![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7291 | . . . 4 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐻) = (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) |
7 | 6 | eqcomd 2803 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷)) = (𝐹 ∘𝑓 + 𝐻)) |
8 | 7 | oveq2d 7039 | . 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 6750 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
14 | 5, 3 | fmpt3d 6750 | . . 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 18767 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘𝑓 + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
18 | 8, 17 | eqtrd 2833 | 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 396 = wceq 1525 ∈ wcel 2083 class class class wbr 4968 ↦ cmpt 5047 ‘cfv 6232 (class class class)co 7023 ∘𝑓 cof 7272 finSupp cfsupp 8686 Basecbs 16316 +gcplusg 16398 0gc0g 16546 Σg cgsu 16547 CMndccmn 18637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-of 7274 df-om 7444 df-1st 7552 df-2nd 7553 df-supp 7689 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-fsupp 8687 df-oi 8827 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-n0 11752 df-z 11836 df-uz 12098 df-fz 12747 df-fzo 12888 df-seq 13224 df-hash 13545 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-0g 16548 df-gsum 16549 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-submnd 17779 df-cntz 18192 df-cmn 18639 |
This theorem is referenced by: gsummptfidmadd 18769 frlmphl 20611 pm2mpghm 21112 lincsum 43986 |
Copyright terms: Public domain | W3C validator |