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Theorem c0snmgmhm 41704
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 17123 . . . . 5 (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
21anim1i 590 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
323adant3 1074 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (#‘𝐵) = 1) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
43ancomd 466 . 2 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (#‘𝐵) = 1) → (𝑇 ∈ Mgm ∧ 𝑆 ∈ Mgm))
5 zrrhm.b . . . . . 6 𝐵 = (Base‘𝑇)
6 fvex 6113 . . . . . 6 (Base‘𝑇) ∈ V
75, 6eqeltri 2684 . . . . 5 𝐵 ∈ V
8 hash1snb 13068 . . . . 5 (𝐵 ∈ V → ((#‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏}))
97, 8ax-mp 5 . . . 4 ((#‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏})
10 eqid 2610 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
11 zrrhm.0 . . . . . . . . . . . 12 0 = (0g𝑆)
1210, 11mndidcl 17131 . . . . . . . . . . 11 (𝑆 ∈ Mnd → 0 ∈ (Base‘𝑆))
1312adantr 480 . . . . . . . . . 10 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 0 ∈ (Base‘𝑆))
1413adantr 480 . . . . . . . . 9 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 0 ∈ (Base‘𝑆))
1514adantr 480 . . . . . . . 8 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥𝐵) → 0 ∈ (Base‘𝑆))
16 zrrhm.h . . . . . . . 8 𝐻 = (𝑥𝐵0 )
1715, 16fmptd 6292 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝐻:𝐵⟶(Base‘𝑆))
1816a1i 11 . . . . . . . . . 10 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝐻 = (𝑥𝐵0 ))
19 eqidd 2611 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥 = 𝑏) → 0 = 0 )
20 vsnid 4156 . . . . . . . . . . . . 13 𝑏 ∈ {𝑏}
2120a1i 11 . . . . . . . . . . . 12 (𝐵 = {𝑏} → 𝑏 ∈ {𝑏})
22 eleq2 2677 . . . . . . . . . . . 12 (𝐵 = {𝑏} → (𝑏𝐵𝑏 ∈ {𝑏}))
2321, 22mpbird 246 . . . . . . . . . . 11 (𝐵 = {𝑏} → 𝑏𝐵)
2423adantl 481 . . . . . . . . . 10 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑏𝐵)
2518, 19, 24, 14fvmptd 6197 . . . . . . . . 9 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻𝑏) = 0 )
26 simpr 476 . . . . . . . . . . 11 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → (𝐻𝑏) = 0 )
2726, 26oveq12d 6567 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → ((𝐻𝑏)(+g𝑆)(𝐻𝑏)) = ( 0 (+g𝑆) 0 ))
28 eqid 2610 . . . . . . . . . . . . . . 15 (+g𝑆) = (+g𝑆)
2910, 28, 11mndlid 17134 . . . . . . . . . . . . . 14 ((𝑆 ∈ Mnd ∧ 0 ∈ (Base‘𝑆)) → ( 0 (+g𝑆) 0 ) = 0 )
3012, 29mpdan 699 . . . . . . . . . . . . 13 (𝑆 ∈ Mnd → ( 0 (+g𝑆) 0 ) = 0 )
3130adantr 480 . . . . . . . . . . . 12 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → ( 0 (+g𝑆) 0 ) = 0 )
3231adantr 480 . . . . . . . . . . 11 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ( 0 (+g𝑆) 0 ) = 0 )
3332adantr 480 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → ( 0 (+g𝑆) 0 ) = 0 )
34 simpr 476 . . . . . . . . . . . . . . . . . 18 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 𝑇 ∈ Mgm)
3534adantr 480 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑇 ∈ Mgm)
3635adantr 480 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → 𝑇 ∈ Mgm)
37 simpr 476 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → 𝑏𝐵)
38 eqid 2610 . . . . . . . . . . . . . . . . 17 (+g𝑇) = (+g𝑇)
395, 38mgmcl 17068 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ Mgm ∧ 𝑏𝐵𝑏𝐵) → (𝑏(+g𝑇)𝑏) ∈ 𝐵)
4036, 37, 37, 39syl3anc 1318 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → (𝑏(+g𝑇)𝑏) ∈ 𝐵)
41 eleq2 2677 . . . . . . . . . . . . . . . . . 18 (𝐵 = {𝑏} → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 ↔ (𝑏(+g𝑇)𝑏) ∈ {𝑏}))
42 elsni 4142 . . . . . . . . . . . . . . . . . 18 ((𝑏(+g𝑇)𝑏) ∈ {𝑏} → (𝑏(+g𝑇)𝑏) = 𝑏)
4341, 42syl6bi 242 . . . . . . . . . . . . . . . . 17 (𝐵 = {𝑏} → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 → (𝑏(+g𝑇)𝑏) = 𝑏))
4443adantl 481 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 → (𝑏(+g𝑇)𝑏) = 𝑏))
4544adantr 480 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 → (𝑏(+g𝑇)𝑏) = 𝑏))
4640, 45mpd 15 . . . . . . . . . . . . . 14 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → (𝑏(+g𝑇)𝑏) = 𝑏)
4724, 46mpdan 699 . . . . . . . . . . . . 13 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝑏(+g𝑇)𝑏) = 𝑏)
4847fveq2d 6107 . . . . . . . . . . . 12 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘(𝑏(+g𝑇)𝑏)) = (𝐻𝑏))
4948adantr 480 . . . . . . . . . . 11 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → (𝐻‘(𝑏(+g𝑇)𝑏)) = (𝐻𝑏))
5049, 26eqtr2d 2645 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → 0 = (𝐻‘(𝑏(+g𝑇)𝑏)))
5127, 33, 503eqtrrd 2649 . . . . . . . . 9 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
5225, 51mpdan 699 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
53 id 22 . . . . . . . . . . 11 (𝐵 = {𝑏} → 𝐵 = {𝑏})
5453raleqdv 3121 . . . . . . . . . . 11 (𝐵 = {𝑏} → (∀𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ ∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
5553, 54raleqbidv 3129 . . . . . . . . . 10 (𝐵 = {𝑏} → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ ∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
5655adantl 481 . . . . . . . . 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))))
57 vex 3176 . . . . . . . . . 10 𝑏 ∈ V
58 oveq1 6556 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝑎(+g𝑇)𝑐) = (𝑏(+g𝑇)𝑐))
5958fveq2d 6107 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝐻‘(𝑎(+g𝑇)𝑐)) = (𝐻‘(𝑏(+g𝑇)𝑐)))
60 fveq2 6103 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝐻𝑎) = (𝐻𝑏))
6160oveq1d 6564 . . . . . . . . . . . 12 (𝑎 = 𝑏 → ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑐)))
6259, 61eqeq12d 2625 . . . . . . . . . . 11 (𝑎 = 𝑏 → ((𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑐))))
63 oveq2 6557 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (𝑏(+g𝑇)𝑐) = (𝑏(+g𝑇)𝑏))
6463fveq2d 6107 . . . . . . . . . . . 12 (𝑐 = 𝑏 → (𝐻‘(𝑏(+g𝑇)𝑐)) = (𝐻‘(𝑏(+g𝑇)𝑏)))
65 fveq2 6103 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (𝐻𝑐) = (𝐻𝑏))
6665oveq2d 6565 . . . . . . . . . . . 12 (𝑐 = 𝑏 → ((𝐻𝑏)(+g𝑆)(𝐻𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
6764, 66eqeq12d 2625 . . . . . . . . . . 11 (𝑐 = 𝑏 → ((𝐻‘(𝑏(+g𝑇)𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏))))
6862, 672ralsng 4167 . . . . . . . . . 10 ((𝑏 ∈ V ∧ 𝑏 ∈ V) → (∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏))))
6957, 57, 68mp2an 704 . . . . . . . . 9 (∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
7056, 69syl6bb 275 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏))))
7152, 70mpbird 246 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))
7217, 71jca 553 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
7372ex 449 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝐵 = {𝑏} → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
7473exlimdv 1848 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (∃𝑏 𝐵 = {𝑏} → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
759, 74syl5bi 231 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → ((#‘𝐵) = 1 → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
76753impia 1253 . 2 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (#‘𝐵) = 1) → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
775, 10, 38, 28ismgmhm 41573 . 2 (𝐻 ∈ (𝑇 MgmHom 𝑆) ↔ ((𝑇 ∈ Mgm ∧ 𝑆 ∈ Mgm) ∧ (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
784, 76, 77sylanbrc 695 1 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (#‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wex 1695  wcel 1977  wral 2896  Vcvv 3173  {csn 4125  cmpt 4643  wf 5800  cfv 5804  (class class class)co 6549  1c1 9816  #chash 12979  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Mgmcmgm 17063  Mndcmnd 17117   MgmHom cmgmhm 41567
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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-nel 2783  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  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-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mgmhm 41569
This theorem is referenced by:  c0snmhm  41705
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