gsumfzmptfidmadd2 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > gsumfzmptfidmadd2		GIF version

Theorem gsumfzmptfidmadd2 13932

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 10694	. . . 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 6251	. . 3 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐻) = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐶 + 𝐷)))
11	10	oveq2d 6034	. 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 13931	. 2 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (𝑀...𝑁) ↦ (𝐶 + 𝐷))) = ((𝐺 Σ_g 𝐹) + (𝐺 Σ_g 𝐻)))
16	11, 15	eqtrd 2264	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 1397 ∈ wcel 2202 ↦ cmpt 4150 ‘cfv 5326 (class class class)co 6018 ∘_𝑓 cof 6233 Fincfn 6909 ℤcz 9479 ...cfz 10243 Basecbs 13087 +_gcplusg 13165 Σ_g cgsu 13345 CMndccmn 13876

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-addcom 8132 ax-addass 8134 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-0id 8140 ax-rnegex 8141 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-of 6235 df-1st 6303 df-2nd 6304 df-recs 6471 df-frec 6557 df-1o 6582 df-er 6702 df-en 6910 df-fin 6912 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-inn 9144 df-2 9202 df-n0 9403 df-z 9480 df-uz 9756 df-fz 10244 df-fzo 10378 df-seqfrec 10711 df-ndx 13090 df-slot 13091 df-base 13093 df-plusg 13178 df-0g 13346 df-igsum 13347 df-mgm 13444 df-sgrp 13490 df-mnd 13505 df-cmn 13878

This theorem is referenced by: lgseisenlem3 15807 lgseisenlem4 15808

W3C validator