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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 2825 | . . 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 | difid 4178 | . . . 4 ⊢ ({𝑀} ∖ {𝑀}) = ∅ | |
9 | gsumdifsnd.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝐴) | |
10 | 9 | snssd 4558 | . . . . 5 ⊢ (𝜑 → {𝑀} ⊆ 𝐴) |
11 | difin2 4119 | . . . . 5 ⊢ ({𝑀} ⊆ 𝐴 → ({𝑀} ∖ {𝑀}) = ((𝐴 ∖ {𝑀}) ∩ {𝑀})) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → ({𝑀} ∖ {𝑀}) = ((𝐴 ∖ {𝑀}) ∩ {𝑀})) |
13 | 8, 12 | syl5reqr 2876 | . . 3 ⊢ (𝜑 → ((𝐴 ∖ {𝑀}) ∩ {𝑀}) = ∅) |
14 | difsnid 4559 | . . . . 5 ⊢ (𝑀 ∈ 𝐴 → ((𝐴 ∖ {𝑀}) ∪ {𝑀}) = 𝐴) | |
15 | 9, 14 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝐴 ∖ {𝑀}) ∪ {𝑀}) = 𝐴) |
16 | 15 | eqcomd 2831 | . . 3 ⊢ (𝜑 → 𝐴 = ((𝐴 ∖ {𝑀}) ∪ {𝑀})) |
17 | 1, 2, 3, 4, 5, 6, 7, 13, 16 | gsumsplit2 18682 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)))) |
18 | cmnmnd 18561 | . . . . 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, 9, 20, 21 | gsumsnd 18705 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)) = 𝑌) |
23 | 22 | oveq2d 6921 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌)) |
24 | 17, 23 | eqtrd 2861 | 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 386 = wceq 1658 ∈ wcel 2166 ∖ cdif 3795 ∪ cun 3796 ∩ cin 3797 ⊆ wss 3798 ∅c0 4144 {csn 4397 class class class wbr 4873 ↦ cmpt 4952 ‘cfv 6123 (class class class)co 6905 finSupp cfsupp 8544 Basecbs 16222 +gcplusg 16305 0gc0g 16453 Σg cgsu 16454 Mndcmnd 17647 CMndccmn 18546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-seq 13096 df-hash 13411 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-0g 16455 df-gsum 16456 df-mre 16599 df-mrc 16600 df-acs 16602 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-mulg 17895 df-cntz 18100 df-cmn 18548 |
This theorem is referenced by: dmatmul 20671 matunitlindflem1 33949 lincdifsn 43060 |
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