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Theorem gsumws4 38500
Description: Valuation of a length 4 word in a monoid. (Contributed by Stanislas Polu, 10-Sep-2020.)
Hypotheses
Ref Expression
gsumws4.0 |- B = ( Base ` G )
gsumws4.1 |- .+ = ( +g ` G )
Assertion
Ref Expression
gsumws4 |- ( ( G e. Mnd /\ ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) ) -> ( G gsumg <" S T U V "> ) = ( S .+ ( T .+ ( U .+ V ) ) ) )
Proof of Theorem gsumws4
StepHypRef Expression
1 s1s3 13669 . . . 4 |- <" S T U V "> = ( <" S "> ++ <" T U V "> )
21a1i 11 . . 3 |- ( ( G e. Mnd /\ ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) ) -> <" S T U V "> = ( <" S "> ++ <" T U V "> ) )
32oveq2d 6666 . 2 |- ( ( G e. Mnd /\ ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) ) -> ( G gsumg <" S T U V "> ) = ( G gsumg ( <" S "> ++ <" T U V "> ) ) )
4 simpl 473 . . 3 |- ( ( G e. Mnd /\ ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) ) -> G e. Mnd )
5 simprl 794 . . . 4 |- ( ( G e. Mnd /\ ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) ) -> S e. B )
65s1cld 13383 . . 3 |- ( ( G e. Mnd /\ ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) ) -> <" S "> e. Word B )
7 simprrl 804 . . . 4 |- ( ( G e. Mnd /\ ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) ) -> T e. B )
8 simprrl 804 . . . . 5 |- ( ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) -> U e. B )
98adantl 482 . . . 4 |- ( ( G e. Mnd /\ ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) ) -> U e. B )
10 simprrr 805 . . . . 5 |- ( ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) -> V e. B )
1110adantl 482 . . . 4 |- ( ( G e. Mnd /\ ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) ) -> V e. B )
127, 9, 11s3cld 13617 . . 3 |- ( ( G e. Mnd /\ ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) ) -> <" T U V "> e. Word B )
13 gsumws4.0 . . . 4 |- B = ( Base ` G )
14 gsumws4.1 . . . 4 |- .+ = ( +g ` G )
1513, 14gsumccat 17378 . . 3 |- ( ( G e. Mnd /\ <" S "> e. Word B /\ <" T U V "> e. Word B ) -> ( G gsumg ( <" S "> ++ <" T U V "> ) ) = ( ( G gsumg <" S "> ) .+ ( G gsumg <" T U V "> ) ) )
164, 6, 12, 15syl3anc 1326 . 2 |- ( ( G e. Mnd /\ ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) ) -> ( G gsumg ( <" S "> ++ <" T U V "> ) ) = ( ( G gsumg <" S "> ) .+ ( G gsumg <" T U V "> ) ) )
1713gsumws1 17376 . . . 4 |- ( S e. B -> ( G gsumg <" S "> ) = S )
1817ad2antrl 764 . . 3 |- ( ( G e. Mnd /\ ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) ) -> ( G gsumg <" S "> ) = S )
1913, 14gsumws3 38499 . . . 4 |- ( ( G e. Mnd /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) -> ( G gsumg <" T U V "> ) = ( T .+ ( U .+ V ) ) )
2019adantrl 752 . . 3 |- ( ( G e. Mnd /\ ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) ) -> ( G gsumg <" T U V "> ) = ( T .+ ( U .+ V ) ) )
2118, 20oveq12d 6668 . 2 |- ( ( G e. Mnd /\ ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) ) -> ( ( G gsumg <" S "> ) .+ ( G gsumg <" T U V "> ) ) = ( S .+ ( T .+ ( U .+ V ) ) ) )
223, 16, 213eqtrd 2660 1 |- ( ( G e. Mnd /\ ( S e. B /\ ( T e. B /\ ( U e. B /\ V e. B ) ) ) ) -> ( G gsumg <" S T U V "> ) = ( S .+ ( T .+ ( U .+ V ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650  Word cword 13291   ++ cconcat 13293   <"cs1 13294   <"cs3 13587   <"cs4 13588   Basecbs 15857   +g cplusg 15941   gsumg cgsu 16101   Mndcmnd 17294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-s4 13595  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336
This theorem is referenced by:  amgm4d  38503
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