gsumfzmptfidmadd2 - Intuitionistic Logic Explorer

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Theorem gsumfzmptfidmadd2 13446

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)
gsumfzmptfidmadd.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
gsumfzmptfidmadd.n	⊢ (𝜑 → 𝑁 ∈ ℤ)
gsumfzmptfidmadd.c	⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐶 ∈ 𝐵)
gsumfzmptfidmadd.d	⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐷 ∈ 𝐵)
gsumfzmptfidmadd.f	⊢ 𝐹 = (𝑥 ∈ (𝑀...𝑁) ↦ 𝐶)
gsumfzmptfidmadd.h	⊢ 𝐻 = (𝑥 ∈ (𝑀...𝑁) ↦ 𝐷)

**Assertion**
Ref	Expression
gsumfzmptfidmadd2	⊢ (𝜑 → (𝐺 Σ_g (𝐹 ∘_𝑓 + 𝐻)) = ((𝐺 Σ_g 𝐹) + (𝐺 Σ_g 𝐻)))

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

**Proof of Theorem gsumfzmptfidmadd2**
Step	Hyp	Ref	Expression
1		gsumfzmptfidmadd.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2		gsumfzmptfidmadd.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
3	1, 2	fzfigd 10508	. . . 4 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
4		gsumfzmptfidmadd.c	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐶 ∈ 𝐵)
5		gsumfzmptfidmadd.d	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐷 ∈ 𝐵)
6		gsumfzmptfidmadd.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (𝑀...𝑁) ↦ 𝐶)
7	6	a1i 9	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝑀...𝑁) ↦ 𝐶))
8		gsumfzmptfidmadd.h	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ (𝑀...𝑁) ↦ 𝐷)
9	8	a1i 9	. . . 4 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝑀...𝑁) ↦ 𝐷))
10	3, 4, 5, 7, 9	offval2 6151	. . 3 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐻) = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐶 + 𝐷)))
11	10	oveq2d 5938	. 2 ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ∘_𝑓 + 𝐻)) = (𝐺 Σ_g (𝑥 ∈ (𝑀...𝑁) ↦ (𝐶 + 𝐷))))
12		gsummptfidmadd.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
13		gsummptfidmadd.p	. . 3 ⊢ + = (+_g‘𝐺)
14		gsummptfidmadd.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd)
15	12, 13, 14, 1, 2, 4, 5, 6, 8	gsumfzmptfidmadd 13445	. 2 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (𝑀...𝑁) ↦ (𝐶 + 𝐷))) = ((𝐺 Σ_g 𝐹) + (𝐺 Σ_g 𝐻)))
16	11, 15	eqtrd 2229	1 ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ∘_𝑓 + 𝐻)) = ((𝐺 Σ_g 𝐹) + (𝐺 Σ_g 𝐻)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2167 ↦ cmpt 4094 ‘cfv 5258 (class class class)co 5922 ∘_𝑓 cof 6133 Fincfn 6799 ℤcz 9323 ...cfz 10080 Basecbs 12654 +_gcplusg 12731 Σ_g cgsu 12904 CMndccmn 13390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7968 ax-resscn 7969 ax-1cn 7970 ax-1re 7971 ax-icn 7972 ax-addcl 7973 ax-addrcl 7974 ax-mulcl 7975 ax-addcom 7977 ax-addass 7979 ax-distr 7981 ax-i2m1 7982 ax-0lt1 7983 ax-0id 7985 ax-rnegex 7986 ax-cnre 7988 ax-pre-ltirr 7989 ax-pre-ltwlin 7990 ax-pre-lttrn 7991 ax-pre-apti 7992 ax-pre-ltadd 7993

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-of 6135 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-1o 6474 df-er 6592 df-en 6800 df-fin 6802 df-pnf 8061 df-mnf 8062 df-xr 8063 df-ltxr 8064 df-le 8065 df-sub 8197 df-neg 8198 df-inn 8988 df-2 9046 df-n0 9247 df-z 9324 df-uz 9599 df-fz 10081 df-fzo 10215 df-seqfrec 10525 df-ndx 12657 df-slot 12658 df-base 12660 df-plusg 12744 df-0g 12905 df-igsum 12906 df-mgm 12975 df-sgrp 13021 df-mnd 13034 df-cmn 13392

This theorem is referenced by: lgseisenlem3 15280 lgseisenlem4 15281

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