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Theorem frmdup1 18787
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 18782 . . 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 14789 . . . . . . 7 ((𝑥 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴𝑥) ∈ Word 𝐵)
117, 9, 10syl2anc 583 . . . . . 6 ((𝜑𝑥 ∈ Word 𝐼) → (𝐴𝑥) ∈ Word 𝐵)
12 frmdup.b . . . . . . 7 𝐵 = (Base‘𝐺)
1312gsumwcl 18762 . . . . . 6 ((𝐺 ∈ Mnd ∧ (𝐴𝑥) ∈ Word 𝐵) → (𝐺 Σg (𝐴𝑥)) ∈ 𝐵)
146, 11, 13syl2anc 583 . . . . 5 ((𝜑𝑥 ∈ Word 𝐼) → (𝐺 Σg (𝐴𝑥)) ∈ 𝐵)
15 frmdup.e . . . . 5 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴𝑥)))
1614, 15fmptd 7115 . . . 4 (𝜑𝐸:Word 𝐼𝐵)
17 eqid 2731 . . . . . . 7 (Base‘𝑀) = (Base‘𝑀)
182, 17frmdbas 18775 . . . . . 6 (𝐼𝑋 → (Base‘𝑀) = Word 𝐼)
191, 18syl 17 . . . . 5 (𝜑 → (Base‘𝑀) = Word 𝐼)
2019feq2d 6703 . . . 4 (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵𝐸:Word 𝐼𝐵))
2116, 20mpbird 257 . . 3 (𝜑𝐸:(Base‘𝑀)⟶𝐵)
222, 17frmdelbas 18776 . . . . . . . . 9 (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
2322ad2antrl 725 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ Word 𝐼)
242, 17frmdelbas 18776 . . . . . . . . 9 (𝑧 ∈ (Base‘𝑀) → 𝑧 ∈ Word 𝐼)
2524ad2antll 726 . . . . . . . 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 14793 . . . . . . . 8 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴𝑦) ++ (𝐴𝑧)))
2823, 25, 26, 27syl3anc 1370 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴𝑦) ++ (𝐴𝑧)))
2928oveq2d 7428 . . . . . 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 14789 . . . . . . . 8 ((𝑦 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴𝑦) ∈ Word 𝐵)
3223, 26, 31syl2anc 583 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴𝑦) ∈ Word 𝐵)
33 wrdco 14789 . . . . . . . 8 ((𝑧 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴𝑧) ∈ Word 𝐵)
3425, 26, 33syl2anc 583 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴𝑧) ∈ Word 𝐵)
35 eqid 2731 . . . . . . . 8 (+g𝐺) = (+g𝐺)
3612, 35gsumccat 18764 . . . . . . 7 ((𝐺 ∈ Mnd ∧ (𝐴𝑦) ∈ Word 𝐵 ∧ (𝐴𝑧) ∈ Word 𝐵) → (𝐺 Σg ((𝐴𝑦) ++ (𝐴𝑧))) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
3730, 32, 34, 36syl3anc 1370 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg ((𝐴𝑦) ++ (𝐴𝑧))) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
3829, 37eqtrd 2771 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
39 eqid 2731 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
402, 17, 39frmdadd 18778 . . . . . . . 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 6895 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g𝑀)𝑧)) = (𝐸‘(𝑦 ++ 𝑧)))
43 ccatcl 14531 . . . . . . . 8 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
4423, 25, 43syl2anc 583 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
45 coeq2 5858 . . . . . . . . 9 (𝑥 = (𝑦 ++ 𝑧) → (𝐴𝑥) = (𝐴 ∘ (𝑦 ++ 𝑧)))
4645oveq2d 7428 . . . . . . . 8 (𝑥 = (𝑦 ++ 𝑧) → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
47 ovex 7445 . . . . . . . 8 (𝐺 Σg (𝐴𝑥)) ∈ V
4846, 15, 47fvmpt3i 7003 . . . . . . 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 2771 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g𝑀)𝑧)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
51 coeq2 5858 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
5251oveq2d 7428 . . . . . . . 8 (𝑥 = 𝑦 → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg (𝐴𝑦)))
5352, 15, 47fvmpt3i 7003 . . . . . . 7 (𝑦 ∈ Word 𝐼 → (𝐸𝑦) = (𝐺 Σg (𝐴𝑦)))
54 coeq2 5858 . . . . . . . . 9 (𝑥 = 𝑧 → (𝐴𝑥) = (𝐴𝑧))
5554oveq2d 7428 . . . . . . . 8 (𝑥 = 𝑧 → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg (𝐴𝑧)))
5655, 15, 47fvmpt3i 7003 . . . . . . 7 (𝑧 ∈ Word 𝐼 → (𝐸𝑧) = (𝐺 Σg (𝐴𝑧)))
5753, 56oveqan12d 7431 . . . . . 6 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → ((𝐸𝑦)(+g𝐺)(𝐸𝑧)) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
5823, 25, 57syl2anc 583 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝐸𝑦)(+g𝐺)(𝐸𝑧)) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
5938, 50, 583eqtr4d 2781 . . . 4 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g𝑀)𝑧)) = ((𝐸𝑦)(+g𝐺)(𝐸𝑧)))
6059ralrimivva 3199 . . 3 (𝜑 → ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g𝑀)𝑧)) = ((𝐸𝑦)(+g𝐺)(𝐸𝑧)))
61 wrd0 14496 . . . 4 ∅ ∈ Word 𝐼
62 coeq2 5858 . . . . . . . 8 (𝑥 = ∅ → (𝐴𝑥) = (𝐴 ∘ ∅))
63 co02 6259 . . . . . . . 8 (𝐴 ∘ ∅) = ∅
6462, 63eqtrdi 2787 . . . . . . 7 (𝑥 = ∅ → (𝐴𝑥) = ∅)
6564oveq2d 7428 . . . . . 6 (𝑥 = ∅ → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg ∅))
66 eqid 2731 . . . . . . 7 (0g𝐺) = (0g𝐺)
6766gsum0 18615 . . . . . 6 (𝐺 Σg ∅) = (0g𝐺)
6865, 67eqtrdi 2787 . . . . 5 (𝑥 = ∅ → (𝐺 Σg (𝐴𝑥)) = (0g𝐺))
6968, 15, 47fvmpt3i 7003 . . . 4 (∅ ∈ Word 𝐼 → (𝐸‘∅) = (0g𝐺))
7061, 69mp1i 13 . . 3 (𝜑 → (𝐸‘∅) = (0g𝐺))
7121, 60, 703jca 1127 . 2 (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵 ∧ ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g𝑀)𝑧)) = ((𝐸𝑦)(+g𝐺)(𝐸𝑧)) ∧ (𝐸‘∅) = (0g𝐺)))
722frmd0 18783 . . 3 ∅ = (0g𝑀)
7317, 12, 39, 35, 72, 66ismhm 18713 . 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 1343 1 (𝜑𝐸 ∈ (𝑀 MndHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  c0 4322  cmpt 5231  ccom 5680  wf 6539  cfv 6543  (class class class)co 7412  Word cword 14471   ++ cconcat 14527  Basecbs 17151  +gcplusg 17204  0gc0g 17392   Σg cgsu 17393  Mndcmnd 18665   MndHom cmhm 18709  freeMndcfrmd 18770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-n0 12480  df-z 12566  df-uz 12830  df-fz 13492  df-fzo 13635  df-seq 13974  df-hash 14298  df-word 14472  df-concat 14528  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-0g 17394  df-gsum 17395  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-mhm 18711  df-submnd 18712  df-frmd 18772
This theorem is referenced by:  frmdup3  18790
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