Theorem gsummptfidmadd 14251

Description: The sum of two group sums expressed as mappings with finite domain. (Contributed by AV, 23-Jul-2019.)

Hypotheses

Ref	Expression
gsummptfidmadd.b	⊢ 𝐵 = (Base‘𝐺)
gsummptfidmadd.p	⊢ + = (+g‘𝐺)
gsummptfidmadd.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
gsummptfidmadd.a	⊢ (𝜑 → 𝐴 ∈ Fin)
gsummptfidmadd.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
gsummptfidmadd.d	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵)
gsummptfidmadd.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
gsummptfidmadd.h	⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷)

Assertion

Ref	Expression
gsummptfidmadd	⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝑥, +

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)   𝐹(𝑥)   𝐺(𝑥)   𝐻(𝑥)

Proof of Theorem gsummptfidmadd

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpteq1 4216	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ 𝑤 ↦ (𝐶 + 𝐷)) = (𝑥 ∈ ∅ ↦ (𝐶 + 𝐷)))
2	1	oveq2d 6104	. . . . . 6 ⊢ (𝑤 = ∅ → (𝐺 Σg (𝑥 ∈ 𝑤 ↦ (𝐶 + 𝐷))) = (𝐺 Σg (𝑥 ∈ ∅ ↦ (𝐶 + 𝐷))))
3		reseq2 5060	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝐹 ↾ 𝑤) = (𝐹 ↾ ∅))
4	3	oveq2d 6104	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝐺 Σg (𝐹 ↾ 𝑤)) = (𝐺 Σg (𝐹 ↾ ∅)))
5		reseq2 5060	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝐻 ↾ 𝑤) = (𝐻 ↾ ∅))
6	5	oveq2d 6104	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝐺 Σg (𝐻 ↾ 𝑤)) = (𝐺 Σg (𝐻 ↾ ∅)))
7	4, 6	oveq12d 6106	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝐺 Σg (𝐹 ↾ 𝑤)) + (𝐺 Σg (𝐻 ↾ 𝑤))) = ((𝐺 Σg (𝐹 ↾ ∅)) + (𝐺 Σg (𝐻 ↾ ∅))))
8	2, 7	eqeq12d 2253	. . . . 5 ⊢ (𝑤 = ∅ → ((𝐺 Σg (𝑥 ∈ 𝑤 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ 𝑤)) + (𝐺 Σg (𝐻 ↾ 𝑤))) ↔ (𝐺 Σg (𝑥 ∈ ∅ ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ ∅)) + (𝐺 Σg (𝐻 ↾ ∅)))))
9		mpteq1 4216	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↦ (𝐶 + 𝐷)) = (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷)))
10	9	oveq2d 6104	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝐺 Σg (𝑥 ∈ 𝑤 ↦ (𝐶 + 𝐷))) = (𝐺 Σg (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷))))
11		reseq2 5060	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦))
12	11	oveq2d 6104	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑤)) = (𝐺 Σg (𝐹 ↾ 𝑦)))
13		reseq2 5060	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝐻 ↾ 𝑤) = (𝐻 ↾ 𝑦))
14	13	oveq2d 6104	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝐺 Σg (𝐻 ↾ 𝑤)) = (𝐺 Σg (𝐻 ↾ 𝑦)))
15	12, 14	oveq12d 6106	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝐺 Σg (𝐹 ↾ 𝑤)) + (𝐺 Σg (𝐻 ↾ 𝑤))) = ((𝐺 Σg (𝐹 ↾ 𝑦)) + (𝐺 Σg (𝐻 ↾ 𝑦))))
16	10, 15	eqeq12d 2253	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝐺 Σg (𝑥 ∈ 𝑤 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ 𝑤)) + (𝐺 Σg (𝐻 ↾ 𝑤))) ↔ (𝐺 Σg (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ 𝑦)) + (𝐺 Σg (𝐻 ↾ 𝑦)))))
17		mpteq1 4216	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ 𝑤 ↦ (𝐶 + 𝐷)) = (𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷)))
18	17	oveq2d 6104	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐺 Σg (𝑥 ∈ 𝑤 ↦ (𝐶 + 𝐷))) = (𝐺 Σg (𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷))))
19		reseq2 5060	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ 𝑤) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
20	19	oveq2d 6104	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐺 Σg (𝐹 ↾ 𝑤)) = (𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))))
21		reseq2 5060	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐻 ↾ 𝑤) = (𝐻 ↾ (𝑦 ∪ {𝑧})))
22	21	oveq2d 6104	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐺 Σg (𝐻 ↾ 𝑤)) = (𝐺 Σg (𝐻 ↾ (𝑦 ∪ {𝑧}))))
23	20, 22	oveq12d 6106	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐺 Σg (𝐹 ↾ 𝑤)) + (𝐺 Σg (𝐻 ↾ 𝑤))) = ((𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))) + (𝐺 Σg (𝐻 ↾ (𝑦 ∪ {𝑧})))))
24	18, 23	eqeq12d 2253	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐺 Σg (𝑥 ∈ 𝑤 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ 𝑤)) + (𝐺 Σg (𝐻 ↾ 𝑤))) ↔ (𝐺 Σg (𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))) + (𝐺 Σg (𝐻 ↾ (𝑦 ∪ {𝑧}))))))
25		mpteq1 4216	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 ∈ 𝑤 ↦ (𝐶 + 𝐷)) = (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷)))
26	25	oveq2d 6104	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝐺 Σg (𝑥 ∈ 𝑤 ↦ (𝐶 + 𝐷))) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))))
27		reseq2 5060	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝐴))
28	27	oveq2d 6104	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝐺 Σg (𝐹 ↾ 𝑤)) = (𝐺 Σg (𝐹 ↾ 𝐴)))
29		reseq2 5060	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝐻 ↾ 𝑤) = (𝐻 ↾ 𝐴))
30	29	oveq2d 6104	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝐺 Σg (𝐻 ↾ 𝑤)) = (𝐺 Σg (𝐻 ↾ 𝐴)))
31	28, 30	oveq12d 6106	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((𝐺 Σg (𝐹 ↾ 𝑤)) + (𝐺 Σg (𝐻 ↾ 𝑤))) = ((𝐺 Σg (𝐹 ↾ 𝐴)) + (𝐺 Σg (𝐻 ↾ 𝐴))))
32	26, 31	eqeq12d 2253	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝐺 Σg (𝑥 ∈ 𝑤 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ 𝑤)) + (𝐺 Σg (𝐻 ↾ 𝑤))) ↔ (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ 𝐴)) + (𝐺 Σg (𝐻 ↾ 𝐴)))))
33		gsummptfidmadd.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ CMnd)
34	33	cmnmndd 14201	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Mnd)
35		gsummptfidmadd.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
36		eqid 2238	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
37	35, 36	mndidcl 13802	. . . . . . 7 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵)
38		gsummptfidmadd.p	. . . . . . . 8 ⊢ + = (+g‘𝐺)
39	35, 38, 36	mndlid 13807	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ (0g‘𝐺) ∈ 𝐵) → ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺))
40	34, 37, 39	syl2anc2 416	. . . . . 6 ⊢ (𝜑 → ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺))
41		res0 5069	. . . . . . . . 9 ⊢ (𝐹 ↾ ∅) = ∅
42	41	oveq2i 6099	. . . . . . . 8 ⊢ (𝐺 Σg (𝐹 ↾ ∅)) = (𝐺 Σg ∅)
43		gsum0cmn 14244	. . . . . . . . 9 ⊢ (𝐺 ∈ CMnd → (𝐺 Σg ∅) = (0g‘𝐺))
44	33, 43	syl 14	. . . . . . . 8 ⊢ (𝜑 → (𝐺 Σg ∅) = (0g‘𝐺))
45	42, 44	eqtrid 2283	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ ∅)) = (0g‘𝐺))
46		res0 5069	. . . . . . . . 9 ⊢ (𝐻 ↾ ∅) = ∅
47	46	oveq2i 6099	. . . . . . . 8 ⊢ (𝐺 Σg (𝐻 ↾ ∅)) = (𝐺 Σg ∅)
48	47, 44	eqtrid 2283	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σg (𝐻 ↾ ∅)) = (0g‘𝐺))
49	45, 48	oveq12d 6106	. . . . . 6 ⊢ (𝜑 → ((𝐺 Σg (𝐹 ↾ ∅)) + (𝐺 Σg (𝐻 ↾ ∅))) = ((0g‘𝐺) + (0g‘𝐺)))
50		mpt0 5514	. . . . . . . 8 ⊢ (𝑥 ∈ ∅ ↦ (𝐶 + 𝐷)) = ∅
51	50	oveq2i 6099	. . . . . . 7 ⊢ (𝐺 Σg (𝑥 ∈ ∅ ↦ (𝐶 + 𝐷))) = (𝐺 Σg ∅)
52	51, 44	eqtrid 2283	. . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ ∅ ↦ (𝐶 + 𝐷))) = (0g‘𝐺))
53	40, 49, 52	3eqtr4rd 2282	. . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ ∅ ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ ∅)) + (𝐺 Σg (𝐻 ↾ ∅))))
54		simpr 110	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ 𝑦)) + (𝐺 Σg (𝐻 ↾ 𝑦)))) → (𝐺 Σg (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ 𝑦)) + (𝐺 Σg (𝐻 ↾ 𝑦))))
55	54	oveq1d 6103	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ 𝑦)) + (𝐺 Σg (𝐻 ↾ 𝑦)))) → ((𝐺 Σg (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷))) + (⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷)) = (((𝐺 Σg (𝐹 ↾ 𝑦)) + (𝐺 Σg (𝐻 ↾ 𝑦))) + (⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷)))
56	33	ad2antrr 492	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝐺 ∈ CMnd)
57	34	ad3antrrr 496	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑥 ∈ (𝑦 ∪ {𝑧})) → 𝐺 ∈ Mnd)
58		simplll 539	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑥 ∈ (𝑦 ∪ {𝑧})) → 𝜑)
59		simprl 535	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ⊆ 𝐴)
60		simprr 537	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
61	60	eldifad 3231	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
62	61	snssd 3860	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → {𝑧} ⊆ 𝐴)
63	59, 62	unssd 3405	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
64	63	sselda 3248	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑥 ∈ (𝑦 ∪ {𝑧})) → 𝑥 ∈ 𝐴)
65		gsummptfidmadd.c	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
66	58, 64, 65	syl2anc 415	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑥 ∈ (𝑦 ∪ {𝑧})) → 𝐶 ∈ 𝐵)
67		gsummptfidmadd.d	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵)
68	58, 64, 67	syl2anc 415	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑥 ∈ (𝑦 ∪ {𝑧})) → 𝐷 ∈ 𝐵)
69	35, 38	mndcl 13795	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Mnd ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐶 + 𝐷) ∈ 𝐵)
70	57, 66, 68, 69	syl3anc 1278	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑥 ∈ (𝑦 ∪ {𝑧})) → (𝐶 + 𝐷) ∈ 𝐵)
71	70	fmpttd 5866	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷)):(𝑦 ∪ {𝑧})⟶𝐵)
72		simplr 533	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
73	60	eldifbd 3232	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
74	35, 38, 56, 71, 72, 61, 73	gsump1 14247	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷))) = ((𝐺 Σg ((𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷)) ↾ 𝑦)) + ((𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷))‘𝑧)))
75		ssun1 3392	. . . . . . . . . . . . 13 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
76		resmpt 5114	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → ((𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷)) ↾ 𝑦) = (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷)))
77	75, 76	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷)) ↾ 𝑦) = (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷))
78	77	oveq2i 6099	. . . . . . . . . . 11 ⊢ (𝐺 Σg ((𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷)) ↾ 𝑦)) = (𝐺 Σg (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷)))
79	78	a1i 9	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg ((𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷)) ↾ 𝑦)) = (𝐺 Σg (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷))))
80		ssun2 3393	. . . . . . . . . . . 12 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
81		vsnid 3741	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ {𝑧}
82	80, 81	sselii 3245	. . . . . . . . . . 11 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
83	34	ad2antrr 492	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝐺 ∈ Mnd)
84	65	ralrimiva 2623	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
85	84	ad2antrr 492	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
86		nfcsb1v 3180	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶
87	86	nfel1 2403	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶 ∈ 𝐵
88		csbeq1a 3156	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑥⦌𝐶)
89	88	eleq1d 2307	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐶 ∈ 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝐶 ∈ 𝐵))
90	87, 89	rspc 2923	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → ⦋𝑧 / 𝑥⦌𝐶 ∈ 𝐵))
91	61, 85, 90	sylc 62	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑥⦌𝐶 ∈ 𝐵)
92	67	ralrimiva 2623	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐷 ∈ 𝐵)
93	92	ad2antrr 492	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑥 ∈ 𝐴 𝐷 ∈ 𝐵)
94		nfcsb1v 3180	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐷
95	94	nfel1 2403	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐷 ∈ 𝐵
96		csbeq1a 3156	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → 𝐷 = ⦋𝑧 / 𝑥⦌𝐷)
97	96	eleq1d 2307	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐷 ∈ 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝐷 ∈ 𝐵))
98	95, 97	rspc 2923	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐷 ∈ 𝐵 → ⦋𝑧 / 𝑥⦌𝐷 ∈ 𝐵))
99	61, 93, 98	sylc 62	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑥⦌𝐷 ∈ 𝐵)
100	35, 38	mndcl 13795	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Mnd ∧ ⦋𝑧 / 𝑥⦌𝐶 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝐷 ∈ 𝐵) → (⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷) ∈ 𝐵)
101	83, 91, 99, 100	syl3anc 1278	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷) ∈ 𝐵)
102		nfcv 2392	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑧
103		nfcv 2392	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 +
104	86, 103, 94	nfov 6118	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷)
105	88, 96	oveq12d 6106	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐶 + 𝐷) = (⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷))
106		eqid 2238	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷)) = (𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷))
107	102, 104, 105, 106	fvmptf 5801	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝑦 ∪ {𝑧}) ∧ (⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷) ∈ 𝐵) → ((𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷))‘𝑧) = (⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷))
108	82, 101, 107	sylancr 418	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷))‘𝑧) = (⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷))
109	79, 108	oveq12d 6106	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝐺 Σg ((𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷)) ↾ 𝑦)) + ((𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷))‘𝑧)) = ((𝐺 Σg (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷))) + (⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷)))
110	74, 109	eqtrd 2271	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷))) + (⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷)))
111	110	adantr 276	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ 𝑦)) + (𝐺 Σg (𝐻 ↾ 𝑦)))) → (𝐺 Σg (𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷))) + (⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷)))
112		gsummptfidmadd.f	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
113	65, 112	fmptd 5865	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
114	113	ad2antrr 492	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝐹:𝐴⟶𝐵)
115	114, 63	fssresd 5569	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐹 ↾ (𝑦 ∪ {𝑧})):(𝑦 ∪ {𝑧})⟶𝐵)
116	35, 38, 56, 115, 72, 61, 73	gsump1 14247	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))) = ((𝐺 Σg ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦)) + ((𝐹 ↾ (𝑦 ∪ {𝑧}))‘𝑧)))
117		resabs1 5094	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦) = (𝐹 ↾ 𝑦))
118	75, 117	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦) = (𝐹 ↾ 𝑦)
119	118	oveq2i 6099	. . . . . . . . . . . . 13 ⊢ (𝐺 Σg ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦)) = (𝐺 Σg (𝐹 ↾ 𝑦))
120	119	a1i 9	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦)) = (𝐺 Σg (𝐹 ↾ 𝑦)))
121		fvres 5722	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) → ((𝐹 ↾ (𝑦 ∪ {𝑧}))‘𝑧) = (𝐹‘𝑧))
122	82, 121	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((𝐹 ↾ (𝑦 ∪ {𝑧}))‘𝑧) = (𝐹‘𝑧)
123	112	fvmpts 5786	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐴 ∧ ⦋𝑧 / 𝑥⦌𝐶 ∈ 𝐵) → (𝐹‘𝑧) = ⦋𝑧 / 𝑥⦌𝐶)
124	61, 91, 123	syl2anc 415	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐹‘𝑧) = ⦋𝑧 / 𝑥⦌𝐶)
125	122, 124	eqtrid 2283	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝐹 ↾ (𝑦 ∪ {𝑧}))‘𝑧) = ⦋𝑧 / 𝑥⦌𝐶)
126	120, 125	oveq12d 6106	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝐺 Σg ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦)) + ((𝐹 ↾ (𝑦 ∪ {𝑧}))‘𝑧)) = ((𝐺 Σg (𝐹 ↾ 𝑦)) + ⦋𝑧 / 𝑥⦌𝐶))
127	116, 126	eqtrd 2271	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))) = ((𝐺 Σg (𝐹 ↾ 𝑦)) + ⦋𝑧 / 𝑥⦌𝐶))
128		gsummptfidmadd.h	. . . . . . . . . . . . . . 15 ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷)
129	67, 128	fmptd 5865	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)
130	129	ad2antrr 492	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝐻:𝐴⟶𝐵)
131	130, 63	fssresd 5569	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐻 ↾ (𝑦 ∪ {𝑧})):(𝑦 ∪ {𝑧})⟶𝐵)
132	35, 38, 56, 131, 72, 61, 73	gsump1 14247	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝐻 ↾ (𝑦 ∪ {𝑧}))) = ((𝐺 Σg ((𝐻 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦)) + ((𝐻 ↾ (𝑦 ∪ {𝑧}))‘𝑧)))
133		resabs1 5094	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → ((𝐻 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦) = (𝐻 ↾ 𝑦))
134	75, 133	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((𝐻 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦) = (𝐻 ↾ 𝑦)
135	134	oveq2i 6099	. . . . . . . . . . . . 13 ⊢ (𝐺 Σg ((𝐻 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦)) = (𝐺 Σg (𝐻 ↾ 𝑦))
136	135	a1i 9	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg ((𝐻 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦)) = (𝐺 Σg (𝐻 ↾ 𝑦)))
137		fvres 5722	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) → ((𝐻 ↾ (𝑦 ∪ {𝑧}))‘𝑧) = (𝐻‘𝑧))
138	82, 137	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((𝐻 ↾ (𝑦 ∪ {𝑧}))‘𝑧) = (𝐻‘𝑧)
139	128	fvmpts 5786	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐴 ∧ ⦋𝑧 / 𝑥⦌𝐷 ∈ 𝐵) → (𝐻‘𝑧) = ⦋𝑧 / 𝑥⦌𝐷)
140	61, 99, 139	syl2anc 415	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐻‘𝑧) = ⦋𝑧 / 𝑥⦌𝐷)
141	138, 140	eqtrid 2283	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝐻 ↾ (𝑦 ∪ {𝑧}))‘𝑧) = ⦋𝑧 / 𝑥⦌𝐷)
142	136, 141	oveq12d 6106	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝐺 Σg ((𝐻 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦)) + ((𝐻 ↾ (𝑦 ∪ {𝑧}))‘𝑧)) = ((𝐺 Σg (𝐻 ↾ 𝑦)) + ⦋𝑧 / 𝑥⦌𝐷))
143	132, 142	eqtrd 2271	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝐻 ↾ (𝑦 ∪ {𝑧}))) = ((𝐺 Σg (𝐻 ↾ 𝑦)) + ⦋𝑧 / 𝑥⦌𝐷))
144	127, 143	oveq12d 6106	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))) + (𝐺 Σg (𝐻 ↾ (𝑦 ∪ {𝑧})))) = (((𝐺 Σg (𝐹 ↾ 𝑦)) + ⦋𝑧 / 𝑥⦌𝐶) + ((𝐺 Σg (𝐻 ↾ 𝑦)) + ⦋𝑧 / 𝑥⦌𝐷)))
145	114, 59	fssresd 5569	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐹 ↾ 𝑦):𝑦⟶𝐵)
146	35, 36, 56, 72, 145	gsumclfi 14249	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝐵)
147	130, 59	fssresd 5569	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐻 ↾ 𝑦):𝑦⟶𝐵)
148	35, 36, 56, 72, 147	gsumclfi 14249	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝐻 ↾ 𝑦)) ∈ 𝐵)
149	35, 38	cmn4 14198	. . . . . . . . . 10 ⊢ ((𝐺 ∈ CMnd ∧ ((𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝐶 ∈ 𝐵) ∧ ((𝐺 Σg (𝐻 ↾ 𝑦)) ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝐷 ∈ 𝐵)) → (((𝐺 Σg (𝐹 ↾ 𝑦)) + ⦋𝑧 / 𝑥⦌𝐶) + ((𝐺 Σg (𝐻 ↾ 𝑦)) + ⦋𝑧 / 𝑥⦌𝐷)) = (((𝐺 Σg (𝐹 ↾ 𝑦)) + (𝐺 Σg (𝐻 ↾ 𝑦))) + (⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷)))
150	56, 146, 91, 148, 99, 149	syl122anc 1287	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (((𝐺 Σg (𝐹 ↾ 𝑦)) + ⦋𝑧 / 𝑥⦌𝐶) + ((𝐺 Σg (𝐻 ↾ 𝑦)) + ⦋𝑧 / 𝑥⦌𝐷)) = (((𝐺 Σg (𝐹 ↾ 𝑦)) + (𝐺 Σg (𝐻 ↾ 𝑦))) + (⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷)))
151	144, 150	eqtrd 2271	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))) + (𝐺 Σg (𝐻 ↾ (𝑦 ∪ {𝑧})))) = (((𝐺 Σg (𝐹 ↾ 𝑦)) + (𝐺 Σg (𝐻 ↾ 𝑦))) + (⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷)))
152	151	adantr 276	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ 𝑦)) + (𝐺 Σg (𝐻 ↾ 𝑦)))) → ((𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))) + (𝐺 Σg (𝐻 ↾ (𝑦 ∪ {𝑧})))) = (((𝐺 Σg (𝐹 ↾ 𝑦)) + (𝐺 Σg (𝐻 ↾ 𝑦))) + (⦋𝑧 / 𝑥⦌𝐶 + ⦋𝑧 / 𝑥⦌𝐷)))
153	55, 111, 152	3eqtr4d 2281	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ 𝑦)) + (𝐺 Σg (𝐻 ↾ 𝑦)))) → (𝐺 Σg (𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))) + (𝐺 Σg (𝐻 ↾ (𝑦 ∪ {𝑧})))))
154	153	ex 115	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝐺 Σg (𝑥 ∈ 𝑦 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ 𝑦)) + (𝐺 Σg (𝐻 ↾ 𝑦))) → (𝐺 Σg (𝑥 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))) + (𝐺 Σg (𝐻 ↾ (𝑦 ∪ {𝑧}))))))
155		gsummptfidmadd.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
156	8, 16, 24, 32, 53, 154, 155	findcard2sd 7200	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝐹 ↾ 𝐴)) + (𝐺 Σg (𝐻 ↾ 𝐴))))
157	112	reseq1i 5061	. . . . . 6 ⊢ (𝐹 ↾ 𝐴) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴)
158	157	oveq2i 6099	. . . . 5 ⊢ (𝐺 Σg (𝐹 ↾ 𝐴)) = (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴))
159	128	reseq1i 5061	. . . . . 6 ⊢ (𝐻 ↾ 𝐴) = ((𝑥 ∈ 𝐴 ↦ 𝐷) ↾ 𝐴)
160	159	oveq2i 6099	. . . . 5 ⊢ (𝐺 Σg (𝐻 ↾ 𝐴)) = (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐷) ↾ 𝐴))
161	158, 160	oveq12i 6100	. . . 4 ⊢ ((𝐺 Σg (𝐹 ↾ 𝐴)) + (𝐺 Σg (𝐻 ↾ 𝐴))) = ((𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴)) + (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐷) ↾ 𝐴)))
162	156, 161	eqtrdi 2287	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴)) + (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐷) ↾ 𝐴))))
163		ssidd 3269	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
164	163	resmptd 5117	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶))
165	164	oveq2d 6104	. . . 4 ⊢ (𝜑 → (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴)) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)))
166	163	resmptd 5117	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐷) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐷))
167	166	oveq2d 6104	. . . 4 ⊢ (𝜑 → (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐷) ↾ 𝐴)) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐷)))
168	165, 167	oveq12d 6106	. . 3 ⊢ (𝜑 → ((𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴)) + (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐷) ↾ 𝐴))) = ((𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) + (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐷))))
169	162, 168	eqtrd 2271	. 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) + (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐷))))
170	112	oveq2i 6099	. . 3 ⊢ (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶))
171	128	oveq2i 6099	. . 3 ⊢ (𝐺 Σg 𝐻) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐷))
172	170, 171	oveq12i 6100	. 2 ⊢ ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)) = ((𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) + (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐷)))
173	169, 172	eqtr4di 2289	1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ⦋csb 3147   ∖ cdif 3217   ∪ cun 3218   ⊆ wss 3220  ∅c0 3520  {csn 3709   ↦ cmpt 4193   ↾ cres 4778  ⟶wf 5376  ‘cfv 5380  (class class class)co 6088  Fincfn 7026  Basecbs 13410  +_gcplusg 13490  0_gc0g 13669  Mndcmnd 13788  CMndccmn 14177   Σ_g cgsu 14240

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-ihash 11237  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-0g 13671  df-gzsum 13672  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-minusg 13868  df-mulg 13982  df-cmn 14179  df-gsumfi 14241

This theorem is used by:  gsummptfidmadd2  14252