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 2821 | . . 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 | difid 4316 | . . . 4 ⊢ ({𝑀} ∖ {𝑀}) = ∅ | |
10 | gsumdifsndf.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝐴) | |
11 | 10 | snssd 4728 | . . . . 5 ⊢ (𝜑 → {𝑀} ⊆ 𝐴) |
12 | difin2 4254 | . . . . 5 ⊢ ({𝑀} ⊆ 𝐴 → ({𝑀} ∖ {𝑀}) = ((𝐴 ∖ {𝑀}) ∩ {𝑀})) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ({𝑀} ∖ {𝑀}) = ((𝐴 ∖ {𝑀}) ∩ {𝑀})) |
14 | 9, 13 | syl5reqr 2871 | . . 3 ⊢ (𝜑 → ((𝐴 ∖ {𝑀}) ∩ {𝑀}) = ∅) |
15 | difsnid 4729 | . . . . 5 ⊢ (𝑀 ∈ 𝐴 → ((𝐴 ∖ {𝑀}) ∪ {𝑀}) = 𝐴) | |
16 | 10, 15 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝐴 ∖ {𝑀}) ∪ {𝑀}) = 𝐴) |
17 | 16 | eqcomd 2827 | . . 3 ⊢ (𝜑 → 𝐴 = ((𝐴 ∖ {𝑀}) ∪ {𝑀})) |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 14, 17 | gsumsplit2f 44172 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)))) |
19 | cmnmnd 18905 | . . . . 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, 10, 21, 22, 1, 23 | gsumsnfd 19054 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)) = 𝑌) |
25 | 24 | oveq2d 7158 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌)) |
26 | 18, 25 | eqtrd 2856 | 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 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2961 ∖ cdif 3921 ∪ cun 3922 ∩ cin 3923 ⊆ wss 3924 ∅c0 4279 {csn 4553 class class class wbr 5052 ↦ cmpt 5132 ‘cfv 6341 (class class class)co 7142 finSupp cfsupp 8819 Basecbs 16466 +gcplusg 16548 0gc0g 16696 Σg cgsu 16697 Mndcmnd 17894 CMndccmn 18889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-of 7395 df-om 7567 df-1st 7675 df-2nd 7676 df-supp 7817 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-fsupp 8820 df-oi 8960 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-n0 11885 df-z 11969 df-uz 12231 df-fz 12883 df-fzo 13024 df-seq 13360 df-hash 13681 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-0g 16698 df-gsum 16699 df-mre 16840 df-mrc 16841 df-acs 16843 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-submnd 17940 df-mulg 18208 df-cntz 18430 df-cmn 18891 |
This theorem is referenced by: (None) |
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