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Theorem gsummptfidmadd2 19881

Description: The sum of two group sums expressed as mappings with finite domain, using a function operation. (Contributed by AV, 23-Jul-2019.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
gsummptfidmadd2	⊢ (𝜑 → (𝐺 Σ_g (𝐹 ∘_f + 𝐻)) = ((𝐺 Σ_g 𝐹) + (𝐺 Σ_g 𝐻)))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝜑,𝑥 𝑥, + Allowed substitution hints: 𝐶(𝑥) 𝐷(𝑥) 𝐹(𝑥) 𝐺(𝑥) 𝐻(𝑥)

**Proof of Theorem gsummptfidmadd2**
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 7705	. . 3 ⊢ (𝜑 → (𝐹 ∘_f + 𝐻) = (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷)))
9	8	oveq2d 7436	. 2 ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ∘_f + 𝐻)) = (𝐺 Σ_g (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 19880	. 2 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σ_g 𝐹) + (𝐺 Σ_g 𝐻)))
14	9, 13	eqtrd 2768	1 ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ∘_f + 𝐻)) = ((𝐺 Σ_g 𝐹) + (𝐺 Σ_g 𝐻)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ↦ cmpt 5231 ‘cfv 6548 (class class class)co 7420 ∘_f cof 7683 Fincfn 8964 Basecbs 17180 +_gcplusg 17233 Σ_g cgsu 17422 CMndccmn 19735

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-0g 17423 df-gsum 17424 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-cntz 19268 df-cmn 19737

This theorem is referenced by: psrdi 21908 psrdir 21909 mamudi 22316 mamudir 22317 mdetrlin 22517 lgseisenlem3 27323 lgseisenlem4 27324

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