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Theorem mhmmulg 18989
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 fvoveq1 7427 . . . . . 6 (𝑛 = 0 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(0 · 𝑋)))
2 oveq1 7411 . . . . . 6 (𝑛 = 0 → (𝑛 × (𝐹𝑋)) = (0 × (𝐹𝑋)))
31, 2eqeq12d 2749 . . . . 5 (𝑛 = 0 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘(0 · 𝑋)) = (0 × (𝐹𝑋))))
43imbi2d 341 . . . 4 (𝑛 = 0 → (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋))) ↔ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0 · 𝑋)) = (0 × (𝐹𝑋)))))
5 fvoveq1 7427 . . . . . 6 (𝑛 = 𝑚 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑚 · 𝑋)))
6 oveq1 7411 . . . . . 6 (𝑛 = 𝑚 → (𝑛 × (𝐹𝑋)) = (𝑚 × (𝐹𝑋)))
75, 6eqeq12d 2749 . . . . 5 (𝑛 = 𝑚 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋))))
87imbi2d 341 . . . 4 (𝑛 = 𝑚 → (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋))) ↔ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋)))))
9 fvoveq1 7427 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘((𝑚 + 1) · 𝑋)))
10 oveq1 7411 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝑛 × (𝐹𝑋)) = ((𝑚 + 1) × (𝐹𝑋)))
119, 10eqeq12d 2749 . . . . 5 (𝑛 = (𝑚 + 1) → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋))))
1211imbi2d 341 . . . 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)))))
13 fvoveq1 7427 . . . . . 6 (𝑛 = 𝑁 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑁 · 𝑋)))
14 oveq1 7411 . . . . . 6 (𝑛 = 𝑁 → (𝑛 × (𝐹𝑋)) = (𝑁 × (𝐹𝑋)))
1513, 14eqeq12d 2749 . . . . 5 (𝑛 = 𝑁 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋))))
1615imbi2d 341 . . . 4 (𝑛 = 𝑁 → (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋))) ↔ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))))
17 eqid 2733 . . . . . . 7 (0g𝐺) = (0g𝐺)
18 eqid 2733 . . . . . . 7 (0g𝐻) = (0g𝐻)
1917, 18mhm0 18676 . . . . . 6 (𝐹 ∈ (𝐺 MndHom 𝐻) → (𝐹‘(0g𝐺)) = (0g𝐻))
2019adantr 482 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0g𝐺)) = (0g𝐻))
21 mhmmulg.b . . . . . . . 8 𝐵 = (Base‘𝐺)
22 mhmmulg.s . . . . . . . 8 · = (.g𝐺)
2321, 17, 22mulg0 18951 . . . . . . 7 (𝑋𝐵 → (0 · 𝑋) = (0g𝐺))
2423adantl 483 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (0 · 𝑋) = (0g𝐺))
2524fveq2d 6892 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0 · 𝑋)) = (𝐹‘(0g𝐺)))
26 eqid 2733 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
2721, 26mhmf 18673 . . . . . . 7 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:𝐵⟶(Base‘𝐻))
2827ffvelcdmda 7082 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ (Base‘𝐻))
29 mhmmulg.t . . . . . . 7 × = (.g𝐻)
3026, 18, 29mulg0 18951 . . . . . 6 ((𝐹𝑋) ∈ (Base‘𝐻) → (0 × (𝐹𝑋)) = (0g𝐻))
3128, 30syl 17 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (0 × (𝐹𝑋)) = (0g𝐻))
3220, 25, 313eqtr4d 2783 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0 · 𝑋)) = (0 × (𝐹𝑋)))
33 oveq1 7411 . . . . . . 7 ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋)) → ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋)))
34 mhmrcl1 18671 . . . . . . . . . . . 12 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐺 ∈ Mnd)
3534ad2antrr 725 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐺 ∈ Mnd)
36 simpr 486 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
37 simplr 768 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝑋𝐵)
38 eqid 2733 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
3921, 22, 38mulgnn0p1 18959 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑚 ∈ ℕ0𝑋𝐵) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g𝐺)𝑋))
4035, 36, 37, 39syl3anc 1372 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g𝐺)𝑋))
4140fveq2d 6892 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 + 1) · 𝑋)) = (𝐹‘((𝑚 · 𝑋)(+g𝐺)𝑋)))
42 simpll 766 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐹 ∈ (𝐺 MndHom 𝐻))
4334ad2antrr 725 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → 𝐺 ∈ Mnd)
44 simplr 768 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → 𝑚 ∈ ℕ0)
45 simpr 486 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → 𝑋𝐵)
4621, 22, 43, 44, 45mulgnn0cld 18969 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
4746an32s 651 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝑚 · 𝑋) ∈ 𝐵)
48 eqid 2733 . . . . . . . . . . 11 (+g𝐻) = (+g𝐻)
4921, 38, 48mhmlin 18675 . . . . . . . . . 10 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ (𝑚 · 𝑋) ∈ 𝐵𝑋𝐵) → (𝐹‘((𝑚 · 𝑋)(+g𝐺)𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)))
5042, 47, 37, 49syl3anc 1372 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 · 𝑋)(+g𝐺)𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)))
5141, 50eqtrd 2773 . . . . . . . 8 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)))
52 mhmrcl2 18672 . . . . . . . . . 10 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐻 ∈ Mnd)
5352ad2antrr 725 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐻 ∈ Mnd)
5428adantr 482 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹𝑋) ∈ (Base‘𝐻))
5526, 29, 48mulgnn0p1 18959 . . . . . . . . 9 ((𝐻 ∈ Mnd ∧ 𝑚 ∈ ℕ0 ∧ (𝐹𝑋) ∈ (Base‘𝐻)) → ((𝑚 + 1) × (𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋)))
5653, 36, 54, 55syl3anc 1372 . . . . . . . 8 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) × (𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋)))
5751, 56eqeq12d 2749 . . . . . . 7 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋)) ↔ ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋))))
5833, 57imbitrrid 245 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋)) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋))))
5958expcom 415 . . . . 5 (𝑚 ∈ ℕ0 → ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋)) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋)))))
6059a2d 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)))))
614, 8, 12, 16, 32, 60nn0ind 12653 . . 3 (𝑁 ∈ ℕ0 → ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋))))
62613impib 1117 . 2 ((𝑁 ∈ ℕ0𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))
63623com12 1124 1 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  cfv 6540  (class class class)co 7404  0cc0 11106  1c1 11107   + caddc 11109  0cn0 12468  Basecbs 17140  +gcplusg 17193  0gc0g 17381  Mndcmnd 18621   MndHom cmhm 18665  .gcmg 18944
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 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-seq 13963  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-mulg 18945
This theorem is referenced by:  pwsmulg  18993  ghmmulg  19098  pwsexpg  20132  evlspw  21638  evls1pw  21827  evl1expd  21846  cayhamlem4  22372  dchrfi  26738  lgsqrlem1  26829  lgseisenlem4  26861  dchrisum0flblem1  26991  fermltlchr  32447  znfermltl  32448  ply1fermltlchr  32609  selvvvval  41107  evlselv  41109
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