gsumfzmptfidmadd2 - Intuitionistic Logic Explorer

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Theorem gsumfzmptfidmadd2 13920

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 10686	. . . 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 6246	. . 3 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐻) = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐶 + 𝐷)))
11	10	oveq2d 6029	. 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 13919	. 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 4148 ‘cfv 5324 (class class class)co 6013 ∘_𝑓 cof 6228 Fincfn 6904 ℤcz 9472 ...cfz 10236 Basecbs 13075 +_gcplusg 13153 Σ_g cgsu 13333 CMndccmn 13864

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 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-addcom 8125 ax-addass 8127 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-0id 8133 ax-rnegex 8134 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141

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 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-of 6230 df-1st 6298 df-2nd 6299 df-recs 6466 df-frec 6552 df-1o 6577 df-er 6697 df-en 6905 df-fin 6907 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-inn 9137 df-2 9195 df-n0 9396 df-z 9473 df-uz 9749 df-fz 10237 df-fzo 10371 df-seqfrec 10703 df-ndx 13078 df-slot 13079 df-base 13081 df-plusg 13166 df-0g 13334 df-igsum 13335 df-mgm 13432 df-sgrp 13478 df-mnd 13493 df-cmn 13866

This theorem is referenced by: lgseisenlem3 15794 lgseisenlem4 15795

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