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Mirrors > Home > MPE Home > Th. List > gsumdifsnd | Structured version Visualization version GIF version |
Description: Extract a summand from a finitely supported group sum. (Contributed by AV, 21-Apr-2019.) (Revised by AV, 28-Jul-2019.) |
Ref | Expression |
---|---|
gsumdifsnd.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumdifsnd.p | ⊢ + = (+g‘𝐺) |
gsumdifsnd.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumdifsnd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑊) |
gsumdifsnd.f | ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp (0g‘𝐺)) |
gsumdifsnd.e | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) |
gsumdifsnd.m | ⊢ (𝜑 → 𝑀 ∈ 𝐴) |
gsumdifsnd.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
gsumdifsnd.s | ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) |
Ref | Expression |
---|---|
gsumdifsnd | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumdifsnd.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2733 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
3 | gsumdifsnd.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | gsumdifsnd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
5 | gsumdifsnd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊) | |
6 | gsumdifsnd.e | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
7 | gsumdifsnd.f | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp (0g‘𝐺)) | |
8 | gsumdifsnd.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝐴) | |
9 | 8 | snssd 4773 | . . . . 5 ⊢ (𝜑 → {𝑀} ⊆ 𝐴) |
10 | difin2 4255 | . . . . 5 ⊢ ({𝑀} ⊆ 𝐴 → ({𝑀} ∖ {𝑀}) = ((𝐴 ∖ {𝑀}) ∩ {𝑀})) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → ({𝑀} ∖ {𝑀}) = ((𝐴 ∖ {𝑀}) ∩ {𝑀})) |
12 | difid 4334 | . . . 4 ⊢ ({𝑀} ∖ {𝑀}) = ∅ | |
13 | 11, 12 | eqtr3di 2788 | . . 3 ⊢ (𝜑 → ((𝐴 ∖ {𝑀}) ∩ {𝑀}) = ∅) |
14 | difsnid 4774 | . . . . 5 ⊢ (𝑀 ∈ 𝐴 → ((𝐴 ∖ {𝑀}) ∪ {𝑀}) = 𝐴) | |
15 | 8, 14 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝐴 ∖ {𝑀}) ∪ {𝑀}) = 𝐴) |
16 | 15 | eqcomd 2739 | . . 3 ⊢ (𝜑 → 𝐴 = ((𝐴 ∖ {𝑀}) ∪ {𝑀})) |
17 | 1, 2, 3, 4, 5, 6, 7, 13, 16 | gsumsplit2 19714 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)))) |
18 | cmnmnd 19587 | . . . . 5 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
19 | 4, 18 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
20 | gsumdifsnd.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
21 | gsumdifsnd.s | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) | |
22 | 1, 19, 8, 20, 21 | gsumsnd 19737 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)) = 𝑌) |
23 | 22 | oveq2d 7377 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌)) |
24 | 17, 23 | eqtrd 2773 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∖ cdif 3911 ∪ cun 3912 ∩ cin 3913 ⊆ wss 3914 ∅c0 4286 {csn 4590 class class class wbr 5109 ↦ cmpt 5192 ‘cfv 6500 (class class class)co 7361 finSupp cfsupp 9311 Basecbs 17091 +gcplusg 17141 0gc0g 17329 Σg cgsu 17330 Mndcmnd 18564 CMndccmn 19570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-seq 13916 df-hash 14240 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-0g 17331 df-gsum 17332 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-mulg 18881 df-cntz 19105 df-cmn 19572 |
This theorem is referenced by: dmatmul 21869 matunitlindflem1 36124 lincdifsn 46595 |
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