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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 2726 | . . 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 4807 | . . . . 5 ⊢ (𝜑 → {𝑀} ⊆ 𝐴) |
10 | difin2 4286 | . . . . 5 ⊢ ({𝑀} ⊆ 𝐴 → ({𝑀} ∖ {𝑀}) = ((𝐴 ∖ {𝑀}) ∩ {𝑀})) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → ({𝑀} ∖ {𝑀}) = ((𝐴 ∖ {𝑀}) ∩ {𝑀})) |
12 | difid 4365 | . . . 4 ⊢ ({𝑀} ∖ {𝑀}) = ∅ | |
13 | 11, 12 | eqtr3di 2781 | . . 3 ⊢ (𝜑 → ((𝐴 ∖ {𝑀}) ∩ {𝑀}) = ∅) |
14 | difsnid 4808 | . . . . 5 ⊢ (𝑀 ∈ 𝐴 → ((𝐴 ∖ {𝑀}) ∪ {𝑀}) = 𝐴) | |
15 | 8, 14 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝐴 ∖ {𝑀}) ∪ {𝑀}) = 𝐴) |
16 | 15 | eqcomd 2732 | . . 3 ⊢ (𝜑 → 𝐴 = ((𝐴 ∖ {𝑀}) ∪ {𝑀})) |
17 | 1, 2, 3, 4, 5, 6, 7, 13, 16 | gsumsplit2 19849 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)))) |
18 | cmnmnd 19717 | . . . . 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 19872 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)) = 𝑌) |
23 | 22 | oveq2d 7421 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌)) |
24 | 17, 23 | eqtrd 2766 | 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 395 = wceq 1533 ∈ wcel 2098 ∖ cdif 3940 ∪ cun 3941 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 {csn 4623 class class class wbr 5141 ↦ cmpt 5224 ‘cfv 6537 (class class class)co 7405 finSupp cfsupp 9363 Basecbs 17153 +gcplusg 17206 0gc0g 17394 Σg cgsu 17395 Mndcmnd 18667 CMndccmn 19700 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-0g 17396 df-gsum 17397 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 |
This theorem is referenced by: dmatmul 22354 matunitlindflem1 36997 lincdifsn 47380 |
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