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

Theorem frmdup1 18674
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 18669 . . 3 (𝐼𝑋𝑀 ∈ Mnd)
41, 3syl 17 . 2 (𝜑𝑀 ∈ Mnd)
5 frmdup.g . 2 (𝜑𝐺 ∈ Mnd)
65adantr 481 . . . . . 6 ((𝜑𝑥 ∈ Word 𝐼) → 𝐺 ∈ Mnd)
7 simpr 485 . . . . . . 7 ((𝜑𝑥 ∈ Word 𝐼) → 𝑥 ∈ Word 𝐼)
8 frmdup.a . . . . . . . 8 (𝜑𝐴:𝐼𝐵)
98adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ Word 𝐼) → 𝐴:𝐼𝐵)
10 wrdco 14720 . . . . . . 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 18649 . . . . . 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 7062 . . . 4 (𝜑𝐸:Word 𝐼𝐵)
17 eqid 2736 . . . . . . 7 (Base‘𝑀) = (Base‘𝑀)
182, 17frmdbas 18662 . . . . . 6 (𝐼𝑋 → (Base‘𝑀) = Word 𝐼)
191, 18syl 17 . . . . 5 (𝜑 → (Base‘𝑀) = Word 𝐼)
2019feq2d 6654 . . . 4 (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵𝐸:Word 𝐼𝐵))
2116, 20mpbird 256 . . 3 (𝜑𝐸:(Base‘𝑀)⟶𝐵)
222, 17frmdelbas 18663 . . . . . . . . 9 (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
2322ad2antrl 726 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ Word 𝐼)
242, 17frmdelbas 18663 . . . . . . . . 9 (𝑧 ∈ (Base‘𝑀) → 𝑧 ∈ Word 𝐼)
2524ad2antll 727 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ Word 𝐼)
268adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐴:𝐼𝐵)
27 ccatco 14724 . . . . . . . 8 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼𝐴:𝐼𝐵) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴𝑦) ++ (𝐴𝑧)))
2823, 25, 26, 27syl3anc 1371 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴𝑦) ++ (𝐴𝑧)))
2928oveq2d 7373 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))) = (𝐺 Σg ((𝐴𝑦) ++ (𝐴𝑧))))
305adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐺 ∈ Mnd)
31 wrdco 14720 . . . . . . . 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 14720 . . . . . . . 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 18651 . . . . . . 7 ((𝐺 ∈ Mnd ∧ (𝐴𝑦) ∈ Word 𝐵 ∧ (𝐴𝑧) ∈ Word 𝐵) → (𝐺 Σg ((𝐴𝑦) ++ (𝐴𝑧))) = ((𝐺 Σg (𝐴𝑦))(+g𝐺)(𝐺 Σg (𝐴𝑧))))
3730, 32, 34, 36syl3anc 1371 . . . . . 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 18665 . . . . . . . 8 ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
4140adantl 482 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
4241fveq2d 6846 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g𝑀)𝑧)) = (𝐸‘(𝑦 ++ 𝑧)))
43 ccatcl 14462 . . . . . . . 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 5814 . . . . . . . . 9 (𝑥 = (𝑦 ++ 𝑧) → (𝐴𝑥) = (𝐴 ∘ (𝑦 ++ 𝑧)))
4645oveq2d 7373 . . . . . . . 8 (𝑥 = (𝑦 ++ 𝑧) → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
47 ovex 7390 . . . . . . . 8 (𝐺 Σg (𝐴𝑥)) ∈ V
4846, 15, 47fvmpt3i 6953 . . . . . . 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 5814 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
5251oveq2d 7373 . . . . . . . 8 (𝑥 = 𝑦 → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg (𝐴𝑦)))
5352, 15, 47fvmpt3i 6953 . . . . . . 7 (𝑦 ∈ Word 𝐼 → (𝐸𝑦) = (𝐺 Σg (𝐴𝑦)))
54 coeq2 5814 . . . . . . . . 9 (𝑥 = 𝑧 → (𝐴𝑥) = (𝐴𝑧))
5554oveq2d 7373 . . . . . . . 8 (𝑥 = 𝑧 → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg (𝐴𝑧)))
5655, 15, 47fvmpt3i 6953 . . . . . . 7 (𝑧 ∈ Word 𝐼 → (𝐸𝑧) = (𝐺 Σg (𝐴𝑧)))
5753, 56oveqan12d 7376 . . . . . 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 3197 . . 3 (𝜑 → ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g𝑀)𝑧)) = ((𝐸𝑦)(+g𝐺)(𝐸𝑧)))
61 wrd0 14427 . . . 4 ∅ ∈ Word 𝐼
62 coeq2 5814 . . . . . . . 8 (𝑥 = ∅ → (𝐴𝑥) = (𝐴 ∘ ∅))
63 co02 6212 . . . . . . . 8 (𝐴 ∘ ∅) = ∅
6462, 63eqtrdi 2792 . . . . . . 7 (𝑥 = ∅ → (𝐴𝑥) = ∅)
6564oveq2d 7373 . . . . . 6 (𝑥 = ∅ → (𝐺 Σg (𝐴𝑥)) = (𝐺 Σg ∅))
66 eqid 2736 . . . . . . 7 (0g𝐺) = (0g𝐺)
6766gsum0 18539 . . . . . 6 (𝐺 Σg ∅) = (0g𝐺)
6865, 67eqtrdi 2792 . . . . 5 (𝑥 = ∅ → (𝐺 Σg (𝐴𝑥)) = (0g𝐺))
6968, 15, 47fvmpt3i 6953 . . . 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 18670 . . 3 ∅ = (0g𝑀)
7317, 12, 39, 35, 72, 66ismhm 18603 . 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 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  c0 4282  cmpt 5188  ccom 5637  wf 6492  cfv 6496  (class class class)co 7357  Word cword 14402   ++ cconcat 14458  Basecbs 17083  +gcplusg 17133  0gc0g 17321   Σg cgsu 17322  Mndcmnd 18556   MndHom cmhm 18599  freeMndcfrmd 18657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-word 14403  df-concat 14459  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-frmd 18659
This theorem is referenced by:  frmdup3  18677
  Copyright terms: Public domain W3C validator