐵 = (Base‘𝑇)
65fvexi 6669 . . . . 5 𝐵 ∈ V
7 hash1snb 13796 . . . . 5 (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏}))
86, 7ax-mp 5 . . . 4 ((♯‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏})
9 eqid 2798 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
10 zrrhm.0 . . . . . . . . . . . 12 0 = (0g𝑆)
119, 10mndidcl 17938 . . . . . . . . . . 11 (𝑆 ∈ Mnd → 0 ∈ (Base‘𝑆))
1211adantr 484 . . . . . . . . . 10 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 0 ∈ (Base‘𝑆))
1312adantr 484 . . . . . . . . 9 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 0 ∈ (Base‘𝑆))
1413adantr 484 . . . . . . . 8 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥𝐵) → 0 ∈ (Base‘𝑆))
15 zrrhm.h . . . . . . . 8 𝐻 = (𝑥𝐵0 )
1614, 15fmptd 6865 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝐻:𝐵⟶(Base‘𝑆))
1715a1i 11 . . . . . . . . . 10 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝐻 = (𝑥𝐵0 ))
18 eqidd 2799 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥 = 𝑏) → 0 = 0 )
19 vsnid 4565 . . . . . . . . . . . . 13 𝑏 ∈ {𝑏}
2019a1i 11 . . . . . . . . . . . 12 (𝐵 = {𝑏} → 𝑏 ∈ {𝑏})
21 eleq2 2878 . . . . . . . . . . . 12 (𝐵 = {𝑏} → (𝑏𝐵𝑏 ∈ {𝑏}))
2220, 21mpbird 260 . . . . . . . . . . 11 (𝐵 = {𝑏} → 𝑏𝐵)
2322adantl 485 . . . . . . . . . 10 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑏𝐵)
2417, 18, 23, 13fvmptd 6762 . . . . . . . . 9 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻𝑏) = 0 )
25 simpr 488 . . . . . . . . . . 11 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → (𝐻𝑏) = 0 )
2625, 25oveq12d 7163 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → ((𝐻𝑏)(+g𝑆)(𝐻𝑏)) = ( 0 (+g𝑆) 0 ))
27 eqid 2798 . . . . . . . . . . . . . . 15 (+g𝑆) = (+g𝑆)
289, 27, 10mndlid 17943 . . . . . . . . . . . . . 14 ((𝑆 ∈ Mnd ∧ 0 ∈ (Base‘𝑆)) → ( 0 (+g𝑆) 0 ) = 0 )
2911, 28mpdan 686 . . . . . . . . . . . . 13 (𝑆 ∈ Mnd → ( 0 (+g𝑆) 0 ) = 0 )
3029adantr 484 . . . . . . . . . . . 12 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → ( 0 (+g𝑆) 0 ) = 0 )
3130adantr 484 . . . . . . . . . . 11 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ( 0 (+g𝑆) 0 ) = 0 )
3231adantr 484 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → ( 0 (+g𝑆) 0 ) = 0 )
33 simpr 488 . . . . . . . . . . . . . . . . . 18 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 𝑇 ∈ Mgm)
3433adantr 484 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑇 ∈ Mgm)
3534adantr 484 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → 𝑇 ∈ Mgm)
36 simpr 488 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → 𝑏𝐵)
37 eqid 2798 . . . . . . . . . . . . . . . . 17 (+g𝑇) = (+g𝑇)
385, 37mgmcl 17867 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ Mgm ∧ 𝑏𝐵𝑏𝐵) → (𝑏(+g𝑇)𝑏) ∈ 𝐵)
3935, 36, 36, 38syl3anc 1368 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → (𝑏(+g𝑇)𝑏) ∈ 𝐵)
40 eleq2 2878 . . . . . . . . . . . . . . . . . 18 (𝐵 = {𝑏} → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 ↔ (𝑏(+g𝑇)𝑏) ∈ {𝑏}))
41 elsni 4545 . . . . . . . . . . . . . . . . . 18 ((𝑏(+g𝑇)𝑏) ∈ {𝑏} → (𝑏(+g𝑇)𝑏) = 𝑏)
4240, 41syl6bi 256 . . . . . . . . . . . . . . . . 17 (𝐵 = {𝑏} → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 → (𝑏(+g𝑇)𝑏) = 𝑏))
4342adantl 485 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 → (𝑏(+g𝑇)𝑏) = 𝑏))
4443adantr 484 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 → (𝑏(+g𝑇)𝑏) = 𝑏))
4539, 44mpd 15 . . . . . . . . . . . . . 14 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → (𝑏(+g𝑇)𝑏) = 𝑏)
4623, 45mpdan 686 . . . . . . . . . . . . 13 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝑏(+g𝑇)𝑏) = 𝑏)
4746fveq2d 6659 . . . . . . . . . . . 12 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘(𝑏(+g𝑇)𝑏)) = (𝐻𝑏))
4847adantr 484 . . . . . . . . . . 11 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → (𝐻‘(𝑏(+g𝑇)𝑏)) = (𝐻𝑏))
4948, 25eqtr2d 2834 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → 0 = (𝐻‘(𝑏(+g𝑇)𝑏)))
5026, 32, 493eqtrrd 2838 . . . . . . . . 9 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
5124, 50mpdan 686 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
52 id 22 . . . . . . . . . . 11 (𝐵 = {𝑏} → 𝐵 = {𝑏})
5352raleqdv 3365 . . . . . . . . . . 11 (𝐵 = {𝑏} → (∀𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ ∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
5452, 53raleqbidv 3355 . . . . . . . . . 10 (𝐵 = {𝑏} → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ ∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
5554adantl 485 . . . . . . . . 9 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ ∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
56 fvoveq1 7168 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝐻‘(𝑎(+g𝑇)𝑐)) = (𝐻‘(𝑏(+g𝑇)𝑐)))
57 fveq2 6655 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝐻𝑎) = (𝐻𝑏))
5857oveq1d 7160 . . . . . . . . . . . 12 (𝑎 = 𝑏 → ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑐)))
5956, 58eqeq12d 2814 . . . . . . . . . . 11 (𝑎 = 𝑏 → ((𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑐))))
60 oveq2 7153 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (𝑏(+g𝑇)𝑐) = (𝑏(+g𝑇)𝑏))
6160fveq2d 6659 . . . . . . . . . . . 12 (𝑐 = 𝑏 → (𝐻‘(𝑏(+g𝑇)𝑐)) = (𝐻‘(𝑏(+g𝑇)𝑏)))
62 fveq2 6655 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (𝐻𝑐) = (𝐻𝑏))
6362oveq2d 7161 . . . . . . . . . . . 12 (𝑐 = 𝑏 → ((𝐻𝑏)(+g𝑆)(𝐻𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
6461, 63eqeq12d 2814 . . . . . . . . . . 11 (𝑐 = 𝑏 → ((𝐻‘(𝑏(+g𝑇)𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏))))
6559, 642ralsng 4579 . . . . . . . . . 10 ((𝑏 ∈ V ∧ 𝑏 ∈ V) → (∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏))))
6665el2v 3449 . . . . . . . . 9 (∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
6755, 66syl6bb 290 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏))))
6851, 67mpbird 260 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))
6916, 68jca 515 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
7069ex 416 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝐵 = {𝑏} → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
7170exlimdv 1934 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (∃𝑏 𝐵 = {𝑏} → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
728, 71syl5bi 245 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → ((♯‘𝐵) = 1 → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
73723impia 1114 . 2 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
745, 9, 37, 27ismgmhm 44571 . 2 (𝐻 ∈ (𝑇 MgmHom 𝑆) ↔ ((𝑇 ∈ Mgm ∧ 𝑆 ∈ Mgm) ∧ (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
754, 73, 74sylanbrc 586 1 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  Vcvv 3442  {csn 4528   ↦ cmpt 5114  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145  1c1 10545  ♯chash 13706  Basecbs 16495  +gcplusg 16577  0gc0g 16725  Mgmcmgm 17862  Mndcmnd 17923   MgmHom cmgmhm 44565 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-oadd 8107  df-er 8290  df-map 8409  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-dju 9332  df-card 9370  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-n0 11904  df-z 11990  df-uz 12252  df-fz 12906  df-hash 13707  df-0g 16727  df-mgm 17864  df-sgrp 17913  df-mnd 17924  df-mgmhm 44567 This theorem is referenced by:  c0snmhm  44707
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