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Theorem ismhmd 18609
Description: Deduction version of ismhm 18608. (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 3194 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
6 ismhmd.h . . 3 (𝜑 → (𝐹0 ) = 𝑍)
73, 5, 63jca 1129 . 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 18608 . 2 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑍)))
151, 2, 7, 14syl21anbrc 1345 1 (𝜑𝐹 ∈ (𝑆 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wf 6493  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  Mndcmnd 18561   MndHom cmhm 18604
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-mhm 18606
This theorem is referenced by:  pwspjmhmmgpd  20048  mhphflem  40813
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