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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 14915 | . . . 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 7442 | . 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 14593 | . . 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 14863 | . . 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 18800 | . . 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 18797 | . . . 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 43657 | . . . 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 7444 | . 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 2772 | 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 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 Word cword 14504 ++ cconcat 14560 ⟨“cs1 14585 ⟨“cs3 14833 ⟨“cs4 14834 Basecbs 17187 +gcplusg 17240 Σg cgsu 17429 Mndcmnd 18701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-word 14505 df-concat 14561 df-s1 14586 df-s2 14839 df-s3 14840 df-s4 14841 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-0g 17430 df-gsum 17431 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 |
This theorem is referenced by: amgm4d 43661 |
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