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Mirrors > Home > MPE Home > Th. List > Mathboxes > gsumsplit2f | Structured version Visualization version GIF version |
Description: Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.) |
Ref | Expression |
---|---|
gsumsplit2f.n | ⊢ Ⅎ𝑘𝜑 |
gsumsplit2f.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumsplit2f.z | ⊢ 0 = (0g‘𝐺) |
gsumsplit2f.p | ⊢ + = (+g‘𝐺) |
gsumsplit2f.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumsplit2f.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumsplit2f.f | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) |
gsumsplit2f.w | ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) |
gsumsplit2f.i | ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) |
gsumsplit2f.u | ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) |
Ref | Expression |
---|---|
gsumsplit2f | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumsplit2f.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumsplit2f.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | gsumsplit2f.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | gsumsplit2f.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
5 | gsumsplit2f.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | gsumsplit2f.n | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
7 | gsumsplit2f.f | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
8 | eqid 2733 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋) | |
9 | 6, 7, 8 | fmptdf 7066 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋):𝐴⟶𝐵) |
10 | gsumsplit2f.w | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) | |
11 | gsumsplit2f.i | . . 3 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) | |
12 | gsumsplit2f.u | . . 3 ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) | |
13 | 1, 2, 3, 4, 5, 9, 10, 11, 12 | gsumsplit 19710 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐶)) + (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐷)))) |
14 | ssun1 4133 | . . . . . 6 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷) | |
15 | 14, 12 | sseqtrrid 3998 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
16 | 15 | resmptd 5995 | . . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐶) = (𝑘 ∈ 𝐶 ↦ 𝑋)) |
17 | 16 | oveq2d 7374 | . . 3 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐶)) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋))) |
18 | ssun2 4134 | . . . . . 6 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷) | |
19 | 18, 12 | sseqtrrid 3998 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
20 | 19 | resmptd 5995 | . . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐷) = (𝑘 ∈ 𝐷 ↦ 𝑋)) |
21 | 20 | oveq2d 7374 | . . 3 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐷)) = (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋))) |
22 | 17, 21 | oveq12d 7376 | . 2 ⊢ (𝜑 → ((𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐶)) + (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐷))) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋)))) |
23 | 13, 22 | eqtrd 2773 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 ∪ cun 3909 ∩ cin 3910 ∅c0 4283 class class class wbr 5106 ↦ cmpt 5189 ↾ cres 5636 ‘cfv 6497 (class class class)co 7358 finSupp cfsupp 9308 Basecbs 17088 +gcplusg 17138 0gc0g 17326 Σg cgsu 17327 CMndccmn 19567 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 df-seq 13913 df-hash 14237 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-0g 17328 df-gsum 17329 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-cntz 19102 df-cmn 19569 |
This theorem is referenced by: gsumdifsndf 46201 |
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