gzsumshift - Intuitionistic Logic Explorer

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Theorem gzsumshift 14239

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

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

**Assertion**
Ref	Expression
gzsumshift	⊢ (𝜑 → (𝐺 Σ_gz 𝐹) = (𝐺 Σ_gz (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 gzsumshift Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gzsumshift.m	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		1zzd 9682	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
3		eluzel2 9942	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
4	1, 3	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5	2, 4	zsubcld 9784	. . . 4 ⊢ (𝜑 → (1 − 𝑀) ∈ ℤ)
6		eluzelz 9947	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
7	1, 6	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
8	5, 4, 7	mptfzshft 12234	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))):((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀)))–1-1-onto→(𝑀...𝑁))
9		gzsumshift.s	. . . . . . . . . . 11 ⊢ 𝑆 = (𝑗 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀)))
10	4	zcnd 9780	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℂ)
11		1cnd 8348	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℂ)
12	10, 11	pncan3d 8647	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 + (1 − 𝑀)) = 1)
13	12	oveq1d 6103	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) = (1...(𝑁 + (1 − 𝑀))))
14	13	mpteq1d 4217	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))) = (𝑗 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))))
15	9, 14	eqtr4id 2290	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 = (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))))
16	13	eqcomd 2244	. . . . . . . . . 10 ⊢ (𝜑 → (1...(𝑁 + (1 − 𝑀))) = ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))))
17		eqidd 2239	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀...𝑁) = (𝑀...𝑁))
18	15, 16, 17	f1oeq123d 5636	. . . . . . . . 9 ⊢ (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	. . . . . . . 8 ⊢ (𝜑 → 𝑆:(1...(𝑁 + (1 − 𝑀)))–1-1-onto→(𝑀...𝑁))
20		f1of 5642	. . . . . . . 8 ⊢ (𝑆:(1...(𝑁 + (1 − 𝑀)))–1-1-onto→(𝑀...𝑁) → 𝑆:(1...(𝑁 + (1 − 𝑀)))⟶(𝑀...𝑁))
21	19, 20	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑆:(1...(𝑁 + (1 − 𝑀)))⟶(𝑀...𝑁))
22	21	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑆:(1...(𝑁 + (1 − 𝑀)))⟶(𝑀...𝑁))
23		1zzd 9682	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 1 ∈ ℤ)
24	7, 5	zaddcld 9783	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + (1 − 𝑀)) ∈ ℤ)
25	24	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑁 + (1 − 𝑀)) ∈ ℤ)
26		elfzelz 10445	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℤ)
27	26	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℤ)
28	5	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (1 − 𝑀) ∈ ℤ)
29	27, 28	zaddcld 9783	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + (1 − 𝑀)) ∈ ℤ)
30	4	zred 9779	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
31	30	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℝ)
32	27	zred 9779	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℝ)
33		1red 8347	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 1 ∈ ℝ)
34		elfzle1 10448	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑘)
35	34	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑘)
36	31, 32, 33, 35	lesub2dd 8897	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (1 − 𝑘) ≤ (1 − 𝑀))
37	33, 31	resubcld 8715	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (1 − 𝑀) ∈ ℝ)
38	33, 32, 37	lesubadd2d 8879	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((1 − 𝑘) ≤ (1 − 𝑀) ↔ 1 ≤ (𝑘 + (1 − 𝑀))))
39	36, 38	mpbid 147	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 1 ≤ (𝑘 + (1 − 𝑀)))
40	7	zred 9779	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
41	40	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℝ)
42		elfzle2 10449	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ≤ 𝑁)
43	42	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ≤ 𝑁)
44	32, 41, 37, 43	leadd1dd 8894	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + (1 − 𝑀)) ≤ (𝑁 + (1 − 𝑀)))
45	23, 25, 29, 39, 44	elfzd 10436	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + (1 − 𝑀)) ∈ (1...(𝑁 + (1 − 𝑀))))
46		fvco3 5779	. . . . . 6 ⊢ ((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 415	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝐹 ∘ 𝑆)‘(𝑘 + (1 − 𝑀))) = (𝐹‘(𝑆‘(𝑘 + (1 − 𝑀)))))
48	15	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑆 = (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))))
49	48	fveq1d 5700	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑆‘(𝑘 + (1 − 𝑀))) = ((𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀)))‘(𝑘 + (1 − 𝑀))))
50		eqid 2238	. . . . . . . . 9 ⊢ (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))) = (𝑗 ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀)))
51		oveq1 6095	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + (1 − 𝑀)) → (𝑗 − (1 − 𝑀)) = ((𝑘 + (1 − 𝑀)) − (1 − 𝑀)))
52		simpr 110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
53	4	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ)
54	7	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℤ)
55		fzaddel 10482	. . . . . . . . . . 11 ⊢ (((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 1279	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝑘 + (1 − 𝑀)) ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀)))))
57	52, 56	mpbid 147	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + (1 − 𝑀)) ∈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))))
58	29, 28	zsubcld 9784	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑘 + (1 − 𝑀)) − (1 − 𝑀)) ∈ ℤ)
59	50, 51, 57, 58	fvmptd3 5802	. . . . . . . 8 ⊢ ((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 2271	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑆‘(𝑘 + (1 − 𝑀))) = ((𝑘 + (1 − 𝑀)) − (1 − 𝑀)))
61	26	zcnd 9780	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℂ)
62	61	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℂ)
63	11, 10	subcld 8644	. . . . . . . . 9 ⊢ (𝜑 → (1 − 𝑀) ∈ ℂ)
64	63	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (1 − 𝑀) ∈ ℂ)
65	62, 64	pncand 8645	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑘 + (1 − 𝑀)) − (1 − 𝑀)) = 𝑘)
66	60, 65	eqtrd 2271	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑆‘(𝑘 + (1 − 𝑀))) = 𝑘)
67	66	fveq2d 5702	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘(𝑆‘(𝑘 + (1 − 𝑀)))) = (𝐹‘𝑘))
68	47, 67	eqtr2d 2272	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = ((𝐹 ∘ 𝑆)‘(𝑘 + (1 − 𝑀))))
69		gzsumshift.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ CMnd)
70		plusgslid 13525	. . . . . 6 ⊢ (+_g = Slot (+_g‘ndx) ∧ (+_g‘ndx) ∈ ℕ)
71	70	slotex 13437	. . . . 5 ⊢ (𝐺 ∈ CMnd → (+_g‘𝐺) ∈ V)
72	69, 71	syl 14	. . . 4 ⊢ (𝜑 → (+_g‘𝐺) ∈ V)
73		gzsumshift.f	. . . . 5 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
74	4, 7	fzfigd 10889	. . . . 5 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
75	73, 74	fexd 5951	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
76	2, 24	fzfigd 10889	. . . . . 6 ⊢ (𝜑 → (1...(𝑁 + (1 − 𝑀))) ∈ Fin)
77		mptexg 5945	. . . . . . 7 ⊢ ((1...(𝑁 + (1 − 𝑀))) ∈ Fin → (𝑗 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))) ∈ V)
78	9, 77	eqeltrid 2325	. . . . . 6 ⊢ ((1...(𝑁 + (1 − 𝑀))) ∈ Fin → 𝑆 ∈ V)
79	76, 78	syl 14	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ V)
80		coexg 5335	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝑆 ∈ V) → (𝐹 ∘ 𝑆) ∈ V)
81	75, 79, 80	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝑆) ∈ V)
82	1, 5, 68, 72, 75, 81	seqshft2g 10940	. . 3 ⊢ (𝜑 → (seq𝑀((+_g‘𝐺), 𝐹)‘𝑁) = (seq(𝑀 + (1 − 𝑀))((+_g‘𝐺), (𝐹 ∘ 𝑆))‘(𝑁 + (1 − 𝑀))))
83	12	seqeq1d 10911	. . . 4 ⊢ (𝜑 → seq(𝑀 + (1 − 𝑀))((+_g‘𝐺), (𝐹 ∘ 𝑆)) = seq1((+_g‘𝐺), (𝐹 ∘ 𝑆)))
84	83	fveq1d 5700	. . 3 ⊢ (𝜑 → (seq(𝑀 + (1 − 𝑀))((+_g‘𝐺), (𝐹 ∘ 𝑆))‘(𝑁 + (1 − 𝑀))) = (seq1((+_g‘𝐺), (𝐹 ∘ 𝑆))‘(𝑁 + (1 − 𝑀))))
85	82, 84	eqtrd 2271	. 2 ⊢ (𝜑 → (seq𝑀((+_g‘𝐺), 𝐹)‘𝑁) = (seq1((+_g‘𝐺), (𝐹 ∘ 𝑆))‘(𝑁 + (1 − 𝑀))))
86		gzsumshift.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
87		eqid 2238	. . 3 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
88	86, 87, 69, 1, 73	gzsumval2 13773	. 2 ⊢ (𝜑 → (𝐺 Σ_gz 𝐹) = (seq𝑀((+_g‘𝐺), 𝐹)‘𝑁))
89		1red 8347	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
90		eluzle 9950	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑁)
91	1, 90	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
92	30, 40, 89, 91	lesub2dd 8897	. . . . 5 ⊢ (𝜑 → (1 − 𝑁) ≤ (1 − 𝑀))
93	89, 30	resubcld 8715	. . . . . 6 ⊢ (𝜑 → (1 − 𝑀) ∈ ℝ)
94	89, 40, 93	lesubadd2d 8879	. . . . 5 ⊢ (𝜑 → ((1 − 𝑁) ≤ (1 − 𝑀) ↔ 1 ≤ (𝑁 + (1 − 𝑀))))
95	92, 94	mpbid 147	. . . 4 ⊢ (𝜑 → 1 ≤ (𝑁 + (1 − 𝑀)))
96		eluz2 9943	. . . 4 ⊢ ((𝑁 + (1 − 𝑀)) ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ (𝑁 + (1 − 𝑀)) ∈ ℤ ∧ 1 ≤ (𝑁 + (1 − 𝑀))))
97	2, 24, 95, 96	syl3anbrc 1212	. . 3 ⊢ (𝜑 → (𝑁 + (1 − 𝑀)) ∈ (ℤ_≥‘1))
98	73, 21	fcod 5556	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝑆):(1...(𝑁 + (1 − 𝑀)))⟶𝐵)
99	86, 87, 69, 97, 98	gzsumval2 13773	. 2 ⊢ (𝜑 → (𝐺 Σ_gz (𝐹 ∘ 𝑆)) = (seq1((+_g‘𝐺), (𝐹 ∘ 𝑆))‘(𝑁 + (1 − 𝑀))))
100	85, 88, 99	3eqtr4d 2281	1 ⊢ (𝜑 → (𝐺 Σ_gz 𝐹) = (𝐺 Σ_gz (𝐹 ∘ 𝑆)))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 Vcvv 2821 class class class wbr 4131 ↦ cmpt 4193 ∘ ccom 4780 ⟶wf 5376 –1-1-onto→wf1o 5379 ‘cfv 5380 (class class class)co 6088 Fincfn 7026 ℂcc 8183 ℝcr 8184 1c1 8186 + caddc 8188 ≤ cle 8367 − cmin 8504 ℤcz 9655 ℤ_≥cuz 9937 ...cfz 10428 seqcseq 10905 Basecbs 13410 +_gcplusg 13490 Σ_gz cgzsu 13670 CMndccmn 14177

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-addcom 8285 ax-addass 8287 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-0id 8293 ax-rnegex 8294 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301

This proof depends on definitions: df-bi 117 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-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-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-frec 6666 df-1o 6691 df-er 6811 df-en 7027 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-inn 9314 df-2 9372 df-n0 9575 df-z 9656 df-uz 9938 df-fz 10429 df-seqfrec 10906 df-ndx 13413 df-slot 13414 df-base 13416 df-plusg 13503 df-0g 13671 df-gzsum 13672

This theorem is used by: gzsumgsum 14245

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