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Theorem ismhmd 39820
Description: Deduction version of ismhm 18074. (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 3103 . . 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 18074 . 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 399  w3a 1088   = wceq 1542  wcel 2114  wral 3053  wf 6335  cfv 6339  (class class class)co 7170  Basecbs 16586  +gcplusg 16668  0gc0g 16816  Mndcmnd 18027   MndHom cmhm 18070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-map 8439  df-mhm 18072
This theorem is referenced by:  pwspjmhmmgpd  39848  mhphflem  39863
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