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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 4808 | . . . . 5 ⊢ (𝜑 → {𝑀} ⊆ 𝐴) |
10 | difin2 4289 | . . . . 5 ⊢ ({𝑀} ⊆ 𝐴 → ({𝑀} ∖ {𝑀}) = ((𝐴 ∖ {𝑀}) ∩ {𝑀})) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → ({𝑀} ∖ {𝑀}) = ((𝐴 ∖ {𝑀}) ∩ {𝑀})) |
12 | difid 4368 | . . . 4 ⊢ ({𝑀} ∖ {𝑀}) = ∅ | |
13 | 11, 12 | eqtr3di 2788 | . . 3 ⊢ (𝜑 → ((𝐴 ∖ {𝑀}) ∩ {𝑀}) = ∅) |
14 | difsnid 4809 | . . . . 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 19780 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)))) |
18 | cmnmnd 19649 | . . . . 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 19803 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)) = 𝑌) |
23 | 22 | oveq2d 7412 | . 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 3943 ∪ cun 3944 ∩ cin 3945 ⊆ wss 3946 ∅c0 4320 {csn 4624 class class class wbr 5144 ↦ cmpt 5227 ‘cfv 6535 (class class class)co 7396 finSupp cfsupp 9349 Basecbs 17131 +gcplusg 17184 0gc0g 17372 Σg cgsu 17373 Mndcmnd 18612 CMndccmn 19632 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-iin 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7657 df-om 7843 df-1st 7962 df-2nd 7963 df-supp 8134 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9350 df-oi 9492 df-card 9921 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-n0 12460 df-z 12546 df-uz 12810 df-fz 13472 df-fzo 13615 df-seq 13954 df-hash 14278 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-0g 17374 df-gsum 17375 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-submnd 18659 df-mulg 18936 df-cntz 19166 df-cmn 19634 |
This theorem is referenced by: dmatmul 21968 matunitlindflem1 36389 lincdifsn 46945 |
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