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Theorem gsummptfsadd 18147

Description: The sum of two group sums expressed as mappings. (Contributed by AV, 4-Apr-2019.) (Revised by AV, 9-Jul-2019.)

**Hypotheses**
Ref	Expression
gsummptfsadd.b	⊢ 𝐵 = (Base‘𝐺)
gsummptfsadd.z	⊢ 0 = (0_g‘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 )

**Assertion**
Ref	Expression
gsummptfsadd	⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((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) 𝑉(𝑥) 0 (𝑥)

**Proof of Theorem gsummptfsadd**
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 6812	. . . 4 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐻) = (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷)))
7	6	eqcomd 2616	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷)) = (𝐹 ∘_𝑓 + 𝐻))
8	7	oveq2d 6565	. 2 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = (𝐺 Σ_g (𝐹 ∘_𝑓 + 𝐻)))
9		gsummptfsadd.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
10		gsummptfsadd.z	. . 3 ⊢ 0 = (0_g‘𝐺)
11		gsummptfsadd.p	. . 3 ⊢ + = (+_g‘𝐺)
12		gsummptfsadd.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd)
13		eqid 2610	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
14	2, 13	fmptd 6292	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵)
15	4	feq1d 5943	. . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵))
16	14, 15	mpbird 246	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
17		eqid 2610	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐷) = (𝑥 ∈ 𝐴 ↦ 𝐷)
18	3, 17	fmptd 6292	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐷):𝐴⟶𝐵)
19	5	feq1d 5943	. . . 4 ⊢ (𝜑 → (𝐻:𝐴⟶𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝐷):𝐴⟶𝐵))
20	18, 19	mpbird 246	. . 3 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)
21		gsummptfsadd.w	. . 3 ⊢ (𝜑 → 𝐹 finSupp 0 )
22		gsummptfsadd.v	. . 3 ⊢ (𝜑 → 𝐻 finSupp 0 )
23	9, 10, 11, 12, 1, 16, 20, 21, 22	gsumadd 18146	. 2 ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ∘_𝑓 + 𝐻)) = ((𝐺 Σ_g 𝐹) + (𝐺 Σ_g 𝐻)))
24	8, 23	eqtrd 2644	1 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σ_g 𝐹) + (𝐺 Σ_g 𝐻)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 finSupp cfsupp 8158 Basecbs 15695 +_gcplusg 15768 0_gc0g 15923 Σ_g cgsu 15924 CMndccmn 18016

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-cntz 17573 df-cmn 18018

This theorem is referenced by: gsummptfidmadd 18148 frlmphl 19939 pm2mpghm 20440 lincsum 42012

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