gsumfzmptfidmadd2 - Intuitionistic Logic Explorer

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Theorem gsumfzmptfidmadd2 13863

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 10640	. . . 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 6224	. . 3 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐻) = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐶 + 𝐷)))
11	10	oveq2d 6010	. 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 13862	. 2 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (𝑀...𝑁) ↦ (𝐶 + 𝐷))) = ((𝐺 Σ_g 𝐹) + (𝐺 Σ_g 𝐻)))
16	11, 15	eqtrd 2262	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 1395 ∈ wcel 2200 ↦ cmpt 4144 ‘cfv 5314 (class class class)co 5994 ∘_𝑓 cof 6206 Fincfn 6877 ℤcz 9434 ...cfz 10192 Basecbs 13018 +_gcplusg 13096 Σ_g cgsu 13276 CMndccmn 13807

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4521 ax-setind 4626 ax-iinf 4677 ax-cnex 8078 ax-resscn 8079 ax-1cn 8080 ax-1re 8081 ax-icn 8082 ax-addcl 8083 ax-addrcl 8084 ax-mulcl 8085 ax-addcom 8087 ax-addass 8089 ax-distr 8091 ax-i2m1 8092 ax-0lt1 8093 ax-0id 8095 ax-rnegex 8096 ax-cnre 8098 ax-pre-ltirr 8099 ax-pre-ltwlin 8100 ax-pre-lttrn 8101 ax-pre-apti 8102 ax-pre-ltadd 8103

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4381 df-iord 4454 df-on 4456 df-ilim 4457 df-suc 4459 df-iom 4680 df-xp 4722 df-rel 4723 df-cnv 4724 df-co 4725 df-dm 4726 df-rn 4727 df-res 4728 df-ima 4729 df-iota 5274 df-fun 5316 df-fn 5317 df-f 5318 df-f1 5319 df-fo 5320 df-f1o 5321 df-fv 5322 df-riota 5947 df-ov 5997 df-oprab 5998 df-mpo 5999 df-of 6208 df-1st 6276 df-2nd 6277 df-recs 6441 df-frec 6527 df-1o 6552 df-er 6670 df-en 6878 df-fin 6880 df-pnf 8171 df-mnf 8172 df-xr 8173 df-ltxr 8174 df-le 8175 df-sub 8307 df-neg 8308 df-inn 9099 df-2 9157 df-n0 9358 df-z 9435 df-uz 9711 df-fz 10193 df-fzo 10327 df-seqfrec 10657 df-ndx 13021 df-slot 13022 df-base 13024 df-plusg 13109 df-0g 13277 df-igsum 13278 df-mgm 13375 df-sgrp 13421 df-mnd 13436 df-cmn 13809

This theorem is referenced by: lgseisenlem3 15736 lgseisenlem4 15737

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