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Theorem c0snmgmhm 41175
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 17216 . . . . 5 (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
21anim1i 591 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
323adant3 1079 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (#‘𝐵) = 1) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
43ancomd 467 . 2 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (#‘𝐵) = 1) → (𝑇 ∈ Mgm ∧ 𝑆 ∈ Mgm))
5 zrrhm.b . . . . . 6 𝐵 = (Base‘𝑇)
6 fvex 6160 . . . . . 6 (Base‘𝑇) ∈ V
75, 6eqeltri 2700 . . . . 5 𝐵 ∈ V
8 hash1snb 13144 . . . . 5 (𝐵 ∈ V → ((#‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏}))
97, 8ax-mp 5 . . . 4 ((#‘𝐵) = 1 ↔ ∃𝑏 𝐵 = {𝑏})
10 eqid 2626 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
11 zrrhm.0 . . . . . . . . . . . 12 0 = (0g𝑆)
1210, 11mndidcl 17224 . . . . . . . . . . 11 (𝑆 ∈ Mnd → 0 ∈ (Base‘𝑆))
1312adantr 481 . . . . . . . . . 10 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 0 ∈ (Base‘𝑆))
1413adantr 481 . . . . . . . . 9 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 0 ∈ (Base‘𝑆))
1514adantr 481 . . . . . . . 8 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥𝐵) → 0 ∈ (Base‘𝑆))
16 zrrhm.h . . . . . . . 8 𝐻 = (𝑥𝐵0 )
1715, 16fmptd 6341 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝐻:𝐵⟶(Base‘𝑆))
1816a1i 11 . . . . . . . . . 10 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝐻 = (𝑥𝐵0 ))
19 eqidd 2627 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑥 = 𝑏) → 0 = 0 )
20 vsnid 4185 . . . . . . . . . . . . 13 𝑏 ∈ {𝑏}
2120a1i 11 . . . . . . . . . . . 12 (𝐵 = {𝑏} → 𝑏 ∈ {𝑏})
22 eleq2 2693 . . . . . . . . . . . 12 (𝐵 = {𝑏} → (𝑏𝐵𝑏 ∈ {𝑏}))
2321, 22mpbird 247 . . . . . . . . . . 11 (𝐵 = {𝑏} → 𝑏𝐵)
2423adantl 482 . . . . . . . . . 10 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑏𝐵)
2518, 19, 24, 14fvmptd 6246 . . . . . . . . 9 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻𝑏) = 0 )
26 simpr 477 . . . . . . . . . . 11 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → (𝐻𝑏) = 0 )
2726, 26oveq12d 6623 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → ((𝐻𝑏)(+g𝑆)(𝐻𝑏)) = ( 0 (+g𝑆) 0 ))
28 eqid 2626 . . . . . . . . . . . . . . 15 (+g𝑆) = (+g𝑆)
2910, 28, 11mndlid 17227 . . . . . . . . . . . . . 14 ((𝑆 ∈ Mnd ∧ 0 ∈ (Base‘𝑆)) → ( 0 (+g𝑆) 0 ) = 0 )
3012, 29mpdan 701 . . . . . . . . . . . . 13 (𝑆 ∈ Mnd → ( 0 (+g𝑆) 0 ) = 0 )
3130adantr 481 . . . . . . . . . . . 12 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → ( 0 (+g𝑆) 0 ) = 0 )
3231adantr 481 . . . . . . . . . . 11 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ( 0 (+g𝑆) 0 ) = 0 )
3332adantr 481 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → ( 0 (+g𝑆) 0 ) = 0 )
34 simpr 477 . . . . . . . . . . . . . . . . . 18 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → 𝑇 ∈ Mgm)
3534adantr 481 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → 𝑇 ∈ Mgm)
3635adantr 481 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → 𝑇 ∈ Mgm)
37 simpr 477 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → 𝑏𝐵)
38 eqid 2626 . . . . . . . . . . . . . . . . 17 (+g𝑇) = (+g𝑇)
395, 38mgmcl 17161 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ Mgm ∧ 𝑏𝐵𝑏𝐵) → (𝑏(+g𝑇)𝑏) ∈ 𝐵)
4036, 37, 37, 39syl3anc 1323 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ 𝑏𝐵) → (𝑏(+g𝑇)𝑏) ∈ 𝐵)
41 eleq2 2693 . . . . . . . . . . . . . . . . . 18 (𝐵 = {𝑏} → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 ↔ (𝑏(+g𝑇)𝑏) ∈ {𝑏}))
42 elsni 4170 . . . . . . . . . . . . . . . . . 18 ((𝑏(+g𝑇)𝑏) ∈ {𝑏} → (𝑏(+g𝑇)𝑏) = 𝑏)
4341, 42syl6bi 243 . . . . . . . . . . . . . . . . 17 (𝐵 = {𝑏} → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 → (𝑏(+g𝑇)𝑏) = 𝑏))
4443adantl 482 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ((𝑏(+g𝑇)𝑏) ∈ 𝐵 → (𝑏(+g𝑇)𝑏) = 𝑏))
4544adantr 481 . . . . . . . . . . . . . . 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 701 . . . . . . . . . . . . 13 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝑏(+g𝑇)𝑏) = 𝑏)
4847fveq2d 6154 . . . . . . . . . . . 12 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘(𝑏(+g𝑇)𝑏)) = (𝐻𝑏))
4948adantr 481 . . . . . . . . . . 11 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → (𝐻‘(𝑏(+g𝑇)𝑏)) = (𝐻𝑏))
5049, 26eqtr2d 2661 . . . . . . . . . 10 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → 0 = (𝐻‘(𝑏(+g𝑇)𝑏)))
5127, 33, 503eqtrrd 2665 . . . . . . . . 9 ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) ∧ (𝐻𝑏) = 0 ) → (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
5225, 51mpdan 701 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
53 id 22 . . . . . . . . . . 11 (𝐵 = {𝑏} → 𝐵 = {𝑏})
5453raleqdv 3138 . . . . . . . . . . 11 (𝐵 = {𝑏} → (∀𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ ∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
5553, 54raleqbidv 3146 . . . . . . . . . 10 (𝐵 = {𝑏} → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ ∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
5655adantl 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))))
57 vex 3194 . . . . . . . . . 10 𝑏 ∈ V
58 oveq1 6612 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝑎(+g𝑇)𝑐) = (𝑏(+g𝑇)𝑐))
5958fveq2d 6154 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝐻‘(𝑎(+g𝑇)𝑐)) = (𝐻‘(𝑏(+g𝑇)𝑐)))
60 fveq2 6150 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝐻𝑎) = (𝐻𝑏))
6160oveq1d 6620 . . . . . . . . . . . 12 (𝑎 = 𝑏 → ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑐)))
6259, 61eqeq12d 2641 . . . . . . . . . . 11 (𝑎 = 𝑏 → ((𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑐))))
63 oveq2 6613 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (𝑏(+g𝑇)𝑐) = (𝑏(+g𝑇)𝑏))
6463fveq2d 6154 . . . . . . . . . . . 12 (𝑐 = 𝑏 → (𝐻‘(𝑏(+g𝑇)𝑐)) = (𝐻‘(𝑏(+g𝑇)𝑏)))
65 fveq2 6150 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (𝐻𝑐) = (𝐻𝑏))
6665oveq2d 6621 . . . . . . . . . . . 12 (𝑐 = 𝑏 → ((𝐻𝑏)(+g𝑆)(𝐻𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
6764, 66eqeq12d 2641 . . . . . . . . . . 11 (𝑐 = 𝑏 → ((𝐻‘(𝑏(+g𝑇)𝑐)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏))))
6862, 672ralsng 4196 . . . . . . . . . 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 707 . . . . . . . . 9 (∀𝑎 ∈ {𝑏}∀𝑐 ∈ {𝑏} (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏)))
7056, 69syl6bb 276 . . . . . . . 8 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)) ↔ (𝐻‘(𝑏(+g𝑇)𝑏)) = ((𝐻𝑏)(+g𝑆)(𝐻𝑏))))
7152, 70mpbird 247 . . . . . . 7 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))
7217, 71jca 554 . . . . . 6 (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) ∧ 𝐵 = {𝑏}) → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
7372ex 450 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (𝐵 = {𝑏} → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
7473exlimdv 1863 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → (∃𝑏 𝐵 = {𝑏} → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
759, 74syl5bi 232 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm) → ((#‘𝐵) = 1 → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
76753impia 1258 . 2 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (#‘𝐵) = 1) → (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐))))
775, 10, 38, 28ismgmhm 41044 . 2 (𝐻 ∈ (𝑇 MgmHom 𝑆) ↔ ((𝑇 ∈ Mgm ∧ 𝑆 ∈ Mgm) ∧ (𝐻:𝐵⟶(Base‘𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(+g𝑇)𝑐)) = ((𝐻𝑎)(+g𝑆)(𝐻𝑐)))))
784, 76, 77sylanbrc 697 1 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (#‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1992  wral 2912  Vcvv 3191  {csn 4153  cmpt 4678  wf 5846  cfv 5850  (class class class)co 6605  1c1 9882  #chash 13054  Basecbs 15776  +gcplusg 15857  0gc0g 16016  Mgmcmgm 17156  Mndcmnd 17210   MgmHom cmgmhm 41038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-hash 13055  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-mgmhm 41040
This theorem is referenced by:  c0snmhm  41176
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