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Theorem frmdup1 18021
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 18016 . . 3 (𝐼𝑋𝑀 ∈ Mnd)
41, 3syl 17 . 2 (𝜑𝑀 ∈ Mnd)
5 frmdup.g . 2 (𝜑𝐺 ∈ Mnd)
65adantr 483 . . . . . 6 ((𝜑𝑥 ∈ Word 𝐼) → 𝐺 ∈ Mnd)
7 simpr 487 . . . . . . 7 ((𝜑𝑥 ∈ Word 𝐼) → 𝑥 ∈ Word 𝐼)
8 frmdup.a . . . . . . . 8 (𝜑𝐴:𝐼𝐵)
98adantr 483 . . . . . . 7 ((𝜑𝑥 ∈ Word 𝐼) → 𝐴:𝐼𝐵)
10 wrdco 14185 . . . . . . 7 ((𝑥 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴𝑥) ∈ Word 𝐵)
117, 9, 10syl2anc 586 . . . . . 6 ((𝜑𝑥 ∈ Word 𝐼) → (𝐴𝑥) ∈ Word 𝐵)
12 frmdup.b . . . . . . 7 𝐵 = (Base‘𝐺)
1312gsumwcl 17995 . . . . . 6 ((𝐺 ∈ Mnd ∧ (𝐴𝑥) ∈ Word 𝐵) → (𝐺 Σg (𝐴𝑥)) ∈ 𝐵)
146, 11, 13syl2anc 586 . . . . 5 ((𝜑𝑥 ∈ Word 𝐼) → (𝐺 Σg (𝐴𝑥)) ∈ 𝐵)
15 frmdup.e . . . . 5 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴𝑥)))
1614, 15fmptd 6871 . . . 4 (𝜑𝐸:Word 𝐼𝐵)
17 eqid 2819 . . . . . . 7 (Base‘𝑀) = (Base‘𝑀)
182, 17frmdbas 18009 . . . . . 6 (𝐼𝑋 → (Base‘𝑀) = Word 𝐼)
191, 18syl 17 . . . . 5 (𝜑 → (Base‘𝑀) = Word 𝐼)
2019feq2d 6493 . . . 4 (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵𝐸:Word 𝐼𝐵))
2116, 20mpbird 259 . . 3 (𝜑𝐸:(Base‘𝑀)⟶𝐵)
222, 17frmdelbas 18010 . . . . . . . . 9 (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
2322ad2antrl 726 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ Word 𝐼)
242, 17frmdelbas 18010 . . . . . . . . 9 (𝑧 ∈ (Base‘𝑀) → 𝑧 ∈ Word 𝐼)
2524ad2antll 727 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ Word 𝐼)
268adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐴:𝐼𝐵)
27 ccatco 14189 . . . . . . . 8 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴𝑦) ++ (𝐴𝑧)))
2823, 25, 26, 27syl3anc 1365 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴𝑦) ++ (𝐴𝑧)))
2928oveq2d 7164 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))) = (𝐺 Σg ((𝐴𝑦) ++ (𝐴𝑧))))
305adantr 483 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐺 ∈ Mnd)
31 wrdco 14185 . . . . . . . 8 ((𝑦 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴𝑦) ∈ Word 𝐵)
3223, 26, 31syl2anc 586 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴𝑦) ∈ Word 𝐵)
33 wrdco 14185 . . . . . . . 8 ((𝑧 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴𝑧) ∈ Word 𝐵)
3425, 26, 33syl2anc 586 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴𝑧) ∈ Word 𝐵)
35 eqid 2819 . . . . . . . 8 (+g𝐺) = (+g𝐺)
3612, 35gsumccat 17998 . . . . . . 7 ((𝐺 ∈ Mnd ∧ (𝐴𝑦) ∈ Word 𝐵 ∧ (𝐴𝑧) ∈ Word 𝐵) → (𝐺 Σg ((𝐴𝑦) ++ (𝐴𝑧))) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
3730, 32, 34, 36syl3anc 1365 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg ((𝐴𝑦) ++ (𝐴𝑧))) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
3829, 37eqtrd 2854 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
39 eqid 2819 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
402, 17, 39frmdadd 18012 . . . . . . . 8 ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
4140adantl 484 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
4241fveq2d 6667 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g𝑀)𝑧)) = (𝐸‘(𝑦 ++ 𝑧)))
43 ccatcl 13918 . . . . . . . 8 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
4423, 25, 43syl2anc 586 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
45 coeq2 5722 . . . . . . . . 9 (𝑥 = (𝑦 ++ 𝑧) → (𝐴𝑥) = (𝐴 ∘ (𝑦 ++ 𝑧)))
4645oveq2d 7164 . . . . . . . 8 (𝑥 = (𝑦 ++ 𝑧) → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
47 ovex 7181 . . . . . . . 8 (𝐺 Σg (𝐴𝑥)) ∈ V
4846, 15, 47fvmpt3i 6766 . . . . . . 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 2854 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g𝑀)𝑧)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
51 coeq2 5722 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
5251oveq2d 7164 . . . . . . . 8 (𝑥 = 𝑦 → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg (𝐴𝑦)))
5352, 15, 47fvmpt3i 6766 . . . . . . 7 (𝑦 ∈ Word 𝐼 → (𝐸𝑦) = (𝐺 Σg (𝐴𝑦)))
54 coeq2 5722 . . . . . . . . 9 (𝑥 = 𝑧 → (𝐴𝑥) = (𝐴𝑧))
5554oveq2d 7164 . . . . . . . 8 (𝑥 = 𝑧 → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg (𝐴𝑧)))
5655, 15, 47fvmpt3i 6766 . . . . . . 7 (𝑧 ∈ Word 𝐼 → (𝐸𝑧) = (𝐺 Σg (𝐴𝑧)))
5753, 56oveqan12d 7167 . . . . . 6 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → ((𝐸𝑦)(+g𝐺)(𝐸𝑧)) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
5823, 25, 57syl2anc 586 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝐸𝑦)(+g𝐺)(𝐸𝑧)) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
5938, 50, 583eqtr4d 2864 . . . 4 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g𝑀)𝑧)) = ((𝐸𝑦)(+g𝐺)(𝐸𝑧)))
6059ralrimivva 3189 . . 3 (𝜑 → ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g𝑀)𝑧)) = ((𝐸𝑦)(+g𝐺)(𝐸𝑧)))
61 wrd0 13881 . . . 4 ∅ ∈ Word 𝐼
62 coeq2 5722 . . . . . . . 8 (𝑥 = ∅ → (𝐴𝑥) = (𝐴 ∘ ∅))
63 co02 6106 . . . . . . . 8 (𝐴 ∘ ∅) = ∅
6462, 63syl6eq 2870 . . . . . . 7 (𝑥 = ∅ → (𝐴𝑥) = ∅)
6564oveq2d 7164 . . . . . 6 (𝑥 = ∅ → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg ∅))
66 eqid 2819 . . . . . . 7 (0g𝐺) = (0g𝐺)
6766gsum0 17886 . . . . . 6 (𝐺 Σg ∅) = (0g𝐺)
6865, 67syl6eq 2870 . . . . 5 (𝑥 = ∅ → (𝐺 Σg (𝐴𝑥)) = (0g𝐺))
6968, 15, 47fvmpt3i 6766 . . . 4 (∅ ∈ Word 𝐼 → (𝐸‘∅) = (0g𝐺))
7061, 69mp1i 13 . . 3 (𝜑 → (𝐸‘∅) = (0g𝐺))
7121, 60, 703jca 1122 . 2 (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵 ∧ ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g𝑀)𝑧)) = ((𝐸𝑦)(+g𝐺)(𝐸𝑧)) ∧ (𝐸‘∅) = (0g𝐺)))
722frmd0 18017 . . 3 ∅ = (0g𝑀)
7317, 12, 39, 35, 72, 66ismhm 17950 . 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 1338 1 (𝜑𝐸 ∈ (𝑀 MndHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1081   = wceq 1530  wcel 2107  wral 3136  c0 4289  cmpt 5137  ccom 5552  wf 6344  cfv 6348  (class class class)co 7148  Word cword 13853   ++ cconcat 13914  Basecbs 16475  +gcplusg 16557  0gc0g 16705   Σg cgsu 16706  Mndcmnd 17903   MndHom cmhm 17946  freeMndcfrmd 18004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885  df-fzo 13026  df-seq 13362  df-hash 13683  df-word 13854  df-concat 13915  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-0g 16707  df-gsum 16708  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-submnd 17949  df-frmd 18006
This theorem is referenced by:  frmdup3  18024
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