Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ismhmd Structured version   Visualization version   GIF version

Theorem ismhmd 40238
Description: Deduction version of ismhm 18432. (Contributed by SN, 27-Jul-2024.)
Hypotheses
Ref Expression
ismhmd.b 𝐵 = (Base‘𝑆)
ismhmd.c 𝐶 = (Base‘𝑇)
ismhmd.p + = (+g𝑆)
ismhmd.q = (+g𝑇)
ismhmd.0 0 = (0g𝑆)
ismhmd.z 𝑍 = (0g𝑇)
ismhmd.s (𝜑𝑆 ∈ Mnd)
ismhmd.t (𝜑𝑇 ∈ Mnd)
ismhmd.f (𝜑𝐹:𝐵𝐶)
ismhmd.a ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
ismhmd.h (𝜑 → (𝐹0 ) = 𝑍)
Assertion
Ref Expression
ismhmd (𝜑𝐹 ∈ (𝑆 MndHom 𝑇))
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   + (𝑥,𝑦)   (𝑥,𝑦)   0 (𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem ismhmd
StepHypRef Expression
1 ismhmd.s . 2 (𝜑𝑆 ∈ Mnd)
2 ismhmd.t . 2 (𝜑𝑇 ∈ Mnd)
3 ismhmd.f . . 3 (𝜑𝐹:𝐵𝐶)
4 ismhmd.a . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
54ralrimivva 3123 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
6 ismhmd.h . . 3 (𝜑 → (𝐹0 ) = 𝑍)
73, 5, 63jca 1127 . 2 (𝜑 → (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑍))
8 ismhmd.b . . 3 𝐵 = (Base‘𝑆)
9 ismhmd.c . . 3 𝐶 = (Base‘𝑇)
10 ismhmd.p . . 3 + = (+g𝑆)
11 ismhmd.q . . 3 = (+g𝑇)
12 ismhmd.0 . . 3 0 = (0g𝑆)
13 ismhmd.z . . 3 𝑍 = (0g𝑇)
148, 9, 10, 11, 12, 13ismhm 18432 . 2 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑍)))
151, 2, 7, 14syl21anbrc 1343 1 (𝜑𝐹 ∈ (𝑆 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wf 6429  cfv 6433  (class class class)co 7275  Basecbs 16912  +gcplusg 16962  0gc0g 17150  Mndcmnd 18385   MndHom cmhm 18428
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-mhm 18430
This theorem is referenced by:  pwspjmhmmgpd  40267  mhphflem  40284
  Copyright terms: Public domain W3C validator