![]() |
Mathbox for Stanislas Polu |
< Previous
Next >
Nearby theorems |
|
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 14926 | . . . 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 7430 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → (𝐺 Σg 〈“𝑆𝑇𝑈𝑉”〉) = (𝐺 Σg (〈“𝑆”〉 ++ 〈“𝑇𝑈𝑉”〉))) |
4 | simpl 481 | . . 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 14604 | . . 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 480 | . . . 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 480 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → 𝑉 ∈ 𝐵) |
12 | 7, 9, 11 | s3cld 14874 | . . 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 18824 | . . 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 1368 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → (𝐺 Σg (〈“𝑆”〉 ++ 〈“𝑇𝑈𝑉”〉)) = ((𝐺 Σg 〈“𝑆”〉) + (𝐺 Σg 〈“𝑇𝑈𝑉”〉))) |
17 | 13 | gsumws1 18821 | . . . 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 43898 | . . . 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 7432 | . 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 2770 | 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 394 = wceq 1534 ∈ wcel 2099 ‘cfv 6544 (class class class)co 7414 Word cword 14515 ++ cconcat 14571 〈“cs1 14596 〈“cs3 14844 〈“cs4 14845 Basecbs 17206 +gcplusg 17259 Σg cgsu 17448 Mndcmnd 18720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9973 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-2 12319 df-n0 12517 df-z 12603 df-uz 12867 df-fz 13531 df-fzo 13674 df-seq 14014 df-hash 14341 df-word 14516 df-concat 14572 df-s1 14597 df-s2 14850 df-s3 14851 df-s4 14852 df-sets 17159 df-slot 17177 df-ndx 17189 df-base 17207 df-ress 17236 df-plusg 17272 df-0g 17449 df-gsum 17450 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-submnd 18767 |
This theorem is referenced by: amgm4d 43902 |
Copyright terms: Public domain | W3C validator |