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Theorem ismhm 18603
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 18601 . . 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 7595 . 2 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd))
3 fveq2 6842 . . . . . . . 8 (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
4 ismhm.c . . . . . . . 8 𝐶 = (Base‘𝑇)
53, 4eqtr4di 2794 . . . . . . 7 (𝑡 = 𝑇 → (Base‘𝑡) = 𝐶)
6 fveq2 6842 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
7 ismhm.b . . . . . . . 8 𝐵 = (Base‘𝑆)
86, 7eqtr4di 2794 . . . . . . 7 (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
95, 8oveqan12rd 7377 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → ((Base‘𝑡) ↑m (Base‘𝑠)) = (𝐶m 𝐵))
108adantr 481 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = 𝐵)
11 fveq2 6842 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → (+g𝑠) = (+g𝑆))
12 ismhm.p . . . . . . . . . . . . 13 + = (+g𝑆)
1311, 12eqtr4di 2794 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (+g𝑠) = + )
1413oveqd 7374 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑥(+g𝑠)𝑦) = (𝑥 + 𝑦))
1514fveq2d 6846 . . . . . . . . . 10 (𝑠 = 𝑆 → (𝑓‘(𝑥(+g𝑠)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
16 fveq2 6842 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (+g𝑡) = (+g𝑇))
17 ismhm.q . . . . . . . . . . . 12 = (+g𝑇)
1816, 17eqtr4di 2794 . . . . . . . . . . 11 (𝑡 = 𝑇 → (+g𝑡) = )
1918oveqd 7374 . . . . . . . . . 10 (𝑡 = 𝑇 → ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) = ((𝑓𝑥) (𝑓𝑦)))
2015, 19eqeqan12d 2750 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
2110, 20raleqbidv 3319 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ ∀𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
2210, 21raleqbidv 3319 . . . . . . 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 6842 . . . . . . . . . 10 (𝑠 = 𝑆 → (0g𝑠) = (0g𝑆))
24 ismhm.z . . . . . . . . . 10 0 = (0g𝑆)
2523, 24eqtr4di 2794 . . . . . . . . 9 (𝑠 = 𝑆 → (0g𝑠) = 0 )
2625fveq2d 6846 . . . . . . . 8 (𝑠 = 𝑆 → (𝑓‘(0g𝑠)) = (𝑓0 ))
27 fveq2 6842 . . . . . . . . 9 (𝑡 = 𝑇 → (0g𝑡) = (0g𝑇))
28 ismhm.y . . . . . . . . 9 𝑌 = (0g𝑇)
2927, 28eqtr4di 2794 . . . . . . . 8 (𝑡 = 𝑇 → (0g𝑡) = 𝑌)
3026, 29eqeqan12d 2750 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑓‘(0g𝑠)) = (0g𝑡) ↔ (𝑓0 ) = 𝑌))
3122, 30anbi12d 631 . . . . . 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 3424 . . . . 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 7390 . . . . . 6 (𝐶m 𝐵) ∈ V
3433rabex 5289 . . . . 5 {𝑓 ∈ (𝐶m 𝐵) ∣ (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌)} ∈ V
3532, 1, 34ovmpoa 7510 . . . 4 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 MndHom 𝑇) = {𝑓 ∈ (𝐶m 𝐵) ∣ (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌)})
3635eleq2d 2823 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝐶m 𝐵) ∣ (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌)}))
374fvexi 6856 . . . . . 6 𝐶 ∈ V
387fvexi 6856 . . . . . 6 𝐵 ∈ V
3937, 38elmap 8809 . . . . 5 (𝐹 ∈ (𝐶m 𝐵) ↔ 𝐹:𝐵𝐶)
4039anbi1i 624 . . . 4 ((𝐹 ∈ (𝐶m 𝐵) ∧ (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)) ↔ (𝐹:𝐵𝐶 ∧ (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
41 fveq1 6841 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
42 fveq1 6841 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
43 fveq1 6841 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
4442, 43oveq12d 7375 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥) (𝑓𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
4541, 44eqeq12d 2752 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
46452ralbidv 3212 . . . . . 6 (𝑓 = 𝐹 → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
47 fveq1 6841 . . . . . . 7 (𝑓 = 𝐹 → (𝑓0 ) = (𝐹0 ))
4847eqeq1d 2738 . . . . . 6 (𝑓 = 𝐹 → ((𝑓0 ) = 𝑌 ↔ (𝐹0 ) = 𝑌))
4946, 48anbi12d 631 . . . . 5 (𝑓 = 𝐹 → ((∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓0 ) = 𝑌) ↔ (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
5049elrab 3645 . . . 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 1095 . . . 4 ((𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌) ↔ (𝐹:𝐵𝐶 ∧ (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
5240, 50, 513bitr4i 302 . . 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 286 . 2 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
542, 53biadanii 820 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 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  {crab 3407  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  Basecbs 17083  +gcplusg 17133  0gc0g 17321  Mndcmnd 18556   MndHom cmhm 18599
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-mhm 18601
This theorem is referenced by:  ismhmd  18604  mhmf  18607  mhmpropd  18608  mhmlin  18609  mhm0  18610  idmhm  18611  mhmf1o  18612  0mhm  18630  resmhm  18631  resmhm2  18632  resmhm2b  18633  mhmco  18634  prdspjmhm  18639  pwsdiagmhm  18641  pwsco1mhm  18642  pwsco2mhm  18643  frmdup1  18674  mhmfmhm  18870  ghmmhm  19018  frgpmhm  19547  mulgmhm  19606  srglmhm  19952  srgrmhm  19953  dfrhm2  20148  isrhm2d  20160  expmhm  20866  mat1mhm  21833  scmatmhm  21883  mat2pmatmhm  22082  pm2mpmhm  22169  dchrelbas3  26586  xrge0iifmhm  32520  esumcocn  32679  elmrsubrn  34114  deg1mhm  41520  ismhm0  46089  mhmismgmhm  46090  c0mhm  46198
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