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Theorem ismhmd 18712
Description: Deduction version of ismhm 18711. (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 3192 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
6 ismhmd.h . . 3 (𝜑 → (𝐹0 ) = 𝑍)
73, 5, 63jca 1125 . 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 18711 . 2 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑍)))
151, 2, 7, 14syl21anbrc 1341 1 (𝜑𝐹 ∈ (𝑆 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053  wf 6530  cfv 6534  (class class class)co 7402  Basecbs 17149  +gcplusg 17202  0gc0g 17390  Mndcmnd 18663   MndHom cmhm 18707
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-mhm 18709
This theorem is referenced by:  pwspjmhmmgpd  20223  imasmhm  32962  mhphflem  41699
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