MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frmdup1 Structured version   Visualization version   GIF version

Theorem frmdup1 18878
Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 27-Sep-2015.)
Hypotheses
Ref Expression
frmdup.m 𝑀 = (freeMnd‘𝐼)
frmdup.b 𝐵 = (Base‘𝐺)
frmdup.e 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴𝑥)))
frmdup.g (𝜑𝐺 ∈ Mnd)
frmdup.i (𝜑𝐼𝑋)
frmdup.a (𝜑𝐴:𝐼𝐵)
Assertion
Ref Expression
frmdup1 (𝜑𝐸 ∈ (𝑀 MndHom 𝐺))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐺   𝜑,𝑥   𝑥,𝐼
Allowed substitution hints:   𝐸(𝑥)   𝑀(𝑥)   𝑋(𝑥)

Proof of Theorem frmdup1
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frmdup.i . . 3 (𝜑𝐼𝑋)
2 frmdup.m . . . 4 𝑀 = (freeMnd‘𝐼)
32frmdmnd 18873 . . 3 (𝐼𝑋𝑀 ∈ Mnd)
41, 3syl 17 . 2 (𝜑𝑀 ∈ Mnd)
5 frmdup.g . 2 (𝜑𝐺 ∈ Mnd)
65adantr 480 . . . . . 6 ((𝜑𝑥 ∈ Word 𝐼) → 𝐺 ∈ Mnd)
7 simpr 484 . . . . . . 7 ((𝜑𝑥 ∈ Word 𝐼) → 𝑥 ∈ Word 𝐼)
8 frmdup.a . . . . . . . 8 (𝜑𝐴:𝐼𝐵)
98adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ Word 𝐼) → 𝐴:𝐼𝐵)
10 wrdco 14871 . . . . . . 7 ((𝑥 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴𝑥) ∈ Word 𝐵)
117, 9, 10syl2anc 584 . . . . . 6 ((𝜑𝑥 ∈ Word 𝐼) → (𝐴𝑥) ∈ Word 𝐵)
12 frmdup.b . . . . . . 7 𝐵 = (Base‘𝐺)
1312gsumwcl 18853 . . . . . 6 ((𝐺 ∈ Mnd ∧ (𝐴𝑥) ∈ Word 𝐵) → (𝐺 Σg (𝐴𝑥)) ∈ 𝐵)
146, 11, 13syl2anc 584 . . . . 5 ((𝜑𝑥 ∈ Word 𝐼) → (𝐺 Σg (𝐴𝑥)) ∈ 𝐵)
15 frmdup.e . . . . 5 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴𝑥)))
1614, 15fmptd 7133 . . . 4 (𝜑𝐸:Word 𝐼𝐵)
17 eqid 2736 . . . . . . 7 (Base‘𝑀) = (Base‘𝑀)
182, 17frmdbas 18866 . . . . . 6 (𝐼𝑋 → (Base‘𝑀) = Word 𝐼)
191, 18syl 17 . . . . 5 (𝜑 → (Base‘𝑀) = Word 𝐼)
2019feq2d 6721 . . . 4 (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵𝐸:Word 𝐼𝐵))
2116, 20mpbird 257 . . 3 (𝜑𝐸:(Base‘𝑀)⟶𝐵)
222, 17frmdelbas 18867 . . . . . . . . 9 (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
2322ad2antrl 728 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ Word 𝐼)
242, 17frmdelbas 18867 . . . . . . . . 9 (𝑧 ∈ (Base‘𝑀) → 𝑧 ∈ Word 𝐼)
2524ad2antll 729 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ Word 𝐼)
268adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐴:𝐼𝐵)
27 ccatco 14875 . . . . . . . 8 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴𝑦) ++ (𝐴𝑧)))
2823, 25, 26, 27syl3anc 1372 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴𝑦) ++ (𝐴𝑧)))
2928oveq2d 7448 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))) = (𝐺 Σg ((𝐴𝑦) ++ (𝐴𝑧))))
305adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐺 ∈ Mnd)
31 wrdco 14871 . . . . . . . 8 ((𝑦 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴𝑦) ∈ Word 𝐵)
3223, 26, 31syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴𝑦) ∈ Word 𝐵)
33 wrdco 14871 . . . . . . . 8 ((𝑧 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴𝑧) ∈ Word 𝐵)
3425, 26, 33syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴𝑧) ∈ Word 𝐵)
35 eqid 2736 . . . . . . . 8 (+g𝐺) = (+g𝐺)
3612, 35gsumccat 18855 . . . . . . 7 ((𝐺 ∈ Mnd ∧ (𝐴𝑦) ∈ Word 𝐵 ∧ (𝐴𝑧) ∈ Word 𝐵) → (𝐺 Σg ((𝐴𝑦) ++ (𝐴𝑧))) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
3730, 32, 34, 36syl3anc 1372 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg ((𝐴𝑦) ++ (𝐴𝑧))) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
3829, 37eqtrd 2776 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
39 eqid 2736 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
402, 17, 39frmdadd 18869 . . . . . . . 8 ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
4140adantl 481 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
4241fveq2d 6909 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g𝑀)𝑧)) = (𝐸‘(𝑦 ++ 𝑧)))
43 ccatcl 14613 . . . . . . . 8 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
4423, 25, 43syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
45 coeq2 5868 . . . . . . . . 9 (𝑥 = (𝑦 ++ 𝑧) → (𝐴𝑥) = (𝐴 ∘ (𝑦 ++ 𝑧)))
4645oveq2d 7448 . . . . . . . 8 (𝑥 = (𝑦 ++ 𝑧) → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
47 ovex 7465 . . . . . . . 8 (𝐺 Σg (𝐴𝑥)) ∈ V
4846, 15, 47fvmpt3i 7020 . . . . . . 7 ((𝑦 ++ 𝑧) ∈ Word 𝐼 → (𝐸‘(𝑦 ++ 𝑧)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
4944, 48syl 17 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦 ++ 𝑧)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
5042, 49eqtrd 2776 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g𝑀)𝑧)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
51 coeq2 5868 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
5251oveq2d 7448 . . . . . . . 8 (𝑥 = 𝑦 → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg (𝐴𝑦)))
5352, 15, 47fvmpt3i 7020 . . . . . . 7 (𝑦 ∈ Word 𝐼 → (𝐸𝑦) = (𝐺 Σg (𝐴𝑦)))
54 coeq2 5868 . . . . . . . . 9 (𝑥 = 𝑧 → (𝐴𝑥) = (𝐴𝑧))
5554oveq2d 7448 . . . . . . . 8 (𝑥 = 𝑧 → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg (𝐴𝑧)))
5655, 15, 47fvmpt3i 7020 . . . . . . 7 (𝑧 ∈ Word 𝐼 → (𝐸𝑧) = (𝐺 Σg (𝐴𝑧)))
5753, 56oveqan12d 7451 . . . . . 6 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → ((𝐸𝑦)(+g𝐺)(𝐸𝑧)) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
5823, 25, 57syl2anc 584 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝐸𝑦)(+g𝐺)(𝐸𝑧)) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
5938, 50, 583eqtr4d 2786 . . . 4 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g𝑀)𝑧)) = ((𝐸𝑦)(+g𝐺)(𝐸𝑧)))
6059ralrimivva 3201 . . 3 (𝜑 → ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g𝑀)𝑧)) = ((𝐸𝑦)(+g𝐺)(𝐸𝑧)))
61 wrd0 14578 . . . 4 ∅ ∈ Word 𝐼
62 coeq2 5868 . . . . . . . 8 (𝑥 = ∅ → (𝐴𝑥) = (𝐴 ∘ ∅))
63 co02 6279 . . . . . . . 8 (𝐴 ∘ ∅) = ∅
6462, 63eqtrdi 2792 . . . . . . 7 (𝑥 = ∅ → (𝐴𝑥) = ∅)
6564oveq2d 7448 . . . . . 6 (𝑥 = ∅ → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg ∅))
66 eqid 2736 . . . . . . 7 (0g𝐺) = (0g𝐺)
6766gsum0 18698 . . . . . 6 (𝐺 Σg ∅) = (0g𝐺)
6865, 67eqtrdi 2792 . . . . 5 (𝑥 = ∅ → (𝐺 Σg (𝐴𝑥)) = (0g𝐺))
6968, 15, 47fvmpt3i 7020 . . . 4 (∅ ∈ Word 𝐼 → (𝐸‘∅) = (0g𝐺))
7061, 69mp1i 13 . . 3 (𝜑 → (𝐸‘∅) = (0g𝐺))
7121, 60, 703jca 1128 . 2 (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵 ∧ ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g𝑀)𝑧)) = ((𝐸𝑦)(+g𝐺)(𝐸𝑧)) ∧ (𝐸‘∅) = (0g𝐺)))
722frmd0 18874 . . 3 ∅ = (0g𝑀)
7317, 12, 39, 35, 72, 66ismhm 18799 . 2 (𝐸 ∈ (𝑀 MndHom 𝐺) ↔ ((𝑀 ∈ Mnd ∧ 𝐺 ∈ Mnd) ∧ (𝐸:(Base‘𝑀)⟶𝐵 ∧ ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g𝑀)𝑧)) = ((𝐸𝑦)(+g𝐺)(𝐸𝑧)) ∧ (𝐸‘∅) = (0g𝐺))))
744, 5, 71, 73syl21anbrc 1344 1 (𝜑𝐸 ∈ (𝑀 MndHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  c0 4332  cmpt 5224  ccom 5688  wf 6556  cfv 6560  (class class class)co 7432  Word cword 14553   ++ cconcat 14609  Basecbs 17248  +gcplusg 17298  0gc0g 17485   Σg cgsu 17486  Mndcmnd 18748   MndHom cmhm 18795  freeMndcfrmd 18861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-word 14554  df-concat 14610  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-0g 17487  df-gsum 17488  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-frmd 18863
This theorem is referenced by:  frmdup3  18881
  Copyright terms: Public domain W3C validator