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Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gsumws4 | Structured version Visualization version GIF version |
Description: Valuation of a length 4 word in a monoid. (Contributed by Stanislas Polu, 10-Sep-2020.) |
Ref | Expression |
---|---|
gsumws4.0 | ⊢ 𝐵 = (Base‘𝐺) |
gsumws4.1 | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
gsumws4 | ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → (𝐺 Σg 〈“𝑆𝑇𝑈𝑉”〉) = (𝑆 + (𝑇 + (𝑈 + 𝑉)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1s3 14771 | . . . 4 ⊢ 〈“𝑆𝑇𝑈𝑉”〉 = (〈“𝑆”〉 ++ 〈“𝑇𝑈𝑉”〉) | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → 〈“𝑆𝑇𝑈𝑉”〉 = (〈“𝑆”〉 ++ 〈“𝑇𝑈𝑉”〉)) |
3 | 2 | oveq2d 7367 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → (𝐺 Σg 〈“𝑆𝑇𝑈𝑉”〉) = (𝐺 Σg (〈“𝑆”〉 ++ 〈“𝑇𝑈𝑉”〉))) |
4 | simpl 483 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → 𝐺 ∈ Mnd) | |
5 | simprl 769 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → 𝑆 ∈ 𝐵) | |
6 | 5 | s1cld 14445 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → 〈“𝑆”〉 ∈ Word 𝐵) |
7 | simprrl 779 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → 𝑇 ∈ 𝐵) | |
8 | simprrl 779 | . . . . 5 ⊢ ((𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵))) → 𝑈 ∈ 𝐵) | |
9 | 8 | adantl 482 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → 𝑈 ∈ 𝐵) |
10 | simprrr 780 | . . . . 5 ⊢ ((𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵))) → 𝑉 ∈ 𝐵) | |
11 | 10 | adantl 482 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → 𝑉 ∈ 𝐵) |
12 | 7, 9, 11 | s3cld 14719 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → 〈“𝑇𝑈𝑉”〉 ∈ Word 𝐵) |
13 | gsumws4.0 | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
14 | gsumws4.1 | . . . 4 ⊢ + = (+g‘𝐺) | |
15 | 13, 14 | gsumccat 18611 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 〈“𝑆”〉 ∈ Word 𝐵 ∧ 〈“𝑇𝑈𝑉”〉 ∈ Word 𝐵) → (𝐺 Σg (〈“𝑆”〉 ++ 〈“𝑇𝑈𝑉”〉)) = ((𝐺 Σg 〈“𝑆”〉) + (𝐺 Σg 〈“𝑇𝑈𝑉”〉))) |
16 | 4, 6, 12, 15 | syl3anc 1371 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → (𝐺 Σg (〈“𝑆”〉 ++ 〈“𝑇𝑈𝑉”〉)) = ((𝐺 Σg 〈“𝑆”〉) + (𝐺 Σg 〈“𝑇𝑈𝑉”〉))) |
17 | 13 | gsumws1 18608 | . . . 4 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg 〈“𝑆”〉) = 𝑆) |
18 | 17 | ad2antrl 726 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → (𝐺 Σg 〈“𝑆”〉) = 𝑆) |
19 | 13, 14 | gsumws3 42380 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵))) → (𝐺 Σg 〈“𝑇𝑈𝑉”〉) = (𝑇 + (𝑈 + 𝑉))) |
20 | 19 | adantrl 714 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → (𝐺 Σg 〈“𝑇𝑈𝑉”〉) = (𝑇 + (𝑈 + 𝑉))) |
21 | 18, 20 | oveq12d 7369 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → ((𝐺 Σg 〈“𝑆”〉) + (𝐺 Σg 〈“𝑇𝑈𝑉”〉)) = (𝑆 + (𝑇 + (𝑈 + 𝑉)))) |
22 | 3, 16, 21 | 3eqtrd 2781 | 1 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → (𝐺 Σg 〈“𝑆𝑇𝑈𝑉”〉) = (𝑆 + (𝑇 + (𝑈 + 𝑉)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 Word cword 14356 ++ cconcat 14412 〈“cs1 14437 〈“cs3 14689 〈“cs4 14690 Basecbs 17043 +gcplusg 17093 Σg cgsu 17282 Mndcmnd 18516 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-fzo 13522 df-seq 13861 df-hash 14185 df-word 14357 df-concat 14413 df-s1 14438 df-s2 14695 df-s3 14696 df-s4 14697 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-0g 17283 df-gsum 17284 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-submnd 18562 |
This theorem is referenced by: amgm4d 42384 |
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