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Theorem mhmmulg 17804
 Description: A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
mhmmulg.b 𝐵 = (Base‘𝐺)
mhmmulg.s · = (.g𝐺)
mhmmulg.t × = (.g𝐻)
Assertion
Ref Expression
mhmmulg ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))

Proof of Theorem mhmmulg
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6821 . . . . . . 7 (𝑛 = 0 → (𝑛 · 𝑋) = (0 · 𝑋))
21fveq2d 6357 . . . . . 6 (𝑛 = 0 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(0 · 𝑋)))
3 oveq1 6821 . . . . . 6 (𝑛 = 0 → (𝑛 × (𝐹𝑋)) = (0 × (𝐹𝑋)))
42, 3eqeq12d 2775 . . . . 5 (𝑛 = 0 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘(0 · 𝑋)) = (0 × (𝐹𝑋))))
54imbi2d 329 . . . 4 (𝑛 = 0 → (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋))) ↔ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0 · 𝑋)) = (0 × (𝐹𝑋)))))
6 oveq1 6821 . . . . . . 7 (𝑛 = 𝑚 → (𝑛 · 𝑋) = (𝑚 · 𝑋))
76fveq2d 6357 . . . . . 6 (𝑛 = 𝑚 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑚 · 𝑋)))
8 oveq1 6821 . . . . . 6 (𝑛 = 𝑚 → (𝑛 × (𝐹𝑋)) = (𝑚 × (𝐹𝑋)))
97, 8eqeq12d 2775 . . . . 5 (𝑛 = 𝑚 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋))))
109imbi2d 329 . . . 4 (𝑛 = 𝑚 → (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋))) ↔ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋)))))
11 oveq1 6821 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑛 · 𝑋) = ((𝑚 + 1) · 𝑋))
1211fveq2d 6357 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘((𝑚 + 1) · 𝑋)))
13 oveq1 6821 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝑛 × (𝐹𝑋)) = ((𝑚 + 1) × (𝐹𝑋)))
1412, 13eqeq12d 2775 . . . . 5 (𝑛 = (𝑚 + 1) → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋))))
1514imbi2d 329 . . . 4 (𝑛 = (𝑚 + 1) → (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋))) ↔ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋)))))
16 oveq1 6821 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 · 𝑋) = (𝑁 · 𝑋))
1716fveq2d 6357 . . . . . 6 (𝑛 = 𝑁 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑁 · 𝑋)))
18 oveq1 6821 . . . . . 6 (𝑛 = 𝑁 → (𝑛 × (𝐹𝑋)) = (𝑁 × (𝐹𝑋)))
1917, 18eqeq12d 2775 . . . . 5 (𝑛 = 𝑁 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋))))
2019imbi2d 329 . . . 4 (𝑛 = 𝑁 → (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋))) ↔ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))))
21 eqid 2760 . . . . . . 7 (0g𝐺) = (0g𝐺)
22 eqid 2760 . . . . . . 7 (0g𝐻) = (0g𝐻)
2321, 22mhm0 17564 . . . . . 6 (𝐹 ∈ (𝐺 MndHom 𝐻) → (𝐹‘(0g𝐺)) = (0g𝐻))
2423adantr 472 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0g𝐺)) = (0g𝐻))
25 mhmmulg.b . . . . . . . 8 𝐵 = (Base‘𝐺)
26 mhmmulg.s . . . . . . . 8 · = (.g𝐺)
2725, 21, 26mulg0 17767 . . . . . . 7 (𝑋𝐵 → (0 · 𝑋) = (0g𝐺))
2827adantl 473 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (0 · 𝑋) = (0g𝐺))
2928fveq2d 6357 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0 · 𝑋)) = (𝐹‘(0g𝐺)))
30 eqid 2760 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
3125, 30mhmf 17561 . . . . . . 7 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:𝐵⟶(Base‘𝐻))
3231ffvelrnda 6523 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ (Base‘𝐻))
33 mhmmulg.t . . . . . . 7 × = (.g𝐻)
3430, 22, 33mulg0 17767 . . . . . 6 ((𝐹𝑋) ∈ (Base‘𝐻) → (0 × (𝐹𝑋)) = (0g𝐻))
3532, 34syl 17 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (0 × (𝐹𝑋)) = (0g𝐻))
3624, 29, 353eqtr4d 2804 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0 · 𝑋)) = (0 × (𝐹𝑋)))
37 oveq1 6821 . . . . . . 7 ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋)) → ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋)))
38 mhmrcl1 17559 . . . . . . . . . . . 12 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐺 ∈ Mnd)
3938ad2antrr 764 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐺 ∈ Mnd)
40 simpr 479 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
41 simplr 809 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝑋𝐵)
42 eqid 2760 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
4325, 26, 42mulgnn0p1 17773 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑚 ∈ ℕ0𝑋𝐵) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g𝐺)𝑋))
4439, 40, 41, 43syl3anc 1477 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g𝐺)𝑋))
4544fveq2d 6357 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 + 1) · 𝑋)) = (𝐹‘((𝑚 · 𝑋)(+g𝐺)𝑋)))
46 simpll 807 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐹 ∈ (𝐺 MndHom 𝐻))
4738ad2antrr 764 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → 𝐺 ∈ Mnd)
48 simplr 809 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → 𝑚 ∈ ℕ0)
49 simpr 479 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → 𝑋𝐵)
5025, 26mulgnn0cl 17779 . . . . . . . . . . . 12 ((𝐺 ∈ Mnd ∧ 𝑚 ∈ ℕ0𝑋𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
5147, 48, 49, 50syl3anc 1477 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
5251an32s 881 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝑚 · 𝑋) ∈ 𝐵)
53 eqid 2760 . . . . . . . . . . 11 (+g𝐻) = (+g𝐻)
5425, 42, 53mhmlin 17563 . . . . . . . . . 10 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ (𝑚 · 𝑋) ∈ 𝐵𝑋𝐵) → (𝐹‘((𝑚 · 𝑋)(+g𝐺)𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)))
5546, 52, 41, 54syl3anc 1477 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 · 𝑋)(+g𝐺)𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)))
5645, 55eqtrd 2794 . . . . . . . 8 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)))
57 mhmrcl2 17560 . . . . . . . . . 10 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐻 ∈ Mnd)
5857ad2antrr 764 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐻 ∈ Mnd)
5932adantr 472 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹𝑋) ∈ (Base‘𝐻))
6030, 33, 53mulgnn0p1 17773 . . . . . . . . 9 ((𝐻 ∈ Mnd ∧ 𝑚 ∈ ℕ0 ∧ (𝐹𝑋) ∈ (Base‘𝐻)) → ((𝑚 + 1) × (𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋)))
6158, 40, 59, 60syl3anc 1477 . . . . . . . 8 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) × (𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋)))
6256, 61eqeq12d 2775 . . . . . . 7 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋)) ↔ ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋))))
6337, 62syl5ibr 236 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋)) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋))))
6463expcom 450 . . . . 5 (𝑚 ∈ ℕ0 → ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋)) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋)))))
6564a2d 29 . . . 4 (𝑚 ∈ ℕ0 → (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋))) → ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋)))))
665, 10, 15, 20, 36, 65nn0ind 11684 . . 3 (𝑁 ∈ ℕ0 → ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋))))
67663impib 1109 . 2 ((𝑁 ∈ ℕ0𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))
68673com12 1118 1 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ‘cfv 6049  (class class class)co 6814  0cc0 10148  1c1 10149   + caddc 10151  ℕ0cn0 11504  Basecbs 16079  +gcplusg 16163  0gc0g 16322  Mndcmnd 17515   MndHom cmhm 17554  .gcmg 17761 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-n0 11505  df-z 11590  df-uz 11900  df-fz 12540  df-seq 13016  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-mhm 17556  df-mulg 17762 This theorem is referenced by:  pwsmulg  17808  ghmmulg  17893  evls1varpw  19913  evl1expd  19931  cayhamlem4  20915  dchrfi  25200  lgsqrlem1  25291  lgseisenlem4  25323  dchrisum0flblem1  25417
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