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Theorem c0snmgmhm 46713
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 18632 . . . . 5 (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
21anim1i 616 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
323adant3 1133 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
43ancomd 463 . 2 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝑇 ∈ Mgm ∧ 𝑆 ∈ Mgm))
5 zrrhm.b . . . . . 6 𝐵 = (Base‘𝑇)
65fvexi 6906 . . . . 5 𝐵 ∈ V
7 hash1snb 14379 . . . . 5 (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏}))
86, 7ax-mp 5 . . . 4 ((♯‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏})
9 eqid 2733 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
10 zrrhm.0 . . . . . . . . . . . 12 0 = (0g𝑆)
119, 10mndidcl 18640 . . . . . . . . . . 11 (𝑆 ∈ Mnd → 0 ∈ (Base‘𝑆))
1211adantr 482 . . . . . . . . . 10 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 0 ∈ (Base‘𝑆))
1312adantr 482 . . . . . . . . 9 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 0 ∈ (Base‘𝑆))
1413adantr 482 . . . . . . . 8 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥𝐵) → 0 ∈ (Base‘𝑆))
15 zrrhm.h . . . . . . . 8 𝐻 = (𝑥𝐵0 )
1614, 15fmptd 7114 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝐻:𝐵⟶(Base‘𝑆))
1715a1i 11 . . . . . . . . . 10 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝐻 = (𝑥𝐵0 ))
18 eqidd 2734 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥 = 𝑏) → 0 = 0 )
19 vsnid 4666 . . . . . . . . . . . . 13 𝑏 ∈ {𝑏}
2019a1i 11 . . . . . . . . . . . 12 (𝐵 = {𝑏} → 𝑏 ∈ {𝑏})
21 eleq2 2823 . . . . . . . . . . . 12 (𝐵 = {𝑏} → (𝑏𝐵𝑏 ∈ {𝑏}))
2220, 21mpbird 257 . . . . . . . . . . 11 (𝐵 = {𝑏} → 𝑏𝐵)
2322adantl 483 . . . . . . . . . 10 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑏𝐵)
2417, 18, 23, 13fvmptd 7006 . . . . . . . . 9 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻𝑏) = 0 )
25 simpr 486 . . . . . . . . . . 11 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → (𝐻𝑏) = 0 )
2625, 25oveq12d 7427 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → ((𝐻𝑏)(+g𝑆)(𝐻𝑏)) = ( 0 (+g𝑆) 0 ))
27 eqid 2733 . . . . . . . . . . . . . . 15 (+g𝑆) = (+g𝑆)
289, 27, 10mndlid 18645 . . . . . . . . . . . . . 14 ((𝑆 ∈ Mnd ∧ 0 ∈ (Base‘𝑆)) → ( 0 (+g𝑆) 0 ) = 0 )
2911, 28mpdan 686 . . . . . . . . . . . . 13 (𝑆 ∈ Mnd → ( 0 (+g𝑆) 0 ) = 0 )
3029adantr 482 . . . . . . . . . . . 12 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → ( 0 (+g𝑆) 0 ) = 0 )
3130adantr 482 . . . . . . . . . . 11 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ( 0 (+g𝑆) 0 ) = 0 )
3231adantr 482 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → ( 0 (+g𝑆) 0 ) = 0 )
33 simpr 486 . . . . . . . . . . . . . . . . . 18 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 𝑇 ∈ Mgm)
3433adantr 482 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑇 ∈ Mgm)
3534adantr 482 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → 𝑇 ∈ Mgm)
36 simpr 486 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → 𝑏𝐵)
37 eqid 2733 . . . . . . . . . . . . . . . . 17 (+g𝑇) = (+g𝑇)
385, 37mgmcl 18564 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ Mgm ∧ 𝑏𝐵𝑏𝐵) → (𝑏(+g𝑇)𝑏) ∈ 𝐵)
3935, 36, 36, 38syl3anc 1372 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → (𝑏(+g𝑇)𝑏) ∈ 𝐵)
40 eleq2 2823 . . . . . . . . . . . . . . . . . 18 (𝐵 = {𝑏} → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 ↔ (𝑏(+g𝑇)𝑏) ∈ {𝑏}))
41 elsni 4646 . . . . . . . . . . . . . . . . . 18 ((𝑏(+g𝑇)𝑏) ∈ {𝑏} → (𝑏(+g𝑇)𝑏) = 𝑏)
4240, 41syl6bi 253 . . . . . . . . . . . . . . . . 17 (𝐵 = {𝑏} → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 → (𝑏(+g𝑇)𝑏) = 𝑏))
4342adantl 483 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 → (𝑏(+g𝑇)𝑏) = 𝑏))
4443adantr 482 . . . . . . . . . . . . . . 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 6896 . . . . . . . . . . . 12 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘(𝑏(+g𝑇)𝑏)) = (𝐻𝑏))
4847adantr 482 . . . . . . . . . . 11 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → (𝐻‘(𝑏(+g𝑇)𝑏)) = (𝐻𝑏))
4948, 25eqtr2d 2774 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → 0 = (𝐻‘(𝑏(+g𝑇)𝑏)))
5026, 32, 493eqtrrd 2778 . . . . . . . . 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 3326 . . . . . . . . . . 11 (𝐵 = {𝑏} → (∀𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ ∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
5452, 53raleqbidv 3343 . . . . . . . . . 10 (𝐵 = {𝑏} → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ ∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
5554adantl 483 . . . . . . . . 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 7432 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝐻‘(𝑎(+g𝑇)𝑐)) = (𝐻‘(𝑏(+g𝑇)𝑐)))
57 fveq2 6892 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝐻𝑎) = (𝐻𝑏))
5857oveq1d 7424 . . . . . . . . . . . 12 (𝑎 = 𝑏 → ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑐)))
5956, 58eqeq12d 2749 . . . . . . . . . . 11 (𝑎 = 𝑏 → ((𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑐))))
60 oveq2 7417 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (𝑏(+g𝑇)𝑐) = (𝑏(+g𝑇)𝑏))
6160fveq2d 6896 . . . . . . . . . . . 12 (𝑐 = 𝑏 → (𝐻‘(𝑏(+g𝑇)𝑐)) = (𝐻‘(𝑏(+g𝑇)𝑏)))
62 fveq2 6892 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (𝐻𝑐) = (𝐻𝑏))
6362oveq2d 7425 . . . . . . . . . . . 12 (𝑐 = 𝑏 → ((𝐻𝑏)(+g𝑆)(𝐻𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
6461, 63eqeq12d 2749 . . . . . . . . . . 11 (𝑐 = 𝑏 → ((𝐻‘(𝑏(+g𝑇)𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏))))
6559, 642ralsng 4681 . . . . . . . . . 10 ((𝑏 ∈ V ∧ 𝑏 ∈ V) → (∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏))))
6665el2v 3483 . . . . . . . . 9 (∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
6755, 66bitrdi 287 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏))))
6851, 67mpbird 257 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))
6916, 68jca 513 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
7069ex 414 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝐵 = {𝑏} → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
7170exlimdv 1937 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (∃𝑏 𝐵 = {𝑏} → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
728, 71biimtrid 241 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → ((♯‘𝐵) = 1 → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
73723impia 1118 . 2 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
745, 9, 37, 27ismgmhm 46553 . 2 (𝐻 ∈ (𝑇 MgmHom 𝑆) ↔ ((𝑇 ∈ Mgm ∧ 𝑆 ∈ Mgm) ∧ (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
754, 73, 74sylanbrc 584 1 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  wral 3062  Vcvv 3475  {csn 4629  cmpt 5232  wf 6540  cfv 6544  (class class class)co 7409  1c1 11111  chash 14290  Basecbs 17144  +gcplusg 17197  0gc0g 17385  Mgmcmgm 18559  Mndcmnd 18625   MgmHom cmgmhm 46547
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  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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-nel 3048  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-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  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-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-hash 14291  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mgmhm 46549
This theorem is referenced by:  c0snmhm  46714
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