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Theorem ismhm 17946
Description: Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
Hypotheses
Ref Expression
ismhm.b 𝐵 = (Base‘𝑆)
ismhm.c 𝐶 = (Base‘𝑇)
ismhm.p + = (+g𝑆)
ismhm.q = (+g𝑇)
ismhm.z 0 = (0g𝑆)
ismhm.y 𝑌 = (0g𝑇)
Assertion
Ref Expression
ismhm (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝑥,𝐹,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   + (𝑥,𝑦)   (𝑥,𝑦)   𝑌(𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem ismhm
Dummy variables 𝑓 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mhm 17944 . . 3 MndHom = (𝑠 ∈ Mnd, 𝑡 ∈ Mnd ↦ {𝑓 ∈ ((Base‘𝑡) ↑m (Base‘𝑠)) ∣ (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ∧ (𝑓‘(0g𝑠)) = (0g𝑡))})
21elmpocl 7377 . 2 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd))
3 fveq2 6667 . . . . . . . 8 (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
4 ismhm.c . . . . . . . 8 𝐶 = (Base‘𝑇)
53, 4syl6eqr 2879 . . . . . . 7 (𝑡 = 𝑇 → (Base‘𝑡) = 𝐶)
6 fveq2 6667 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
7 ismhm.b . . . . . . . 8 𝐵 = (Base‘𝑆)
86, 7syl6eqr 2879 . . . . . . 7 (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
95, 8oveqan12rd 7168 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → ((Base‘𝑡) ↑m (Base‘𝑠)) = (𝐶m 𝐵))
108adantr 481 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = 𝐵)
11 fveq2 6667 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → (+g𝑠) = (+g𝑆))
12 ismhm.p . . . . . . . . . . . . 13 + = (+g𝑆)
1311, 12syl6eqr 2879 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (+g𝑠) = + )
1413oveqd 7165 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑥(+g𝑠)𝑦) = (𝑥 + 𝑦))
1514fveq2d 6671 . . . . . . . . . 10 (𝑠 = 𝑆 → (𝑓‘(𝑥(+g𝑠)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
16 fveq2 6667 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (+g𝑡) = (+g𝑇))
17 ismhm.q . . . . . . . . . . . 12 = (+g𝑇)
1816, 17syl6eqr 2879 . . . . . . . . . . 11 (𝑡 = 𝑇 → (+g𝑡) = )
1918oveqd 7165 . . . . . . . . . 10 (𝑡 = 𝑇 → ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) = ((𝑓𝑥) (𝑓𝑦)))
2015, 19eqeqan12d 2843 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
2110, 20raleqbidv 3407 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ ∀𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
2210, 21raleqbidv 3407 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
23 fveq2 6667 . . . . . . . . . 10 (𝑠 = 𝑆 → (0g𝑠) = (0g𝑆))
24 ismhm.z . . . . . . . . . 10 0 = (0g𝑆)
2523, 24syl6eqr 2879 . . . . . . . . 9 (𝑠 = 𝑆 → (0g𝑠) = 0 )
2625fveq2d 6671 . . . . . . . 8 (𝑠 = 𝑆 → (𝑓‘(0g𝑠)) = (𝑓0 ))
27 fveq2 6667 . . . . . . . . 9 (𝑡 = 𝑇 → (0g𝑡) = (0g𝑇))
28 ismhm.y . . . . . . . . 9 𝑌 = (0g𝑇)
2927, 28syl6eqr 2879 . . . . . . . 8 (𝑡 = 𝑇 → (0g𝑡) = 𝑌)
3026, 29eqeqan12d 2843 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑓‘(0g𝑠)) = (0g𝑡) ↔ (𝑓0 ) = 𝑌))
3122, 30anbi12d 630 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → ((∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ∧ (𝑓‘(0g𝑠)) = (0g𝑡)) ↔ (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌)))
329, 31rabeqbidv 3491 . . . . 5 ((𝑠 = 𝑆𝑡 = 𝑇) → {𝑓 ∈ ((Base‘𝑡) ↑m (Base‘𝑠)) ∣ (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ∧ (𝑓‘(0g𝑠)) = (0g𝑡))} = {𝑓 ∈ (𝐶m 𝐵) ∣ (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌)})
33 ovex 7181 . . . . . 6 (𝐶m 𝐵) ∈ V
3433rabex 5232 . . . . 5 {𝑓 ∈ (𝐶m 𝐵) ∣ (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌)} ∈ V
3532, 1, 34ovmpoa 7295 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 MndHom 𝑇) = {𝑓 ∈ (𝐶m 𝐵) ∣ (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌)})
3635eleq2d 2903 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝐶m 𝐵) ∣ (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌)}))
374fvexi 6681 . . . . . 6 𝐶 ∈ V
387fvexi 6681 . . . . . 6 𝐵 ∈ V
3937, 38elmap 8425 . . . . 5 (𝐹 ∈ (𝐶m 𝐵) ↔ 𝐹:𝐵𝐶)
4039anbi1i 623 . . . 4 ((𝐹 ∈ (𝐶m 𝐵) ∧ (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)) ↔ (𝐹:𝐵𝐶 ∧ (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
41 fveq1 6666 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
42 fveq1 6666 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
43 fveq1 6666 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
4442, 43oveq12d 7166 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥) (𝑓𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
4541, 44eqeq12d 2842 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
46452ralbidv 3204 . . . . . 6 (𝑓 = 𝐹 → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
47 fveq1 6666 . . . . . . 7 (𝑓 = 𝐹 → (𝑓0 ) = (𝐹0 ))
4847eqeq1d 2828 . . . . . 6 (𝑓 = 𝐹 → ((𝑓0 ) = 𝑌 ↔ (𝐹0 ) = 𝑌))
4946, 48anbi12d 630 . . . . 5 (𝑓 = 𝐹 → ((∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌) ↔ (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
5049elrab 3684 . . . 4 (𝐹 ∈ {𝑓 ∈ (𝐶m 𝐵) ∣ (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌)} ↔ (𝐹 ∈ (𝐶m 𝐵) ∧ (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
51 3anass 1089 . . . 4 ((𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌) ↔ (𝐹:𝐵𝐶 ∧ (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
5240, 50, 513bitr4i 304 . . 3 (𝐹 ∈ {𝑓 ∈ (𝐶m 𝐵) ∣ (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌)} ↔ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌))
5336, 52syl6bb 288 . 2 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
542, 53biadanii 819 1 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3143  {crab 3147  wf 6348  cfv 6352  (class class class)co 7148  m cmap 8396  Basecbs 16473  +gcplusg 16555  0gc0g 16703  Mndcmnd 17900   MndHom cmhm 17942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8398  df-mhm 17944
This theorem is referenced by:  mhmf  17949  mhmpropd  17950  mhmlin  17951  mhm0  17952  idmhm  17953  mhmf1o  17954  0mhm  17967  resmhm  17968  resmhm2  17969  resmhm2b  17970  mhmco  17971  prdspjmhm  17976  pwsdiagmhm  17978  pwsco1mhm  17979  pwsco2mhm  17980  frmdup1  18012  mhmfmhm  18152  ghmmhm  18298  frgpmhm  18811  mulgmhm  18868  srglmhm  19205  srgrmhm  19206  dfrhm2  19389  isrhm2d  19400  expmhm  20530  mat1mhm  21009  scmatmhm  21059  mat2pmatmhm  21257  pm2mpmhm  21344  dchrelbas3  25728  xrge0iifmhm  31068  esumcocn  31225  elmrsubrn  32651  deg1mhm  39672  ismhm0  43904  mhmismgmhm  43905  c0mhm  44013
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