gsumfzmptfidmadd2 - Intuitionistic Logic Explorer

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Theorem gsumfzmptfidmadd2 13480

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 10525	. . . 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 6152	. . 3 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐻) = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐶 + 𝐷)))
11	10	oveq2d 5939	. 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 13479	. 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 4095 ‘cfv 5259 (class class class)co 5923 ∘_𝑓 cof 6134 Fincfn 6800 ℤcz 9328 ...cfz 10085 Basecbs 12688 +_gcplusg 12765 Σ_g cgsu 12938 CMndccmn 13424

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 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7972 ax-resscn 7973 ax-1cn 7974 ax-1re 7975 ax-icn 7976 ax-addcl 7977 ax-addrcl 7978 ax-mulcl 7979 ax-addcom 7981 ax-addass 7983 ax-distr 7985 ax-i2m1 7986 ax-0lt1 7987 ax-0id 7989 ax-rnegex 7990 ax-cnre 7992 ax-pre-ltirr 7993 ax-pre-ltwlin 7994 ax-pre-lttrn 7995 ax-pre-apti 7996 ax-pre-ltadd 7997

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 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-riota 5878 df-ov 5926 df-oprab 5927 df-mpo 5928 df-of 6136 df-1st 6199 df-2nd 6200 df-recs 6364 df-frec 6450 df-1o 6475 df-er 6593 df-en 6801 df-fin 6803 df-pnf 8065 df-mnf 8066 df-xr 8067 df-ltxr 8068 df-le 8069 df-sub 8201 df-neg 8202 df-inn 8993 df-2 9051 df-n0 9252 df-z 9329 df-uz 9604 df-fz 10086 df-fzo 10220 df-seqfrec 10542 df-ndx 12691 df-slot 12692 df-base 12694 df-plusg 12778 df-0g 12939 df-igsum 12940 df-mgm 13009 df-sgrp 13055 df-mnd 13068 df-cmn 13426

This theorem is referenced by: lgseisenlem3 15323 lgseisenlem4 15324

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