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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 14959 | . . . 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 7446 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → (𝐺 Σg 〈“𝑆𝑇𝑈𝑉”〉) = (𝐺 Σg (〈“𝑆”〉 ++ 〈“𝑇𝑈𝑉”〉))) |
4 | simpl 482 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → 𝐺 ∈ Mnd) | |
5 | simprl 771 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → 𝑆 ∈ 𝐵) | |
6 | 5 | s1cld 14637 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → 〈“𝑆”〉 ∈ Word 𝐵) |
7 | simprrl 781 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → 𝑇 ∈ 𝐵) | |
8 | simprrl 781 | . . . . 5 ⊢ ((𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵))) → 𝑈 ∈ 𝐵) | |
9 | 8 | adantl 481 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → 𝑈 ∈ 𝐵) |
10 | simprrr 782 | . . . . 5 ⊢ ((𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵))) → 𝑉 ∈ 𝐵) | |
11 | 10 | adantl 481 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → 𝑉 ∈ 𝐵) |
12 | 7, 9, 11 | s3cld 14907 | . . 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 18866 | . . 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 1370 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → (𝐺 Σg (〈“𝑆”〉 ++ 〈“𝑇𝑈𝑉”〉)) = ((𝐺 Σg 〈“𝑆”〉) + (𝐺 Σg 〈“𝑇𝑈𝑉”〉))) |
17 | 13 | gsumws1 18863 | . . . 4 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg 〈“𝑆”〉) = 𝑆) |
18 | 17 | ad2antrl 728 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → (𝐺 Σg 〈“𝑆”〉) = 𝑆) |
19 | 13, 14 | gsumws3 44185 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵))) → (𝐺 Σg 〈“𝑇𝑈𝑉”〉) = (𝑇 + (𝑈 + 𝑉))) |
20 | 19 | adantrl 716 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → (𝐺 Σg 〈“𝑇𝑈𝑉”〉) = (𝑇 + (𝑈 + 𝑉))) |
21 | 18, 20 | oveq12d 7448 | . 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 2778 | 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 395 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 Word cword 14548 ++ cconcat 14604 〈“cs1 14629 〈“cs3 14877 〈“cs4 14878 Basecbs 17244 +gcplusg 17297 Σg cgsu 17486 Mndcmnd 18759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-word 14549 df-concat 14605 df-s1 14630 df-s2 14883 df-s3 14884 df-s4 14885 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-0g 17487 df-gsum 17488 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 |
This theorem is referenced by: amgm4d 44189 |
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