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Theorem ismhm 18673
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 18671 . . 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 7648 . 2 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd))
3 fveq2 6892 . . . . . . . 8 (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
4 ismhm.c . . . . . . . 8 𝐶 = (Base‘𝑇)
53, 4eqtr4di 2791 . . . . . . 7 (𝑡 = 𝑇 → (Base‘𝑡) = 𝐶)
6 fveq2 6892 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
7 ismhm.b . . . . . . . 8 𝐵 = (Base‘𝑆)
86, 7eqtr4di 2791 . . . . . . 7 (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
95, 8oveqan12rd 7429 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → ((Base‘𝑡) ↑m (Base‘𝑠)) = (𝐶m 𝐵))
108adantr 482 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = 𝐵)
11 fveq2 6892 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → (+g𝑠) = (+g𝑆))
12 ismhm.p . . . . . . . . . . . . 13 + = (+g𝑆)
1311, 12eqtr4di 2791 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (+g𝑠) = + )
1413oveqd 7426 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑥(+g𝑠)𝑦) = (𝑥 + 𝑦))
1514fveq2d 6896 . . . . . . . . . 10 (𝑠 = 𝑆 → (𝑓‘(𝑥(+g𝑠)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
16 fveq2 6892 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (+g𝑡) = (+g𝑇))
17 ismhm.q . . . . . . . . . . . 12 = (+g𝑇)
1816, 17eqtr4di 2791 . . . . . . . . . . 11 (𝑡 = 𝑇 → (+g𝑡) = )
1918oveqd 7426 . . . . . . . . . 10 (𝑡 = 𝑇 → ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) = ((𝑓𝑥) (𝑓𝑦)))
2015, 19eqeqan12d 2747 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
2110, 20raleqbidv 3343 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ ∀𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
2210, 21raleqbidv 3343 . . . . . . 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 6892 . . . . . . . . . 10 (𝑠 = 𝑆 → (0g𝑠) = (0g𝑆))
24 ismhm.z . . . . . . . . . 10 0 = (0g𝑆)
2523, 24eqtr4di 2791 . . . . . . . . 9 (𝑠 = 𝑆 → (0g𝑠) = 0 )
2625fveq2d 6896 . . . . . . . 8 (𝑠 = 𝑆 → (𝑓‘(0g𝑠)) = (𝑓0 ))
27 fveq2 6892 . . . . . . . . 9 (𝑡 = 𝑇 → (0g𝑡) = (0g𝑇))
28 ismhm.y . . . . . . . . 9 𝑌 = (0g𝑇)
2927, 28eqtr4di 2791 . . . . . . . 8 (𝑡 = 𝑇 → (0g𝑡) = 𝑌)
3026, 29eqeqan12d 2747 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑓‘(0g𝑠)) = (0g𝑡) ↔ (𝑓0 ) = 𝑌))
3122, 30anbi12d 632 . . . . . 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 3450 . . . . 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 7442 . . . . . 6 (𝐶m 𝐵) ∈ V
3433rabex 5333 . . . . 5 {𝑓 ∈ (𝐶m 𝐵) ∣ (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌)} ∈ V
3532, 1, 34ovmpoa 7563 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 MndHom 𝑇) = {𝑓 ∈ (𝐶m 𝐵) ∣ (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌)})
3635eleq2d 2820 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝐶m 𝐵) ∣ (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌)}))
374fvexi 6906 . . . . . 6 𝐶 ∈ V
387fvexi 6906 . . . . . 6 𝐵 ∈ V
3937, 38elmap 8865 . . . . 5 (𝐹 ∈ (𝐶m 𝐵) ↔ 𝐹:𝐵𝐶)
4039anbi1i 625 . . . 4 ((𝐹 ∈ (𝐶m 𝐵) ∧ (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)) ↔ (𝐹:𝐵𝐶 ∧ (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
41 fveq1 6891 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
42 fveq1 6891 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
43 fveq1 6891 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
4442, 43oveq12d 7427 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥) (𝑓𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
4541, 44eqeq12d 2749 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
46452ralbidv 3219 . . . . . 6 (𝑓 = 𝐹 → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
47 fveq1 6891 . . . . . . 7 (𝑓 = 𝐹 → (𝑓0 ) = (𝐹0 ))
4847eqeq1d 2735 . . . . . 6 (𝑓 = 𝐹 → ((𝑓0 ) = 𝑌 ↔ (𝐹0 ) = 𝑌))
4946, 48anbi12d 632 . . . . 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 1096 . . . 4 ((𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌) ↔ (𝐹:𝐵𝐶 ∧ (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
5240, 50, 513bitr4i 303 . . 3 (𝐹 ∈ {𝑓 ∈ (𝐶m 𝐵) ∣ (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌)} ↔ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌))
5336, 52bitrdi 287 . 2 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
542, 53biadanii 821 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  {crab 3433  wf 6540  cfv 6544  (class class class)co 7409  m cmap 8820  Basecbs 17144  +gcplusg 17197  0gc0g 17385  Mndcmnd 18625   MndHom cmhm 18669
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
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-mhm 18671
This theorem is referenced by:  ismhmd  18674  mhmf  18677  mhmpropd  18678  mhmlin  18679  mhm0  18680  idmhm  18681  mhmf1o  18682  0mhm  18700  resmhm  18701  resmhm2  18702  resmhm2b  18703  mhmco  18704  prdspjmhm  18710  pwsdiagmhm  18712  pwsco1mhm  18713  pwsco2mhm  18714  frmdup1  18745  mhmfmhm  18948  ghmmhm  19102  frgpmhm  19633  mulgmhm  19695  srglmhm  20044  srgrmhm  20045  dfrhm2  20253  isrhm2d  20265  expmhm  21014  mat1mhm  21986  scmatmhm  22036  mat2pmatmhm  22235  pm2mpmhm  22322  dchrelbas3  26741  xrge0iifmhm  32919  esumcocn  33078  elmrsubrn  34511  deg1mhm  41949  ismhm0  46575  mhmismgmhm  46576  c0mhm  46709
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