MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mhmmulg Structured version   Visualization version   GIF version

Theorem mhmmulg 17349
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 6531 . . . . . . 7 (𝑛 = 0 → (𝑛 · 𝑋) = (0 · 𝑋))
21fveq2d 6089 . . . . . 6 (𝑛 = 0 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(0 · 𝑋)))
3 oveq1 6531 . . . . . 6 (𝑛 = 0 → (𝑛 × (𝐹𝑋)) = (0 × (𝐹𝑋)))
42, 3eqeq12d 2621 . . . . 5 (𝑛 = 0 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘(0 · 𝑋)) = (0 × (𝐹𝑋))))
54imbi2d 328 . . . 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 6531 . . . . . . 7 (𝑛 = 𝑚 → (𝑛 · 𝑋) = (𝑚 · 𝑋))
76fveq2d 6089 . . . . . 6 (𝑛 = 𝑚 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑚 · 𝑋)))
8 oveq1 6531 . . . . . 6 (𝑛 = 𝑚 → (𝑛 × (𝐹𝑋)) = (𝑚 × (𝐹𝑋)))
97, 8eqeq12d 2621 . . . . 5 (𝑛 = 𝑚 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋))))
109imbi2d 328 . . . 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 6531 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑛 · 𝑋) = ((𝑚 + 1) · 𝑋))
1211fveq2d 6089 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘((𝑚 + 1) · 𝑋)))
13 oveq1 6531 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝑛 × (𝐹𝑋)) = ((𝑚 + 1) × (𝐹𝑋)))
1412, 13eqeq12d 2621 . . . . 5 (𝑛 = (𝑚 + 1) → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋))))
1514imbi2d 328 . . . 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 6531 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 · 𝑋) = (𝑁 · 𝑋))
1716fveq2d 6089 . . . . . 6 (𝑛 = 𝑁 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑁 · 𝑋)))
18 oveq1 6531 . . . . . 6 (𝑛 = 𝑁 → (𝑛 × (𝐹𝑋)) = (𝑁 × (𝐹𝑋)))
1917, 18eqeq12d 2621 . . . . 5 (𝑛 = 𝑁 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹𝑋)) ↔ (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋))))
2019imbi2d 328 . . . 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 2606 . . . . . . 7 (0g𝐺) = (0g𝐺)
22 eqid 2606 . . . . . . 7 (0g𝐻) = (0g𝐻)
2321, 22mhm0 17109 . . . . . 6 (𝐹 ∈ (𝐺 MndHom 𝐻) → (𝐹‘(0g𝐺)) = (0g𝐻))
2423adantr 479 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0g𝐺)) = (0g𝐻))
25 mhmmulg.b . . . . . . . 8 𝐵 = (Base‘𝐺)
26 mhmmulg.s . . . . . . . 8 · = (.g𝐺)
2725, 21, 26mulg0 17312 . . . . . . 7 (𝑋𝐵 → (0 · 𝑋) = (0g𝐺))
2827adantl 480 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (0 · 𝑋) = (0g𝐺))
2928fveq2d 6089 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0 · 𝑋)) = (𝐹‘(0g𝐺)))
30 eqid 2606 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
3125, 30mhmf 17106 . . . . . . 7 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:𝐵⟶(Base‘𝐻))
3231ffvelrnda 6249 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ (Base‘𝐻))
33 mhmmulg.t . . . . . . 7 × = (.g𝐻)
3430, 22, 33mulg0 17312 . . . . . 6 ((𝐹𝑋) ∈ (Base‘𝐻) → (0 × (𝐹𝑋)) = (0g𝐻))
3532, 34syl 17 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (0 × (𝐹𝑋)) = (0g𝐻))
3624, 29, 353eqtr4d 2650 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(0 · 𝑋)) = (0 × (𝐹𝑋)))
37 oveq1 6531 . . . . . . 7 ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋)) → ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋)))
38 mhmrcl1 17104 . . . . . . . . . . . 12 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐺 ∈ Mnd)
3938ad2antrr 757 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐺 ∈ Mnd)
40 simpr 475 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
41 simplr 787 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝑋𝐵)
42 eqid 2606 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
4325, 26, 42mulgnn0p1 17318 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑚 ∈ ℕ0𝑋𝐵) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g𝐺)𝑋))
4439, 40, 41, 43syl3anc 1317 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g𝐺)𝑋))
4544fveq2d 6089 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 + 1) · 𝑋)) = (𝐹‘((𝑚 · 𝑋)(+g𝐺)𝑋)))
46 simpll 785 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐹 ∈ (𝐺 MndHom 𝐻))
4738ad2antrr 757 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → 𝐺 ∈ Mnd)
48 simplr 787 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → 𝑚 ∈ ℕ0)
49 simpr 475 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → 𝑋𝐵)
5025, 26mulgnn0cl 17324 . . . . . . . . . . . 12 ((𝐺 ∈ Mnd ∧ 𝑚 ∈ ℕ0𝑋𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
5147, 48, 49, 50syl3anc 1317 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
5251an32s 841 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝑚 · 𝑋) ∈ 𝐵)
53 eqid 2606 . . . . . . . . . . 11 (+g𝐻) = (+g𝐻)
5425, 42, 53mhmlin 17108 . . . . . . . . . 10 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ (𝑚 · 𝑋) ∈ 𝐵𝑋𝐵) → (𝐹‘((𝑚 · 𝑋)(+g𝐺)𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)))
5546, 52, 41, 54syl3anc 1317 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 · 𝑋)(+g𝐺)𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)))
5645, 55eqtrd 2640 . . . . . . . 8 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g𝐻)(𝐹𝑋)))
57 mhmrcl2 17105 . . . . . . . . . 10 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐻 ∈ Mnd)
5857ad2antrr 757 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐻 ∈ Mnd)
5932adantr 479 . . . . . . . . 9 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹𝑋) ∈ (Base‘𝐻))
6030, 33, 53mulgnn0p1 17318 . . . . . . . . 9 ((𝐻 ∈ Mnd ∧ 𝑚 ∈ ℕ0 ∧ (𝐹𝑋) ∈ (Base‘𝐻)) → ((𝑚 + 1) × (𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋)))
6158, 40, 59, 60syl3anc 1317 . . . . . . . 8 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) × (𝐹𝑋)) = ((𝑚 × (𝐹𝑋))(+g𝐻)(𝐹𝑋)))
6256, 61eqeq12d 2621 . . . . . . 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 234 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹𝑋)) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹𝑋))))
6463expcom 449 . . . . 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 11301 . . 3 (𝑁 ∈ ℕ0 → ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋))))
67663impib 1253 . 2 ((𝑁 ∈ ℕ0𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))
68673com12 1260 1 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  cfv 5787  (class class class)co 6524  0cc0 9789  1c1 9790   + caddc 9792  0cn0 11136  Basecbs 15638  +gcplusg 15711  0gc0g 15866  Mndcmnd 17060   MndHom cmhm 17099  .gcmg 17306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-er 7603  df-map 7720  df-en 7816  df-dom 7817  df-sdom 7818  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-n0 11137  df-z 11208  df-uz 11517  df-fz 12150  df-seq 12616  df-0g 15868  df-mgm 17008  df-sgrp 17050  df-mnd 17061  df-mhm 17101  df-mulg 17307
This theorem is referenced by:  pwsmulg  17353  ghmmulg  17438  evls1varpw  19455  evl1expd  19473  cayhamlem4  20451  dchrfi  24694  lgsqrlem1  24785  lgseisenlem4  24817  dchrisum0flblem1  24911
  Copyright terms: Public domain W3C validator