Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gsumdifsndf | Structured version Visualization version GIF version |
Description: Extract a summand from a finitely supported group sum. (Contributed by AV, 4-Sep-2019.) |
Ref | Expression |
---|---|
gsumdifsndf.k | ⊢ Ⅎ𝑘𝑌 |
gsumdifsndf.n | ⊢ Ⅎ𝑘𝜑 |
gsumdifsndf.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumdifsndf.p | ⊢ + = (+g‘𝐺) |
gsumdifsndf.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumdifsndf.a | ⊢ (𝜑 → 𝐴 ∈ 𝑊) |
gsumdifsndf.f | ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp (0g‘𝐺)) |
gsumdifsndf.e | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) |
gsumdifsndf.m | ⊢ (𝜑 → 𝑀 ∈ 𝐴) |
gsumdifsndf.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
gsumdifsndf.s | ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) |
Ref | Expression |
---|---|
gsumdifsndf | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumdifsndf.n | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | gsumdifsndf.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | eqid 2738 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | gsumdifsndf.p | . . 3 ⊢ + = (+g‘𝐺) | |
5 | gsumdifsndf.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
6 | gsumdifsndf.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊) | |
7 | gsumdifsndf.e | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
8 | gsumdifsndf.f | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp (0g‘𝐺)) | |
9 | gsumdifsndf.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝐴) | |
10 | 9 | snssd 4743 | . . . . 5 ⊢ (𝜑 → {𝑀} ⊆ 𝐴) |
11 | difin2 4226 | . . . . 5 ⊢ ({𝑀} ⊆ 𝐴 → ({𝑀} ∖ {𝑀}) = ((𝐴 ∖ {𝑀}) ∩ {𝑀})) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → ({𝑀} ∖ {𝑀}) = ((𝐴 ∖ {𝑀}) ∩ {𝑀})) |
13 | difid 4305 | . . . 4 ⊢ ({𝑀} ∖ {𝑀}) = ∅ | |
14 | 12, 13 | eqtr3di 2793 | . . 3 ⊢ (𝜑 → ((𝐴 ∖ {𝑀}) ∩ {𝑀}) = ∅) |
15 | difsnid 4744 | . . . . 5 ⊢ (𝑀 ∈ 𝐴 → ((𝐴 ∖ {𝑀}) ∪ {𝑀}) = 𝐴) | |
16 | 9, 15 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝐴 ∖ {𝑀}) ∪ {𝑀}) = 𝐴) |
17 | 16 | eqcomd 2744 | . . 3 ⊢ (𝜑 → 𝐴 = ((𝐴 ∖ {𝑀}) ∪ {𝑀})) |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 14, 17 | gsumsplit2f 45330 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)))) |
19 | cmnmnd 19390 | . . . . 5 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
20 | 5, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
21 | gsumdifsndf.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
22 | gsumdifsndf.s | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) | |
23 | gsumdifsndf.k | . . . 4 ⊢ Ⅎ𝑘𝑌 | |
24 | 2, 20, 9, 21, 22, 1, 23 | gsumsnfd 19540 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)) = 𝑌) |
25 | 24 | oveq2d 7284 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌)) |
26 | 18, 25 | eqtrd 2778 | 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 396 = wceq 1539 Ⅎwnf 1786 ∈ wcel 2106 Ⅎwnfc 2887 ∖ cdif 3884 ∪ cun 3885 ∩ cin 3886 ⊆ wss 3887 ∅c0 4257 {csn 4562 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6427 (class class class)co 7268 finSupp cfsupp 9116 Basecbs 16900 +gcplusg 16950 0gc0g 17138 Σg cgsu 17139 Mndcmnd 18373 CMndccmn 19374 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-iin 4928 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-se 5541 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-isom 6436 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7704 df-1st 7821 df-2nd 7822 df-supp 7966 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-fsupp 9117 df-oi 9257 df-card 9685 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-nn 11962 df-2 12024 df-n0 12222 df-z 12308 df-uz 12571 df-fz 13228 df-fzo 13371 df-seq 13710 df-hash 14033 df-sets 16853 df-slot 16871 df-ndx 16883 df-base 16901 df-ress 16930 df-plusg 16963 df-0g 17140 df-gsum 17141 df-mre 17283 df-mrc 17284 df-acs 17286 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-submnd 18419 df-mulg 18689 df-cntz 18911 df-cmn 19376 |
This theorem is referenced by: (None) |
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