gsumshift - Intuitionistic Logic Explorer

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

Theorem gsumshift 14105

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

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

**Assertion**
Ref	Expression
gsumshift	⊢ (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 gsumshift Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumshift.m	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		1zzd 9621	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
3		eluzel2 9876	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
4	1, 3	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5	2, 4	zsubcld 9723	. . . 4 ⊢ (𝜑 → (1 − 𝑀) ∈ ℤ)
6		eluzelz 9881	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
7	1, 6	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
8	5, 4, 7	mptfzshft 12153	. . . . . . . . . 10 ⊢ (𝜑 → (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))):((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀)))–1-1-onto→(𝑀...𝑁))
9		gsumshift.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (𝑗 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀)))
10	4	zcnd 9719	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℂ)
11		1cnd 8306	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
12	10, 11	pncan3d 8603	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 + (1 − 𝑀)) = 1)
13	12	oveq1d 6073	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) = (1...(𝑁 + (1 − 𝑀))))
14	13	mpteq1d 4200	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))) = (𝑗 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))))
15	9, 14	eqtr4id 2286	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 = (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))))
16	13	eqcomd 2240	. . . . . . . . . . 11 ⊢ (𝜑 → (1...(𝑁 + (1 − 𝑀))) = ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))))
17		eqidd 2235	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀...𝑁) = (𝑀...𝑁))
18	15, 16, 17	f1oeq123d 5613	. . . . . . . . . 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 5619	. . . . . . . . 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 9621	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 1 ∈ ℤ)
24	7, 5	zaddcld 9722	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + (1 − 𝑀)) ∈ ℤ)
25	24	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑁 + (1 − 𝑀)) ∈ ℤ)
26		elfzelz 10378	. . . . . . . . . 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 9722	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + (1 − 𝑀)) ∈ ℤ)
30	4	zred 9718	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
31	30	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℝ)
32	27	zred 9718	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℝ)
33		1red 8305	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 1 ∈ ℝ)
34		elfzle1 10381	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑘)
35	34	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑘)
36	31, 32, 33, 35	lesub2dd 8853	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (1 − 𝑘) ≤ (1 − 𝑀))
37	33, 31	resubcld 8671	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (1 − 𝑀) ∈ ℝ)
38	33, 32, 37	lesubadd2d 8835	. . . . . . . . 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 9718	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
41	40	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℝ)
42		elfzle2 10382	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ≤ 𝑁)
43	42	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ≤ 𝑁)
44	32, 41, 37, 43	leadd1dd 8850	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + (1 − 𝑀)) ≤ (𝑁 + (1 − 𝑀)))
45	23, 25, 29, 39, 44	elfzd 10369	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + (1 − 𝑀)) ∈ (1...(𝑁 + (1 − 𝑀))))
46		fvco3 5753	. . . . . . 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 5677	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑆‘(𝑘 + (1 − 𝑀))) = ((𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀)))‘(𝑘 + (1 − 𝑀))))
50		eqid 2234	. . . . . . . . . 10 ⊢ (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))) = (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀)))
51		oveq1 6065	. . . . . . . . . 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 10414	. . . . . . . . . . . 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 1275	. . . . . . . . . . 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 9723	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑘 + (1 − 𝑀)) − (1 − 𝑀)) ∈ ℤ)
59	50, 51, 57, 58	fvmptd3 5776	. . . . . . . . 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 2267	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑆‘(𝑘 + (1 − 𝑀))) = ((𝑘 + (1 − 𝑀)) − (1 − 𝑀)))
61	26	zcnd 9719	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℂ)
62	61	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℂ)
63	11, 10	subcld 8600	. . . . . . . . . 10 ⊢ (𝜑 → (1 − 𝑀) ∈ ℂ)
64	63	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (1 − 𝑀) ∈ ℂ)
65	62, 64	pncand 8601	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑘 + (1 − 𝑀)) − (1 − 𝑀)) = 𝑘)
66	60, 65	eqtrd 2267	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑆‘(𝑘 + (1 − 𝑀))) = 𝑘)
67	66	fveq2d 5679	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘(𝑆‘(𝑘 + (1 − 𝑀)))) = (𝐹‘𝑘))
68	47, 67	eqtrd 2267	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝐹 ∘ 𝑆)‘(𝑘 + (1 − 𝑀))) = (𝐹‘𝑘))
69	68	eqcomd 2240	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = ((𝐹 ∘ 𝑆)‘(𝑘 + (1 − 𝑀))))
70		gsumshift.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ CMnd)
71		plusgslid 13409	. . . . . 6 ⊢ (+_g = Slot (+_g‘ndx) ∧ (+_g‘ndx) ∈ ℕ)
72	71	slotex 13323	. . . . 5 ⊢ (𝐺 ∈ CMnd → (+_g‘𝐺) ∈ V)
73	70, 72	syl 14	. . . 4 ⊢ (𝜑 → (+_g‘𝐺) ∈ V)
74		gsumshift.f	. . . . 5 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
75	4, 7	fzfigd 10817	. . . . 5 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
76	74, 75	fexd 5921	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
77	2, 24	fzfigd 10817	. . . . . 6 ⊢ (𝜑 → (1...(𝑁 + (1 − 𝑀))) ∈ Fin)
78		mptexg 5916	. . . . . . 7 ⊢ ((1...(𝑁 + (1 − 𝑀))) ∈ Fin → (𝑗 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))) ∈ V)
79	9, 78	eqeltrid 2321	. . . . . 6 ⊢ ((1...(𝑁 + (1 − 𝑀))) ∈ Fin → 𝑆 ∈ V)
80	77, 79	syl 14	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ V)
81		coexg 5312	. . . . 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 10868	. . 3 ⊢ (𝜑 → (seq𝑀((+_g‘𝐺), 𝐹)‘𝑁) = (seq(𝑀 + (1 − 𝑀))((+_g‘𝐺), (𝐹 ∘ 𝑆))‘(𝑁 + (1 − 𝑀))))
84	12	seqeq1d 10839	. . . 4 ⊢ (𝜑 → seq(𝑀 + (1 − 𝑀))((+_g‘𝐺), (𝐹 ∘ 𝑆)) = seq1((+_g‘𝐺), (𝐹 ∘ 𝑆)))
85	84	fveq1d 5677	. . 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 2267	. 2 ⊢ (𝜑 → (seq𝑀((+_g‘𝐺), 𝐹)‘𝑁) = (seq1((+_g‘𝐺), (𝐹 ∘ 𝑆))‘(𝑁 + (1 − 𝑀))))
87		gsumshift.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
88		eqid 2234	. . 3 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
89	87, 88, 70, 1, 74	gsumval2 13660	. 2 ⊢ (𝜑 → (𝐺 Σ_g 𝐹) = (seq𝑀((+_g‘𝐺), 𝐹)‘𝑁))
90		1red 8305	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
91		eluzle 9884	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑁)
92	1, 91	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
93	30, 40, 90, 92	lesub2dd 8853	. . . . 5 ⊢ (𝜑 → (1 − 𝑁) ≤ (1 − 𝑀))
94	90, 30	resubcld 8671	. . . . . 6 ⊢ (𝜑 → (1 − 𝑀) ∈ ℝ)
95	90, 40, 94	lesubadd2d 8835	. . . . 5 ⊢ (𝜑 → ((1 − 𝑁) ≤ (1 − 𝑀) ↔ 1 ≤ (𝑁 + (1 − 𝑀))))
96	93, 95	mpbid 147	. . . 4 ⊢ (𝜑 → 1 ≤ (𝑁 + (1 − 𝑀)))
97		eluz2 9877	. . . 4 ⊢ ((𝑁 + (1 − 𝑀)) ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ (𝑁 + (1 − 𝑀)) ∈ ℤ ∧ 1 ≤ (𝑁 + (1 − 𝑀))))
98	2, 24, 96, 97	syl3anbrc 1208	. . 3 ⊢ (𝜑 → (𝑁 + (1 − 𝑀)) ∈ (ℤ_≥‘1))
99		fco 5532	. . . 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 13660	. 2 ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ∘ 𝑆)) = (seq1((+_g‘𝐺), (𝐹 ∘ 𝑆))‘(𝑁 + (1 − 𝑀))))
102	86, 89, 101	3eqtr4d 2277	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 1398 ∈ wcel 2205 Vcvv 2815 class class class wbr 4114 ↦ cmpt 4176 ∘ ccom 4758 ⟶wf 5353 –1-1-onto→wf1o 5356 ‘cfv 5357 (class class class)co 6058 Fincfn 6988 ℂcc 8141 ℝcr 8142 1c1 8144 + caddc 8146 ≤ cle 8325 − cmin 8460 ℤcz 9594 ℤ_≥cuz 9871 ...cfz 10361 seqcseq 10833 Basecbs 13296 +_gcplusg 13374 Σ_g cgsu 13554 CMndccmn 14037

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-addcom 8243 ax-addass 8245 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-0id 8251 ax-rnegex 8252 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-1o 6660 df-er 6780 df-en 6989 df-fin 6991 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-inn 9255 df-2 9313 df-n0 9514 df-z 9595 df-uz 9872 df-fz 10362 df-seqfrec 10834 df-ndx 13299 df-slot 13300 df-base 13302 df-plusg 13387 df-0g 13555 df-igsum 13556

This theorem is referenced by: gsumgfsum 14106

W3C validator