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| Mirrors > Home > MPE Home > Th. List > gsumsplit2 | Structured version Visualization version GIF version | ||
| Description: Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 5-Jun-2019.) | 
| Ref | Expression | 
|---|---|
| gsumsplit2.b | ⊢ 𝐵 = (Base‘𝐺) | 
| gsumsplit2.z | ⊢ 0 = (0g‘𝐺) | 
| gsumsplit2.p | ⊢ + = (+g‘𝐺) | 
| gsumsplit2.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) | 
| gsumsplit2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| gsumsplit2.f | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | 
| gsumsplit2.w | ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) | 
| gsumsplit2.i | ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) | 
| gsumsplit2.u | ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) | 
| Ref | Expression | 
|---|---|
| gsumsplit2 | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋)))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | gsumsplit2.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | gsumsplit2.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
| 3 | gsumsplit2.p | . . 3 ⊢ + = (+g‘𝐺) | |
| 4 | gsumsplit2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
| 5 | gsumsplit2.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 6 | gsumsplit2.f | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
| 7 | 6 | fmpttd 7135 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋):𝐴⟶𝐵) | 
| 8 | gsumsplit2.w | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) | |
| 9 | gsumsplit2.i | . . 3 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) | |
| 10 | gsumsplit2.u | . . 3 ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) | |
| 11 | 1, 2, 3, 4, 5, 7, 8, 9, 10 | gsumsplit 19946 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐶)) + (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐷)))) | 
| 12 | ssun1 4178 | . . . . . 6 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷) | |
| 13 | 12, 10 | sseqtrrid 4027 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | 
| 14 | 13 | resmptd 6058 | . . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐶) = (𝑘 ∈ 𝐶 ↦ 𝑋)) | 
| 15 | 14 | oveq2d 7447 | . . 3 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐶)) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋))) | 
| 16 | ssun2 4179 | . . . . . 6 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷) | |
| 17 | 16, 10 | sseqtrrid 4027 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) | 
| 18 | 17 | resmptd 6058 | . . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐷) = (𝑘 ∈ 𝐷 ↦ 𝑋)) | 
| 19 | 18 | oveq2d 7447 | . . 3 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐷)) = (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋))) | 
| 20 | 15, 19 | oveq12d 7449 | . 2 ⊢ (𝜑 → ((𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐶)) + (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐷))) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋)))) | 
| 21 | 11, 20 | eqtrd 2777 | 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 395 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 class class class wbr 5143 ↦ cmpt 5225 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 finSupp cfsupp 9401 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Σg cgsu 17485 CMndccmn 19798 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-0g 17486 df-gsum 17487 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-cntz 19335 df-cmn 19800 | 
| This theorem is referenced by: gsummptfidmsplit 19948 gsumdifsnd 19979 psdmul 22170 chfacfscmulgsum 22866 chfacfpmmulgsum 22870 tdeglem4 26099 gsummptres 33055 gsummptres2 33056 elrspunsn 33457 evl1deg1 33601 evl1deg2 33602 evl1deg3 33603 lbsdiflsp0 33677 | 
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