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Theorem c0snmgmhm 46792
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 18634 . . . . 5 (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
21anim1i 615 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
323adant3 1132 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
43ancomd 462 . 2 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝑇 ∈ Mgm ∧ 𝑆 ∈ Mgm))
5 zrrhm.b . . . . . 6 𝐵 = (Base‘𝑇)
65fvexi 6905 . . . . 5 𝐵 ∈ V
7 hash1snb 14381 . . . . 5 (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏}))
86, 7ax-mp 5 . . . 4 ((♯‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏})
9 eqid 2732 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
10 zrrhm.0 . . . . . . . . . . . 12 0 = (0g𝑆)
119, 10mndidcl 18642 . . . . . . . . . . 11 (𝑆 ∈ Mnd → 0 ∈ (Base‘𝑆))
1211adantr 481 . . . . . . . . . 10 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 0 ∈ (Base‘𝑆))
1312adantr 481 . . . . . . . . 9 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 0 ∈ (Base‘𝑆))
1413adantr 481 . . . . . . . 8 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥𝐵) → 0 ∈ (Base‘𝑆))
15 zrrhm.h . . . . . . . 8 𝐻 = (𝑥𝐵0 )
1614, 15fmptd 7115 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝐻:𝐵⟶(Base‘𝑆))
1715a1i 11 . . . . . . . . . 10 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝐻 = (𝑥𝐵0 ))
18 eqidd 2733 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥 = 𝑏) → 0 = 0 )
19 vsnid 4665 . . . . . . . . . . . . 13 𝑏 ∈ {𝑏}
2019a1i 11 . . . . . . . . . . . 12 (𝐵 = {𝑏} → 𝑏 ∈ {𝑏})
21 eleq2 2822 . . . . . . . . . . . 12 (𝐵 = {𝑏} → (𝑏𝐵𝑏 ∈ {𝑏}))
2220, 21mpbird 256 . . . . . . . . . . 11 (𝐵 = {𝑏} → 𝑏𝐵)
2322adantl 482 . . . . . . . . . 10 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑏𝐵)
2417, 18, 23, 13fvmptd 7005 . . . . . . . . 9 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻𝑏) = 0 )
25 simpr 485 . . . . . . . . . . 11 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → (𝐻𝑏) = 0 )
2625, 25oveq12d 7429 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → ((𝐻𝑏)(+g𝑆)(𝐻𝑏)) = ( 0 (+g𝑆) 0 ))
27 eqid 2732 . . . . . . . . . . . . . . 15 (+g𝑆) = (+g𝑆)
289, 27, 10mndlid 18647 . . . . . . . . . . . . . 14 ((𝑆 ∈ Mnd ∧ 0 ∈ (Base‘𝑆)) → ( 0 (+g𝑆) 0 ) = 0 )
2911, 28mpdan 685 . . . . . . . . . . . . 13 (𝑆 ∈ Mnd → ( 0 (+g𝑆) 0 ) = 0 )
3029adantr 481 . . . . . . . . . . . 12 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → ( 0 (+g𝑆) 0 ) = 0 )
3130adantr 481 . . . . . . . . . . 11 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ( 0 (+g𝑆) 0 ) = 0 )
3231adantr 481 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → ( 0 (+g𝑆) 0 ) = 0 )
33 simpr 485 . . . . . . . . . . . . . . . . . 18 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 𝑇 ∈ Mgm)
3433adantr 481 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑇 ∈ Mgm)
3534adantr 481 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → 𝑇 ∈ Mgm)
36 simpr 485 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → 𝑏𝐵)
37 eqid 2732 . . . . . . . . . . . . . . . . 17 (+g𝑇) = (+g𝑇)
385, 37mgmcl 18566 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ Mgm ∧ 𝑏𝐵𝑏𝐵) → (𝑏(+g𝑇)𝑏) ∈ 𝐵)
3935, 36, 36, 38syl3anc 1371 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → (𝑏(+g𝑇)𝑏) ∈ 𝐵)
40 eleq2 2822 . . . . . . . . . . . . . . . . . 18 (𝐵 = {𝑏} → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 ↔ (𝑏(+g𝑇)𝑏) ∈ {𝑏}))
41 elsni 4645 . . . . . . . . . . . . . . . . . 18 ((𝑏(+g𝑇)𝑏) ∈ {𝑏} → (𝑏(+g𝑇)𝑏) = 𝑏)
4240, 41syl6bi 252 . . . . . . . . . . . . . . . . 17 (𝐵 = {𝑏} → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 → (𝑏(+g𝑇)𝑏) = 𝑏))
4342adantl 482 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 → (𝑏(+g𝑇)𝑏) = 𝑏))
4443adantr 481 . . . . . . . . . . . . . . 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 685 . . . . . . . . . . . . 13 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝑏(+g𝑇)𝑏) = 𝑏)
4746fveq2d 6895 . . . . . . . . . . . 12 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘(𝑏(+g𝑇)𝑏)) = (𝐻𝑏))
4847adantr 481 . . . . . . . . . . 11 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → (𝐻‘(𝑏(+g𝑇)𝑏)) = (𝐻𝑏))
4948, 25eqtr2d 2773 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → 0 = (𝐻‘(𝑏(+g𝑇)𝑏)))
5026, 32, 493eqtrrd 2777 . . . . . . . . 9 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
5124, 50mpdan 685 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
52 id 22 . . . . . . . . . . 11 (𝐵 = {𝑏} → 𝐵 = {𝑏})
5352raleqdv 3325 . . . . . . . . . . 11 (𝐵 = {𝑏} → (∀𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ ∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
5452, 53raleqbidv 3342 . . . . . . . . . 10 (𝐵 = {𝑏} → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ ∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
5554adantl 482 . . . . . . . . 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 7434 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝐻‘(𝑎(+g𝑇)𝑐)) = (𝐻‘(𝑏(+g𝑇)𝑐)))
57 fveq2 6891 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝐻𝑎) = (𝐻𝑏))
5857oveq1d 7426 . . . . . . . . . . . 12 (𝑎 = 𝑏 → ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑐)))
5956, 58eqeq12d 2748 . . . . . . . . . . 11 (𝑎 = 𝑏 → ((𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑐))))
60 oveq2 7419 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (𝑏(+g𝑇)𝑐) = (𝑏(+g𝑇)𝑏))
6160fveq2d 6895 . . . . . . . . . . . 12 (𝑐 = 𝑏 → (𝐻‘(𝑏(+g𝑇)𝑐)) = (𝐻‘(𝑏(+g𝑇)𝑏)))
62 fveq2 6891 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (𝐻𝑐) = (𝐻𝑏))
6362oveq2d 7427 . . . . . . . . . . . 12 (𝑐 = 𝑏 → ((𝐻𝑏)(+g𝑆)(𝐻𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
6461, 63eqeq12d 2748 . . . . . . . . . . 11 (𝑐 = 𝑏 → ((𝐻‘(𝑏(+g𝑇)𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏))))
6559, 642ralsng 4680 . . . . . . . . . 10 ((𝑏 ∈ V ∧ 𝑏 ∈ V) → (∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏))))
6665el2v 3482 . . . . . . . . 9 (∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
6755, 66bitrdi 286 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏))))
6851, 67mpbird 256 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))
6916, 68jca 512 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
7069ex 413 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝐵 = {𝑏} → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
7170exlimdv 1936 . . . 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 1117 . 2 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
745, 9, 37, 27ismgmhm 46632 . 2 (𝐻 ∈ (𝑇 MgmHom 𝑆) ↔ ((𝑇 ∈ Mgm ∧ 𝑆 ∈ Mgm) ∧ (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
754, 73, 74sylanbrc 583 1 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wral 3061  Vcvv 3474  {csn 4628  cmpt 5231  wf 6539  cfv 6543  (class class class)co 7411  1c1 11113  chash 14292  Basecbs 17146  +gcplusg 17199  0gc0g 17387  Mgmcmgm 18561  Mndcmnd 18627   MgmHom cmgmhm 46626
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-n0 12475  df-z 12561  df-uz 12825  df-fz 13487  df-hash 14293  df-0g 17389  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-mgmhm 46628
This theorem is referenced by:  c0snmhm  46793
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