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Mirrors > Home > MPE Home > Th. List > gsumspl | Structured version Visualization version Unicode version |
Description: The primary purpose of the splice construction is to enable local rewrites. Thus, in any monoidal valuation, if a splice does not cause a local change it does not cause a global change. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
Ref | Expression |
---|---|
gsumspl.b | |
gsumspl.m | |
gsumspl.s | Word |
gsumspl.f | |
gsumspl.t | |
gsumspl.x | Word |
gsumspl.y | Word |
gsumspl.eq | g g |
Ref | Expression |
---|---|
gsumspl | g splice g splice |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumspl.eq | . . . 4 g g | |
2 | 1 | oveq2d 6666 | . . 3 g substr g g substr g |
3 | 2 | oveq1d 6665 | . 2 g substr g g substr g substr g g substr |
4 | gsumspl.s | . . . . 5 Word | |
5 | gsumspl.f | . . . . 5 | |
6 | gsumspl.t | . . . . 5 | |
7 | gsumspl.x | . . . . 5 Word | |
8 | splval 13502 | . . . . 5 Word Word splice substr ++ ++ substr | |
9 | 4, 5, 6, 7, 8 | syl13anc 1328 | . . . 4 splice substr ++ ++ substr |
10 | 9 | oveq2d 6666 | . . 3 g splice g substr ++ ++ substr |
11 | gsumspl.m | . . . 4 | |
12 | swrdcl 13419 | . . . . . 6 Word substr Word | |
13 | 4, 12 | syl 17 | . . . . 5 substr Word |
14 | ccatcl 13359 | . . . . 5 substr Word Word substr ++ Word | |
15 | 13, 7, 14 | syl2anc 693 | . . . 4 substr ++ Word |
16 | swrdcl 13419 | . . . . 5 Word substr Word | |
17 | 4, 16 | syl 17 | . . . 4 substr Word |
18 | gsumspl.b | . . . . 5 | |
19 | eqid 2622 | . . . . 5 | |
20 | 18, 19 | gsumccat 17378 | . . . 4 substr ++ Word substr Word g substr ++ ++ substr g substr ++ g substr |
21 | 11, 15, 17, 20 | syl3anc 1326 | . . 3 g substr ++ ++ substr g substr ++ g substr |
22 | 18, 19 | gsumccat 17378 | . . . . 5 substr Word Word g substr ++ g substr g |
23 | 11, 13, 7, 22 | syl3anc 1326 | . . . 4 g substr ++ g substr g |
24 | 23 | oveq1d 6665 | . . 3 g substr ++ g substr g substr g g substr |
25 | 10, 21, 24 | 3eqtrd 2660 | . 2 g splice g substr g g substr |
26 | gsumspl.y | . . . . 5 Word | |
27 | splval 13502 | . . . . 5 Word Word splice substr ++ ++ substr | |
28 | 4, 5, 6, 26, 27 | syl13anc 1328 | . . . 4 splice substr ++ ++ substr |
29 | 28 | oveq2d 6666 | . . 3 g splice g substr ++ ++ substr |
30 | ccatcl 13359 | . . . . 5 substr Word Word substr ++ Word | |
31 | 13, 26, 30 | syl2anc 693 | . . . 4 substr ++ Word |
32 | 18, 19 | gsumccat 17378 | . . . 4 substr ++ Word substr Word g substr ++ ++ substr g substr ++ g substr |
33 | 11, 31, 17, 32 | syl3anc 1326 | . . 3 g substr ++ ++ substr g substr ++ g substr |
34 | 18, 19 | gsumccat 17378 | . . . . 5 substr Word Word g substr ++ g substr g |
35 | 11, 13, 26, 34 | syl3anc 1326 | . . . 4 g substr ++ g substr g |
36 | 35 | oveq1d 6665 | . . 3 g substr ++ g substr g substr g g substr |
37 | 29, 33, 36 | 3eqtrd 2660 | . 2 g splice g substr g g substr |
38 | 3, 25, 37 | 3eqtr4d 2666 | 1 g splice g splice |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 cop 4183 cotp 4185 cfv 5888 (class class class)co 6650 cc0 9936 cfz 12326 chash 13117 Word cword 13291 ++ cconcat 13293 substr csubstr 13295 splice csplice 13296 cbs 15857 cplusg 15941 g cgsu 16101 cmnd 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-ot 4186 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-substr 13303 df-splice 13304 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: psgnunilem2 17915 |
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