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Theorem mhmmulg 17523
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 6622 . . . . . . 7 (𝑛 = 0 → (𝑛 · 𝑋) = (0 · 𝑋))
21fveq2d 6162 . . . . . 6 (𝑛 = 0 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(0 · 𝑋)))
3 oveq1 6622 . . . . . 6 (𝑛 = 0 → (𝑛 × (𝐹𝑋)) = (0 × (𝐹𝑋)))
42, 3eqeq12d 2636 . . . . 5 (𝑛 = 0 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘(0 · 𝑋)) = (0 × (𝐹𝑋))))
54imbi2d 330 . . . 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 6622 . . . . . . 7 (𝑛 = 𝑚 → (𝑛 · 𝑋) = (𝑚 · 𝑋))
76fveq2d 6162 . . . . . 6 (𝑛 = 𝑚 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑚 · 𝑋)))
8 oveq1 6622 . . . . . 6 (𝑛 = 𝑚 → (𝑛 × (𝐹𝑋)) = (𝑚 × (𝐹𝑋)))
97, 8eqeq12d 2636 . . . . 5 (𝑛 = 𝑚 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋))))
109imbi2d 330 . . . 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 6622 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑛 · 𝑋) = ((𝑚 + 1) · 𝑋))
1211fveq2d 6162 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘((𝑚 + 1) · 𝑋)))
13 oveq1 6622 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝑛 × (𝐹𝑋)) = ((𝑚 + 1) × (𝐹𝑋)))
1412, 13eqeq12d 2636 . . . . 5 (𝑛 = (𝑚 + 1) → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋))))
1514imbi2d 330 . . . 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 6622 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 · 𝑋) = (𝑁 · 𝑋))
1716fveq2d 6162 . . . . . 6 (𝑛 = 𝑁 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑁 · 𝑋)))
18 oveq1 6622 . . . . . 6 (𝑛 = 𝑁 → (𝑛 × (𝐹𝑋)) = (𝑁 × (𝐹𝑋)))
1917, 18eqeq12d 2636 . . . . 5 (𝑛 = 𝑁 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋))))
2019imbi2d 330 . . . 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 2621 . . . . . . 7 (0g𝐺) = (0g𝐺)
22 eqid 2621 . . . . . . 7 (0g𝐻) = (0g𝐻)
2321, 22mhm0 17283 . . . . . 6 (𝐹 ∈ (𝐺 MndHom 𝐻) → (𝐹‘(0g𝐺)) = (0g𝐻))
2423adantr 481 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0g𝐺)) = (0g𝐻))
25 mhmmulg.b . . . . . . . 8 𝐵 = (Base‘𝐺)
26 mhmmulg.s . . . . . . . 8 · = (.g𝐺)
2725, 21, 26mulg0 17486 . . . . . . 7 (𝑋𝐵 → (0 · 𝑋) = (0g𝐺))
2827adantl 482 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (0 · 𝑋) = (0g𝐺))
2928fveq2d 6162 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0 · 𝑋)) = (𝐹‘(0g𝐺)))
30 eqid 2621 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
3125, 30mhmf 17280 . . . . . . 7 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:𝐵⟶(Base‘𝐻))
3231ffvelrnda 6325 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ (Base‘𝐻))
33 mhmmulg.t . . . . . . 7 × = (.g𝐻)
3430, 22, 33mulg0 17486 . . . . . 6 ((𝐹𝑋) ∈ (Base‘𝐻) → (0 × (𝐹𝑋)) = (0g𝐻))
3532, 34syl 17 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (0 × (𝐹𝑋)) = (0g𝐻))
3624, 29, 353eqtr4d 2665 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0 · 𝑋)) = (0 × (𝐹𝑋)))
37 oveq1 6622 . . . . . . 7 ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋)) → ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋)))
38 mhmrcl1 17278 . . . . . . . . . . . 12 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐺 ∈ Mnd)
3938ad2antrr 761 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐺 ∈ Mnd)
40 simpr 477 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
41 simplr 791 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝑋𝐵)
42 eqid 2621 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
4325, 26, 42mulgnn0p1 17492 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑚 ∈ ℕ0𝑋𝐵) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g𝐺)𝑋))
4439, 40, 41, 43syl3anc 1323 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g𝐺)𝑋))
4544fveq2d 6162 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 + 1) · 𝑋)) = (𝐹‘((𝑚 · 𝑋)(+g𝐺)𝑋)))
46 simpll 789 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐹 ∈ (𝐺 MndHom 𝐻))
4738ad2antrr 761 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → 𝐺 ∈ Mnd)
48 simplr 791 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → 𝑚 ∈ ℕ0)
49 simpr 477 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → 𝑋𝐵)
5025, 26mulgnn0cl 17498 . . . . . . . . . . . 12 ((𝐺 ∈ Mnd ∧ 𝑚 ∈ ℕ0𝑋𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
5147, 48, 49, 50syl3anc 1323 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
5251an32s 845 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝑚 · 𝑋) ∈ 𝐵)
53 eqid 2621 . . . . . . . . . . 11 (+g𝐻) = (+g𝐻)
5425, 42, 53mhmlin 17282 . . . . . . . . . 10 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ (𝑚 · 𝑋) ∈ 𝐵𝑋𝐵) → (𝐹‘((𝑚 · 𝑋)(+g𝐺)𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)))
5546, 52, 41, 54syl3anc 1323 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 · 𝑋)(+g𝐺)𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)))
5645, 55eqtrd 2655 . . . . . . . 8 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)))
57 mhmrcl2 17279 . . . . . . . . . 10 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐻 ∈ Mnd)
5857ad2antrr 761 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐻 ∈ Mnd)
5932adantr 481 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹𝑋) ∈ (Base‘𝐻))
6030, 33, 53mulgnn0p1 17492 . . . . . . . . 9 ((𝐻 ∈ Mnd ∧ 𝑚 ∈ ℕ0 ∧ (𝐹𝑋) ∈ (Base‘𝐻)) → ((𝑚 + 1) × (𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋)))
6158, 40, 59, 60syl3anc 1323 . . . . . . . 8 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) × (𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋)))
6256, 61eqeq12d 2636 . . . . . . 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 451 . . . . 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 11432 . . 3 (𝑁 ∈ ℕ0 → ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋))))
67663impib 1259 . 2 ((𝑁 ∈ ℕ0𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))
68673com12 1266 1 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  cfv 5857  (class class class)co 6615  0cc0 9896  1c1 9897   + caddc 9899  0cn0 11252  Basecbs 15800  +gcplusg 15881  0gc0g 16040  Mndcmnd 17234   MndHom cmhm 17273  .gcmg 17480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-seq 12758  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-mhm 17275  df-mulg 17481
This theorem is referenced by:  pwsmulg  17527  ghmmulg  17612  evls1varpw  19631  evl1expd  19649  cayhamlem4  20633  dchrfi  24914  lgsqrlem1  25005  lgseisenlem4  25037  dchrisum0flblem1  25131
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