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Theorem frmdup1 17602
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 . . . 4 (𝜑𝐼𝑋)
2 frmdup.m . . . . 5 𝑀 = (freeMnd‘𝐼)
32frmdmnd 17597 . . . 4 (𝐼𝑋𝑀 ∈ Mnd)
41, 3syl 17 . . 3 (𝜑𝑀 ∈ Mnd)
5 frmdup.g . . 3 (𝜑𝐺 ∈ Mnd)
64, 5jca 555 . 2 (𝜑 → (𝑀 ∈ Mnd ∧ 𝐺 ∈ Mnd))
75adantr 472 . . . . . 6 ((𝜑𝑥 ∈ Word 𝐼) → 𝐺 ∈ Mnd)
8 simpr 479 . . . . . . 7 ((𝜑𝑥 ∈ Word 𝐼) → 𝑥 ∈ Word 𝐼)
9 frmdup.a . . . . . . . 8 (𝜑𝐴:𝐼𝐵)
109adantr 472 . . . . . . 7 ((𝜑𝑥 ∈ Word 𝐼) → 𝐴:𝐼𝐵)
11 wrdco 13777 . . . . . . 7 ((𝑥 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴𝑥) ∈ Word 𝐵)
128, 10, 11syl2anc 696 . . . . . 6 ((𝜑𝑥 ∈ Word 𝐼) → (𝐴𝑥) ∈ Word 𝐵)
13 frmdup.b . . . . . . 7 𝐵 = (Base‘𝐺)
1413gsumwcl 17578 . . . . . 6 ((𝐺 ∈ Mnd ∧ (𝐴𝑥) ∈ Word 𝐵) → (𝐺 Σg (𝐴𝑥)) ∈ 𝐵)
157, 12, 14syl2anc 696 . . . . 5 ((𝜑𝑥 ∈ Word 𝐼) → (𝐺 Σg (𝐴𝑥)) ∈ 𝐵)
16 frmdup.e . . . . 5 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴𝑥)))
1715, 16fmptd 6548 . . . 4 (𝜑𝐸:Word 𝐼𝐵)
18 eqid 2760 . . . . . . 7 (Base‘𝑀) = (Base‘𝑀)
192, 18frmdbas 17590 . . . . . 6 (𝐼𝑋 → (Base‘𝑀) = Word 𝐼)
201, 19syl 17 . . . . 5 (𝜑 → (Base‘𝑀) = Word 𝐼)
2120feq2d 6192 . . . 4 (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵𝐸:Word 𝐼𝐵))
2217, 21mpbird 247 . . 3 (𝜑𝐸:(Base‘𝑀)⟶𝐵)
232, 18frmdelbas 17591 . . . . . . . . 9 (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
2423ad2antrl 766 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ Word 𝐼)
252, 18frmdelbas 17591 . . . . . . . . 9 (𝑧 ∈ (Base‘𝑀) → 𝑧 ∈ Word 𝐼)
2625ad2antll 767 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ Word 𝐼)
279adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐴:𝐼𝐵)
28 ccatco 13781 . . . . . . . 8 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴𝑦) ++ (𝐴𝑧)))
2924, 26, 27, 28syl3anc 1477 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴𝑦) ++ (𝐴𝑧)))
3029oveq2d 6829 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))) = (𝐺 Σg ((𝐴𝑦) ++ (𝐴𝑧))))
315adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐺 ∈ Mnd)
32 wrdco 13777 . . . . . . . 8 ((𝑦 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴𝑦) ∈ Word 𝐵)
3324, 27, 32syl2anc 696 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴𝑦) ∈ Word 𝐵)
34 wrdco 13777 . . . . . . . 8 ((𝑧 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴𝑧) ∈ Word 𝐵)
3526, 27, 34syl2anc 696 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴𝑧) ∈ Word 𝐵)
36 eqid 2760 . . . . . . . 8 (+g𝐺) = (+g𝐺)
3713, 36gsumccat 17579 . . . . . . 7 ((𝐺 ∈ Mnd ∧ (𝐴𝑦) ∈ Word 𝐵 ∧ (𝐴𝑧) ∈ Word 𝐵) → (𝐺 Σg ((𝐴𝑦) ++ (𝐴𝑧))) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
3831, 33, 35, 37syl3anc 1477 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg ((𝐴𝑦) ++ (𝐴𝑧))) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
3930, 38eqtrd 2794 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
40 eqid 2760 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
412, 18, 40frmdadd 17593 . . . . . . . 8 ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
4241adantl 473 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
4342fveq2d 6356 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g𝑀)𝑧)) = (𝐸‘(𝑦 ++ 𝑧)))
44 ccatcl 13546 . . . . . . . 8 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
4524, 26, 44syl2anc 696 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
46 coeq2 5436 . . . . . . . . 9 (𝑥 = (𝑦 ++ 𝑧) → (𝐴𝑥) = (𝐴 ∘ (𝑦 ++ 𝑧)))
4746oveq2d 6829 . . . . . . . 8 (𝑥 = (𝑦 ++ 𝑧) → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
48 ovex 6841 . . . . . . . 8 (𝐺 Σg (𝐴𝑥)) ∈ V
4947, 16, 48fvmpt3i 6449 . . . . . . 7 ((𝑦 ++ 𝑧) ∈ Word 𝐼 → (𝐸‘(𝑦 ++ 𝑧)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
5045, 49syl 17 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦 ++ 𝑧)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
5143, 50eqtrd 2794 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g𝑀)𝑧)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
52 coeq2 5436 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
5352oveq2d 6829 . . . . . . . 8 (𝑥 = 𝑦 → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg (𝐴𝑦)))
5453, 16, 48fvmpt3i 6449 . . . . . . 7 (𝑦 ∈ Word 𝐼 → (𝐸𝑦) = (𝐺 Σg (𝐴𝑦)))
55 coeq2 5436 . . . . . . . . 9 (𝑥 = 𝑧 → (𝐴𝑥) = (𝐴𝑧))
5655oveq2d 6829 . . . . . . . 8 (𝑥 = 𝑧 → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg (𝐴𝑧)))
5756, 16, 48fvmpt3i 6449 . . . . . . 7 (𝑧 ∈ Word 𝐼 → (𝐸𝑧) = (𝐺 Σg (𝐴𝑧)))
5854, 57oveqan12d 6832 . . . . . 6 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → ((𝐸𝑦)(+g𝐺)(𝐸𝑧)) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
5924, 26, 58syl2anc 696 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝐸𝑦)(+g𝐺)(𝐸𝑧)) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
6039, 51, 593eqtr4d 2804 . . . 4 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g𝑀)𝑧)) = ((𝐸𝑦)(+g𝐺)(𝐸𝑧)))
6160ralrimivva 3109 . . 3 (𝜑 → ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g𝑀)𝑧)) = ((𝐸𝑦)(+g𝐺)(𝐸𝑧)))
62 wrd0 13516 . . . 4 ∅ ∈ Word 𝐼
63 coeq2 5436 . . . . . . . 8 (𝑥 = ∅ → (𝐴𝑥) = (𝐴 ∘ ∅))
64 co02 5810 . . . . . . . 8 (𝐴 ∘ ∅) = ∅
6563, 64syl6eq 2810 . . . . . . 7 (𝑥 = ∅ → (𝐴𝑥) = ∅)
6665oveq2d 6829 . . . . . 6 (𝑥 = ∅ → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg ∅))
67 eqid 2760 . . . . . . 7 (0g𝐺) = (0g𝐺)
6867gsum0 17479 . . . . . 6 (𝐺 Σg ∅) = (0g𝐺)
6966, 68syl6eq 2810 . . . . 5 (𝑥 = ∅ → (𝐺 Σg (𝐴𝑥)) = (0g𝐺))
7069, 16, 48fvmpt3i 6449 . . . 4 (∅ ∈ Word 𝐼 → (𝐸‘∅) = (0g𝐺))
7162, 70mp1i 13 . . 3 (𝜑 → (𝐸‘∅) = (0g𝐺))
7222, 61, 713jca 1123 . 2 (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵 ∧ ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g𝑀)𝑧)) = ((𝐸𝑦)(+g𝐺)(𝐸𝑧)) ∧ (𝐸‘∅) = (0g𝐺)))
732frmd0 17598 . . 3 ∅ = (0g𝑀)
7418, 13, 40, 36, 73, 67ismhm 17538 . 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))))
756, 72, 74sylanbrc 701 1 (𝜑𝐸 ∈ (𝑀 MndHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  c0 4058  cmpt 4881  ccom 5270  wf 6045  cfv 6049  (class class class)co 6813  Word cword 13477   ++ cconcat 13479  Basecbs 16059  +gcplusg 16143  0gc0g 16302   Σg cgsu 16303  Mndcmnd 17495   MndHom cmhm 17534  freeMndcfrmd 17585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-seq 12996  df-hash 13312  df-word 13485  df-concat 13487  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-0g 16304  df-gsum 16305  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-submnd 17537  df-frmd 17587
This theorem is referenced by:  frmdup3  17605
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