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Mirrors > Home > MPE Home > Th. List > gsummptfsadd | Structured version Visualization version GIF version |
Description: The sum of two group sums expressed as mappings. (Contributed by AV, 4-Apr-2019.) (Revised by AV, 9-Jul-2019.) |
Ref | Expression |
---|---|
gsummptfsadd.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptfsadd.z | ⊢ 0 = (0g‘𝐺) |
gsummptfsadd.p | ⊢ + = (+g‘𝐺) |
gsummptfsadd.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptfsadd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsummptfsadd.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
gsummptfsadd.d | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) |
gsummptfsadd.f | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
gsummptfsadd.h | ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷)) |
gsummptfsadd.w | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
gsummptfsadd.v | ⊢ (𝜑 → 𝐻 finSupp 0 ) |
Ref | Expression |
---|---|
gsummptfsadd | ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptfsadd.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | gsummptfsadd.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) | |
3 | gsummptfsadd.d | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) | |
4 | gsummptfsadd.f | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
5 | gsummptfsadd.h | . . . . 5 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷)) | |
6 | 1, 2, 3, 4, 5 | offval2 7687 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f + 𝐻) = (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) |
7 | 6 | eqcomd 2732 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷)) = (𝐹 ∘f + 𝐻)) |
8 | 7 | oveq2d 7421 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = (𝐺 Σg (𝐹 ∘f + 𝐻))) |
9 | gsummptfsadd.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
10 | gsummptfsadd.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
11 | gsummptfsadd.p | . . 3 ⊢ + = (+g‘𝐺) | |
12 | gsummptfsadd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
13 | 4, 2 | fmpt3d 7111 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
14 | 5, 3 | fmpt3d 7111 | . . 3 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) |
15 | gsummptfsadd.w | . . 3 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
16 | gsummptfsadd.v | . . 3 ⊢ (𝜑 → 𝐻 finSupp 0 ) | |
17 | 9, 10, 11, 12, 1, 13, 14, 15, 16 | gsumadd 19843 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
18 | 8, 17 | eqtrd 2766 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 ↦ cmpt 5224 ‘cfv 6537 (class class class)co 7405 ∘f cof 7665 finSupp cfsupp 9363 Basecbs 17153 +gcplusg 17206 0gc0g 17394 Σg cgsu 17395 CMndccmn 19700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-0g 17396 df-gsum 17397 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-cntz 19233 df-cmn 19702 |
This theorem is referenced by: gsummptfidmadd 19845 frlmphl 21676 pm2mpghm 22673 mhphf 41726 lincsum 47385 |
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