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Mirrors > Home > MPE Home > Th. List > gsummptfidmadd2 | Structured version Visualization version GIF version |
Description: The sum of two group sums expressed as mappings with finite domain, using a function operation. (Contributed by AV, 23-Jul-2019.) |
Ref | Expression |
---|---|
gsummptfidmadd.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptfidmadd.p | ⊢ + = (+g‘𝐺) |
gsummptfidmadd.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptfidmadd.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
gsummptfidmadd.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
gsummptfidmadd.d | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) |
gsummptfidmadd.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) |
gsummptfidmadd.h | ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷) |
Ref | Expression |
---|---|
gsummptfidmadd2 | ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘𝑓 + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptfidmadd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
2 | gsummptfidmadd.c | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) | |
3 | gsummptfidmadd.d | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) | |
4 | gsummptfidmadd.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
6 | gsummptfidmadd.h | . . . . 5 ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷) | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷)) |
8 | 1, 2, 3, 5, 7 | offval2 7148 | . . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐻) = (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) |
9 | 8 | oveq2d 6894 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘𝑓 + 𝐻)) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷)))) |
10 | gsummptfidmadd.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
11 | gsummptfidmadd.p | . . 3 ⊢ + = (+g‘𝐺) | |
12 | gsummptfidmadd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
13 | 10, 11, 12, 1, 2, 3, 4, 6 | gsummptfidmadd 18640 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
14 | 9, 13 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘𝑓 + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ↦ cmpt 4922 ‘cfv 6101 (class class class)co 6878 ∘𝑓 cof 7129 Fincfn 8195 Basecbs 16184 +gcplusg 16267 Σg cgsu 16416 CMndccmn 18508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-oi 8657 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-fzo 12721 df-seq 13056 df-hash 13371 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-0g 16417 df-gsum 16418 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-submnd 17651 df-cntz 18062 df-cmn 18510 |
This theorem is referenced by: psrdi 19729 psrdir 19730 mamudi 20534 mamudir 20535 mdetrlin 20734 lgseisenlem3 25454 lgseisenlem4 25455 |
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