gsumgfsumlem - Mathbox for Jim Kingdon

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Theorem gsumgfsumlem 16751

Description: Shifting the indexes of a group sum indexed by consecutive integers. (Contributed by Jim Kingdon, 26-Mar-2026.)

**Hypotheses**
Ref	Expression
gsumgfsumlem.b	⊢ 𝐵 = (Base‘𝐺)
gsumgfsumlem.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
gsumgfsumlem.m	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
gsumgfsumlem.f	⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
gsumgfsumlem.s	⊢ 𝑆 = (𝑗 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀)))

**Assertion**
Ref	Expression
gsumgfsumlem	⊢ (𝜑 → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝐹 ∘ 𝑆)))

Distinct variable groups: 𝑗,𝑀 𝑗,𝑁 𝜑,𝑗 Allowed substitution hints: 𝐵(𝑗) 𝑆(𝑗) 𝐹(𝑗) 𝐺(𝑗)

Proof of Theorem gsumgfsumlem Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumgfsumlem.m	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		1zzd 9511	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
3		eluzel2 9765	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
4	1, 3	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5	2, 4	zsubcld 9612	. . . 4 ⊢ (𝜑 → (1 − 𝑀) ∈ ℤ)
6		eluzelz 9770	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
7	1, 6	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
8	5, 4, 7	mptfzshft 12026	. . . . . . . . . 10 ⊢ (𝜑 → (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))):((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀)))–1-1-onto→(𝑀...𝑁))
9		gsumgfsumlem.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (𝑗 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀)))
10	4	zcnd 9608	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℂ)
11		1cnd 8200	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
12	10, 11	pncan3d 8498	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 + (1 − 𝑀)) = 1)
13	12	oveq1d 6038	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) = (1...(𝑁 + (1 − 𝑀))))
14	13	mpteq1d 4175	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))) = (𝑗 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))))
15	9, 14	eqtr4id 2282	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 = (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))))
16	13	eqcomd 2236	. . . . . . . . . . 11 ⊢ (𝜑 → (1...(𝑁 + (1 − 𝑀))) = ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))))
17		eqidd 2231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀...𝑁) = (𝑀...𝑁))
18	15, 16, 17	f1oeq123d 5580	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆:(1...(𝑁 + (1 − 𝑀)))–1-1-onto→(𝑀...𝑁) ↔ (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))):((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀)))–1-1-onto→(𝑀...𝑁)))
19	8, 18	mpbird 167	. . . . . . . . 9 ⊢ (𝜑 → 𝑆:(1...(𝑁 + (1 − 𝑀)))–1-1-onto→(𝑀...𝑁))
20		f1of 5586	. . . . . . . . 9 ⊢ (𝑆:(1...(𝑁 + (1 − 𝑀)))–1-1-onto→(𝑀...𝑁) → 𝑆:(1...(𝑁 + (1 − 𝑀)))⟶(𝑀...𝑁))
21	19, 20	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑆:(1...(𝑁 + (1 − 𝑀)))⟶(𝑀...𝑁))
22	21	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑆:(1...(𝑁 + (1 − 𝑀)))⟶(𝑀...𝑁))
23		1zzd 9511	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 1 ∈ ℤ)
24	7, 5	zaddcld 9611	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + (1 − 𝑀)) ∈ ℤ)
25	24	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑁 + (1 − 𝑀)) ∈ ℤ)
26		elfzelz 10265	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℤ)
27	26	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℤ)
28	5	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (1 − 𝑀) ∈ ℤ)
29	27, 28	zaddcld 9611	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + (1 − 𝑀)) ∈ ℤ)
30	4	zred 9607	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
31	30	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℝ)
32	27	zred 9607	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℝ)
33		1red 8199	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 1 ∈ ℝ)
34		elfzle1 10267	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑘)
35	34	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑘)
36	31, 32, 33, 35	lesub2dd 8747	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (1 − 𝑘) ≤ (1 − 𝑀))
37	33, 31	resubcld 8565	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (1 − 𝑀) ∈ ℝ)
38	33, 32, 37	lesubadd2d 8729	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((1 − 𝑘) ≤ (1 − 𝑀) ↔ 1 ≤ (𝑘 + (1 − 𝑀))))
39	36, 38	mpbid 147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 1 ≤ (𝑘 + (1 − 𝑀)))
40	7	zred 9607	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
41	40	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℝ)
42		elfzle2 10268	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ≤ 𝑁)
43	42	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ≤ 𝑁)
44	32, 41, 37, 43	leadd1dd 8744	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + (1 − 𝑀)) ≤ (𝑁 + (1 − 𝑀)))
45	23, 25, 29, 39, 44	elfzd 10256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + (1 − 𝑀)) ∈ (1...(𝑁 + (1 − 𝑀))))
46		fvco3 5720	. . . . . . 7 ⊢ ((𝑆:(1...(𝑁 + (1 − 𝑀)))⟶(𝑀...𝑁) ∧ (𝑘 + (1 − 𝑀)) ∈ (1...(𝑁 + (1 − 𝑀)))) → ((𝐹 ∘ 𝑆)‘(𝑘 + (1 − 𝑀))) = (𝐹‘(𝑆‘(𝑘 + (1 − 𝑀)))))
47	22, 45, 46	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝐹 ∘ 𝑆)‘(𝑘 + (1 − 𝑀))) = (𝐹‘(𝑆‘(𝑘 + (1 − 𝑀)))))
48	15	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑆 = (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))))
49	48	fveq1d 5644	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑆‘(𝑘 + (1 − 𝑀))) = ((𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀)))‘(𝑘 + (1 − 𝑀))))
50		eqid 2230	. . . . . . . . . 10 ⊢ (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))) = (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀)))
51		oveq1 6030	. . . . . . . . . 10 ⊢ (𝑗 = (𝑘 + (1 − 𝑀)) → (𝑗 − (1 − 𝑀)) = ((𝑘 + (1 − 𝑀)) − (1 − 𝑀)))
52		simpr 110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
53	4	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ)
54	7	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℤ)
55		fzaddel 10299	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ (1 − 𝑀) ∈ ℤ)) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝑘 + (1 − 𝑀)) ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀)))))
56	53, 54, 27, 28, 55	syl22anc 1274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝑘 + (1 − 𝑀)) ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀)))))
57	52, 56	mpbid 147	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + (1 − 𝑀)) ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))))
58	29, 28	zsubcld 9612	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑘 + (1 − 𝑀)) − (1 − 𝑀)) ∈ ℤ)
59	50, 51, 57, 58	fvmptd3 5743	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀)))‘(𝑘 + (1 − 𝑀))) = ((𝑘 + (1 − 𝑀)) − (1 − 𝑀)))
60	49, 59	eqtrd 2263	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑆‘(𝑘 + (1 − 𝑀))) = ((𝑘 + (1 − 𝑀)) − (1 − 𝑀)))
61	26	zcnd 9608	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℂ)
62	61	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℂ)
63	11, 10	subcld 8495	. . . . . . . . . 10 ⊢ (𝜑 → (1 − 𝑀) ∈ ℂ)
64	63	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (1 − 𝑀) ∈ ℂ)
65	62, 64	pncand 8496	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑘 + (1 − 𝑀)) − (1 − 𝑀)) = 𝑘)
66	60, 65	eqtrd 2263	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑆‘(𝑘 + (1 − 𝑀))) = 𝑘)
67	66	fveq2d 5646	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘(𝑆‘(𝑘 + (1 − 𝑀)))) = (𝐹‘𝑘))
68	47, 67	eqtrd 2263	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝐹 ∘ 𝑆)‘(𝑘 + (1 − 𝑀))) = (𝐹‘𝑘))
69	68	eqcomd 2236	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = ((𝐹 ∘ 𝑆)‘(𝑘 + (1 − 𝑀))))
70		gsumgfsumlem.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ CMnd)
71		plusgslid 13218	. . . . . 6 ⊢ (+_g = Slot (+_g‘ndx) ∧ (+_g‘ndx) ∈ ℕ)
72	71	slotex 13132	. . . . 5 ⊢ (𝐺 ∈ CMnd → (+_g‘𝐺) ∈ V)
73	70, 72	syl 14	. . . 4 ⊢ (𝜑 → (+_g‘𝐺) ∈ V)
74		gsumgfsumlem.f	. . . . 5 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
75	4, 7	fzfigd 10699	. . . . 5 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
76	74, 75	fexd 5889	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
77	2, 24	fzfigd 10699	. . . . . 6 ⊢ (𝜑 → (1...(𝑁 + (1 − 𝑀))) ∈ Fin)
78		mptexg 5884	. . . . . . 7 ⊢ ((1...(𝑁 + (1 − 𝑀))) ∈ Fin → (𝑗 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))) ∈ V)
79	9, 78	eqeltrid 2317	. . . . . 6 ⊢ ((1...(𝑁 + (1 − 𝑀))) ∈ Fin → 𝑆 ∈ V)
80	77, 79	syl 14	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ V)
81		coexg 5283	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝑆 ∈ V) → (𝐹 ∘ 𝑆) ∈ V)
82	76, 80, 81	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝑆) ∈ V)
83	1, 5, 69, 73, 76, 82	seqshft2g 10750	. . 3 ⊢ (𝜑 → (seq𝑀((+_g‘𝐺), 𝐹)‘𝑁) = (seq(𝑀 + (1 − 𝑀))((+_g‘𝐺), (𝐹 ∘ 𝑆))‘(𝑁 + (1 − 𝑀))))
84	12	seqeq1d 10721	. . . 4 ⊢ (𝜑 → seq(𝑀 + (1 − 𝑀))((+_g‘𝐺), (𝐹 ∘ 𝑆)) = seq1((+_g‘𝐺), (𝐹 ∘ 𝑆)))
85	84	fveq1d 5644	. . 3 ⊢ (𝜑 → (seq(𝑀 + (1 − 𝑀))((+_g‘𝐺), (𝐹 ∘ 𝑆))‘(𝑁 + (1 − 𝑀))) = (seq1((+_g‘𝐺), (𝐹 ∘ 𝑆))‘(𝑁 + (1 − 𝑀))))
86	83, 85	eqtrd 2263	. 2 ⊢ (𝜑 → (seq𝑀((+_g‘𝐺), 𝐹)‘𝑁) = (seq1((+_g‘𝐺), (𝐹 ∘ 𝑆))‘(𝑁 + (1 − 𝑀))))
87		gsumgfsumlem.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
88		eqid 2230	. . 3 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
89	87, 88, 70, 1, 74	gsumval2 13503	. 2 ⊢ (𝜑 → (𝐺 Σ_g 𝐹) = (seq𝑀((+_g‘𝐺), 𝐹)‘𝑁))
90		1red 8199	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
91		eluzle 9773	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑁)
92	1, 91	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
93	30, 40, 90, 92	lesub2dd 8747	. . . . 5 ⊢ (𝜑 → (1 − 𝑁) ≤ (1 − 𝑀))
94	90, 30	resubcld 8565	. . . . . 6 ⊢ (𝜑 → (1 − 𝑀) ∈ ℝ)
95	90, 40, 94	lesubadd2d 8729	. . . . 5 ⊢ (𝜑 → ((1 − 𝑁) ≤ (1 − 𝑀) ↔ 1 ≤ (𝑁 + (1 − 𝑀))))
96	93, 95	mpbid 147	. . . 4 ⊢ (𝜑 → 1 ≤ (𝑁 + (1 − 𝑀)))
97		eluz2 9766	. . . 4 ⊢ ((𝑁 + (1 − 𝑀)) ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ (𝑁 + (1 − 𝑀)) ∈ ℤ ∧ 1 ≤ (𝑁 + (1 − 𝑀))))
98	2, 24, 96, 97	syl3anbrc 1207	. . 3 ⊢ (𝜑 → (𝑁 + (1 − 𝑀)) ∈ (ℤ_≥‘1))
99		fco 5502	. . . 4 ⊢ ((𝐹:(𝑀...𝑁)⟶𝐵 ∧ 𝑆:(1...(𝑁 + (1 − 𝑀)))⟶(𝑀...𝑁)) → (𝐹 ∘ 𝑆):(1...(𝑁 + (1 − 𝑀)))⟶𝐵)
100	74, 21, 99	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝑆):(1...(𝑁 + (1 − 𝑀)))⟶𝐵)
101	87, 88, 70, 98, 100	gsumval2 13503	. 2 ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ∘ 𝑆)) = (seq1((+_g‘𝐺), (𝐹 ∘ 𝑆))‘(𝑁 + (1 − 𝑀))))
102	86, 89, 101	3eqtr4d 2273	1 ⊢ (𝜑 → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝐹 ∘ 𝑆)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1397 ∈ wcel 2201 Vcvv 2801 class class class wbr 4089 ↦ cmpt 4151 ∘ ccom 4731 ⟶wf 5324 –1-1-onto→wf1o 5327 ‘cfv 5328 (class class class)co 6023 Fincfn 6914 ℂcc 8035 ℝcr 8036 1c1 8038 + caddc 8040 ≤ cle 8220 − cmin 8355 ℤcz 9484 ℤ_≥cuz 9760 ...cfz 10248 seqcseq 10715 Basecbs 13105 +_gcplusg 13183 Σ_g cgsu 13363 CMndccmn 13894

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 2203 ax-14 2204 ax-ext 2212 ax-coll 4205 ax-sep 4208 ax-nul 4216 ax-pow 4266 ax-pr 4301 ax-un 4532 ax-setind 4637 ax-iinf 4688 ax-cnex 8128 ax-resscn 8129 ax-1cn 8130 ax-1re 8131 ax-icn 8132 ax-addcl 8133 ax-addrcl 8134 ax-mulcl 8135 ax-addcom 8137 ax-addass 8139 ax-distr 8141 ax-i2m1 8142 ax-0lt1 8143 ax-0id 8145 ax-rnegex 8146 ax-cnre 8148 ax-pre-ltirr 8149 ax-pre-ltwlin 8150 ax-pre-lttrn 8151 ax-pre-apti 8152 ax-pre-ltadd 8153

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 1810 df-eu 2081 df-mo 2082 df-clab 2217 df-cleq 2223 df-clel 2226 df-nfc 2362 df-ne 2402 df-nel 2497 df-ral 2514 df-rex 2515 df-reu 2516 df-rab 2518 df-v 2803 df-sbc 3031 df-csb 3127 df-dif 3201 df-un 3203 df-in 3205 df-ss 3212 df-nul 3494 df-pw 3655 df-sn 3676 df-pr 3677 df-op 3679 df-uni 3895 df-int 3930 df-iun 3973 df-br 4090 df-opab 4152 df-mpt 4153 df-tr 4189 df-id 4392 df-iord 4465 df-on 4467 df-ilim 4468 df-suc 4470 df-iom 4691 df-xp 4733 df-rel 4734 df-cnv 4735 df-co 4736 df-dm 4737 df-rn 4738 df-res 4739 df-ima 4740 df-iota 5288 df-fun 5330 df-fn 5331 df-f 5332 df-f1 5333 df-fo 5334 df-f1o 5335 df-fv 5336 df-riota 5976 df-ov 6026 df-oprab 6027 df-mpo 6028 df-1st 6308 df-2nd 6309 df-recs 6476 df-frec 6562 df-1o 6587 df-er 6707 df-en 6915 df-fin 6917 df-pnf 8221 df-mnf 8222 df-xr 8223 df-ltxr 8224 df-le 8225 df-sub 8357 df-neg 8358 df-inn 9149 df-2 9207 df-n0 9408 df-z 9485 df-uz 9761 df-fz 10249 df-seqfrec 10716 df-ndx 13108 df-slot 13109 df-base 13111 df-plusg 13196 df-0g 13364 df-igsum 13365

This theorem is referenced by: gsumgfsum 16752

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