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Mirrors > Home > MPE Home > Th. List > gsumws2 | Structured version Visualization version GIF version |
Description: Valuation of a pair in a monoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
Ref | Expression |
---|---|
gsumccat.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumccat.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
gsumws2 | ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → (𝐺 Σg 〈“𝑆𝑇”〉) = (𝑆 + 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-s2 14201 | . . . 4 ⊢ 〈“𝑆𝑇”〉 = (〈“𝑆”〉 ++ 〈“𝑇”〉) | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → 〈“𝑆𝑇”〉 = (〈“𝑆”〉 ++ 〈“𝑇”〉)) |
3 | 2 | oveq2d 7151 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → (𝐺 Σg 〈“𝑆𝑇”〉) = (𝐺 Σg (〈“𝑆”〉 ++ 〈“𝑇”〉))) |
4 | id 22 | . . 3 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mnd) | |
5 | s1cl 13947 | . . 3 ⊢ (𝑆 ∈ 𝐵 → 〈“𝑆”〉 ∈ Word 𝐵) | |
6 | s1cl 13947 | . . 3 ⊢ (𝑇 ∈ 𝐵 → 〈“𝑇”〉 ∈ Word 𝐵) | |
7 | gsumccat.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
8 | gsumccat.p | . . . 4 ⊢ + = (+g‘𝐺) | |
9 | 7, 8 | gsumccat 17998 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 〈“𝑆”〉 ∈ Word 𝐵 ∧ 〈“𝑇”〉 ∈ Word 𝐵) → (𝐺 Σg (〈“𝑆”〉 ++ 〈“𝑇”〉)) = ((𝐺 Σg 〈“𝑆”〉) + (𝐺 Σg 〈“𝑇”〉))) |
10 | 4, 5, 6, 9 | syl3an 1157 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → (𝐺 Σg (〈“𝑆”〉 ++ 〈“𝑇”〉)) = ((𝐺 Σg 〈“𝑆”〉) + (𝐺 Σg 〈“𝑇”〉))) |
11 | 7 | gsumws1 17994 | . . . 4 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg 〈“𝑆”〉) = 𝑆) |
12 | 7 | gsumws1 17994 | . . . 4 ⊢ (𝑇 ∈ 𝐵 → (𝐺 Σg 〈“𝑇”〉) = 𝑇) |
13 | 11, 12 | oveqan12d 7154 | . . 3 ⊢ ((𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → ((𝐺 Σg 〈“𝑆”〉) + (𝐺 Σg 〈“𝑇”〉)) = (𝑆 + 𝑇)) |
14 | 13 | 3adant1 1127 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → ((𝐺 Σg 〈“𝑆”〉) + (𝐺 Σg 〈“𝑇”〉)) = (𝑆 + 𝑇)) |
15 | 3, 10, 14 | 3eqtrd 2837 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → (𝐺 Σg 〈“𝑆𝑇”〉) = (𝑆 + 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Word cword 13857 ++ cconcat 13913 〈“cs1 13940 〈“cs2 14194 Basecbs 16475 +gcplusg 16557 Σg cgsu 16706 Mndcmnd 17903 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 |
This theorem is referenced by: psgnunilem2 18615 frgpuplem 18890 cyc3genpmlem 30843 cyc3genpm 30844 gsumws3 40902 amgm2d 40904 amgmw2d 45332 |
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