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Theorem mhmmulg 12899
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 5891 . . . . . 6 (𝑛 = 0 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(0 · 𝑋)))
2 oveq1 5875 . . . . . 6 (𝑛 = 0 → (𝑛 × (𝐹𝑋)) = (0 × (𝐹𝑋)))
31, 2eqeq12d 2192 . . . . 5 (𝑛 = 0 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘(0 · 𝑋)) = (0 × (𝐹𝑋))))
43imbi2d 230 . . . 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 5891 . . . . . 6 (𝑛 = 𝑚 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑚 · 𝑋)))
6 oveq1 5875 . . . . . 6 (𝑛 = 𝑚 → (𝑛 × (𝐹𝑋)) = (𝑚 × (𝐹𝑋)))
75, 6eqeq12d 2192 . . . . 5 (𝑛 = 𝑚 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋))))
87imbi2d 230 . . . 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 5891 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘((𝑚 + 1) · 𝑋)))
10 oveq1 5875 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝑛 × (𝐹𝑋)) = ((𝑚 + 1) × (𝐹𝑋)))
119, 10eqeq12d 2192 . . . . 5 (𝑛 = (𝑚 + 1) → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋))))
1211imbi2d 230 . . . 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 5891 . . . . . 6 (𝑛 = 𝑁 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑁 · 𝑋)))
14 oveq1 5875 . . . . . 6 (𝑛 = 𝑁 → (𝑛 × (𝐹𝑋)) = (𝑁 × (𝐹𝑋)))
1513, 14eqeq12d 2192 . . . . 5 (𝑛 = 𝑁 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋))))
1615imbi2d 230 . . . 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 2177 . . . . . . 7 (0g𝐺) = (0g𝐺)
18 eqid 2177 . . . . . . 7 (0g𝐻) = (0g𝐻)
1917, 18mhm0 12736 . . . . . 6 (𝐹 ∈ (𝐺 MndHom 𝐻) → (𝐹‘(0g𝐺)) = (0g𝐻))
2019adantr 276 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0g𝐺)) = (0g𝐻))
21 mhmmulg.b . . . . . . . 8 𝐵 = (Base‘𝐺)
22 mhmmulg.s . . . . . . . 8 · = (.g𝐺)
2321, 17, 22mulg0 12864 . . . . . . 7 (𝑋𝐵 → (0 · 𝑋) = (0g𝐺))
2423adantl 277 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (0 · 𝑋) = (0g𝐺))
2524fveq2d 5514 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0 · 𝑋)) = (𝐹‘(0g𝐺)))
26 eqid 2177 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
2721, 26mhmf 12733 . . . . . . 7 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:𝐵⟶(Base‘𝐻))
2827ffvelcdmda 5646 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ (Base‘𝐻))
29 mhmmulg.t . . . . . . 7 × = (.g𝐻)
3026, 18, 29mulg0 12864 . . . . . 6 ((𝐹𝑋) ∈ (Base‘𝐻) → (0 × (𝐹𝑋)) = (0g𝐻))
3128, 30syl 14 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (0 × (𝐹𝑋)) = (0g𝐻))
3220, 25, 313eqtr4d 2220 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0 · 𝑋)) = (0 × (𝐹𝑋)))
33 oveq1 5875 . . . . . . 7 ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋)) → ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋)))
34 mhmrcl1 12731 . . . . . . . . . . . 12 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐺 ∈ Mnd)
3534ad2antrr 488 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐺 ∈ Mnd)
36 simpr 110 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
37 simplr 528 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝑋𝐵)
38 eqid 2177 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
3921, 22, 38mulgnn0p1 12870 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑚 ∈ ℕ0𝑋𝐵) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g𝐺)𝑋))
4035, 36, 37, 39syl3anc 1238 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g𝐺)𝑋))
4140fveq2d 5514 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 + 1) · 𝑋)) = (𝐹‘((𝑚 · 𝑋)(+g𝐺)𝑋)))
42 simpll 527 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐹 ∈ (𝐺 MndHom 𝐻))
4334ad2antrr 488 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → 𝐺 ∈ Mnd)
44 simplr 528 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → 𝑚 ∈ ℕ0)
45 simpr 110 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → 𝑋𝐵)
4621, 22mulgnn0cl 12875 . . . . . . . . . . . 12 ((𝐺 ∈ Mnd ∧ 𝑚 ∈ ℕ0𝑋𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
4743, 44, 45, 46syl3anc 1238 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
4847an32s 568 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝑚 · 𝑋) ∈ 𝐵)
49 eqid 2177 . . . . . . . . . . 11 (+g𝐻) = (+g𝐻)
5021, 38, 49mhmlin 12735 . . . . . . . . . 10 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ (𝑚 · 𝑋) ∈ 𝐵𝑋𝐵) → (𝐹‘((𝑚 · 𝑋)(+g𝐺)𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)))
5142, 48, 37, 50syl3anc 1238 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 · 𝑋)(+g𝐺)𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)))
5241, 51eqtrd 2210 . . . . . . . 8 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)))
53 mhmrcl2 12732 . . . . . . . . . 10 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐻 ∈ Mnd)
5453ad2antrr 488 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐻 ∈ Mnd)
5528adantr 276 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹𝑋) ∈ (Base‘𝐻))
5626, 29, 49mulgnn0p1 12870 . . . . . . . . 9 ((𝐻 ∈ Mnd ∧ 𝑚 ∈ ℕ0 ∧ (𝐹𝑋) ∈ (Base‘𝐻)) → ((𝑚 + 1) × (𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋)))
5754, 36, 55, 56syl3anc 1238 . . . . . . . 8 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) × (𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋)))
5852, 57eqeq12d 2192 . . . . . . 7 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋)) ↔ ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋))))
5933, 58syl5ibr 156 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋)) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋))))
6059expcom 116 . . . . 5 (𝑚 ∈ ℕ0 → ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋)) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋)))))
6160a2d 26 . . . 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)))))
624, 8, 12, 16, 32, 61nn0ind 9343 . . 3 (𝑁 ∈ ℕ0 → ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋))))
63623impib 1201 . 2 ((𝑁 ∈ ℕ0𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))
64633com12 1207 1 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  cfv 5211  (class class class)co 5868  0cc0 7789  1c1 7790   + caddc 7792  0cn0 9152  Basecbs 12432  +gcplusg 12505  0gc0g 12640  Mndcmnd 12696   MndHom cmhm 12726  .gcmg 12859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-map 6643  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-inn 8896  df-2 8954  df-n0 9153  df-z 9230  df-uz 9505  df-seqfrec 10419  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-mhm 12728  df-minusg 12758  df-mulg 12860
This theorem is referenced by: (None)
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