gzsumshift - Intuitionistic Logic Explorer

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Theorem gzsumshift 14149

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 9671	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
3		eluzel2 9926	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
4	1, 3	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5	2, 4	zsubcld 9773	. . . 4 ⊢ (𝜑 → (1 − 𝑀) ∈ ℤ)
6		eluzelz 9931	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
7	1, 6	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
8	5, 4, 7	mptfzshft 12209	. . . . . . . . 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 9769	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℂ)
11		1cnd 8342	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℂ)
12	10, 11	pncan3d 8640	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 + (1 − 𝑀)) = 1)
13	12	oveq1d 6100	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) = (1...(𝑁 + (1 − 𝑀))))
14	13	mpteq1d 4216	. . . . . . . . . . 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 5633	. . . . . . . . 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 5639	. . . . . . . 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 9671	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 1 ∈ ℤ)
24	7, 5	zaddcld 9772	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + (1 − 𝑀)) ∈ ℤ)
25	24	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑁 + (1 − 𝑀)) ∈ ℤ)
26		elfzelz 10428	. . . . . . . . 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 9772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + (1 − 𝑀)) ∈ ℤ)
30	4	zred 9768	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
31	30	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℝ)
32	27	zred 9768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℝ)
33		1red 8341	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 1 ∈ ℝ)
34		elfzle1 10431	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑘)
35	34	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑘)
36	31, 32, 33, 35	lesub2dd 8890	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (1 − 𝑘) ≤ (1 − 𝑀))
37	33, 31	resubcld 8708	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (1 − 𝑀) ∈ ℝ)
38	33, 32, 37	lesubadd2d 8872	. . . . . . . 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 9768	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
41	40	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℝ)
42		elfzle2 10432	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ≤ 𝑁)
43	42	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ≤ 𝑁)
44	32, 41, 37, 43	leadd1dd 8887	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + (1 − 𝑀)) ≤ (𝑁 + (1 − 𝑀)))
45	23, 25, 29, 39, 44	elfzd 10419	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + (1 − 𝑀)) ∈ (1...(𝑁 + (1 − 𝑀))))
46		fvco3 5776	. . . . . 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 5697	. . . . . . . 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 6092	. . . . . . . . 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 10465	. . . . . . . . . . 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 9773	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑘 + (1 − 𝑀)) − (1 − 𝑀)) ∈ ℤ)
59	50, 51, 57, 58	fvmptd3 5799	. . . . . . . 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 9769	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℂ)
62	61	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℂ)
63	11, 10	subcld 8637	. . . . . . . . 9 ⊢ (𝜑 → (1 − 𝑀) ∈ ℂ)
64	63	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (1 − 𝑀) ∈ ℂ)
65	62, 64	pncand 8638	. . . . . . 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 5699	. . . . 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 13466	. . . . . 6 ⊢ (+_g = Slot (+_g‘ndx) ∧ (+_g‘ndx) ∈ ℕ)
71	70	slotex 13379	. . . . 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 10868	. . . . 5 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
75	73, 74	fexd 5948	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
76	2, 24	fzfigd 10868	. . . . . 6 ⊢ (𝜑 → (1...(𝑁 + (1 − 𝑀))) ∈ Fin)
77		mptexg 5942	. . . . . . 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 5332	. . . . 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 10919	. . 3 ⊢ (𝜑 → (seq𝑀((+_g‘𝐺), 𝐹)‘𝑁) = (seq(𝑀 + (1 − 𝑀))((+_g‘𝐺), (𝐹 ∘ 𝑆))‘(𝑁 + (1 − 𝑀))))
83	12	seqeq1d 10890	. . . 4 ⊢ (𝜑 → seq(𝑀 + (1 − 𝑀))((+_g‘𝐺), (𝐹 ∘ 𝑆)) = seq1((+_g‘𝐺), (𝐹 ∘ 𝑆)))
84	83	fveq1d 5697	. . 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 13714	. 2 ⊢ (𝜑 → (𝐺 Σ_gz 𝐹) = (seq𝑀((+_g‘𝐺), 𝐹)‘𝑁))
89		1red 8341	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
90		eluzle 9934	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑁)
91	1, 90	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
92	30, 40, 89, 91	lesub2dd 8890	. . . . 5 ⊢ (𝜑 → (1 − 𝑁) ≤ (1 − 𝑀))
93	89, 30	resubcld 8708	. . . . . 6 ⊢ (𝜑 → (1 − 𝑀) ∈ ℝ)
94	89, 40, 93	lesubadd2d 8872	. . . . 5 ⊢ (𝜑 → ((1 − 𝑁) ≤ (1 − 𝑀) ↔ 1 ≤ (𝑁 + (1 − 𝑀))))
95	92, 94	mpbid 147	. . . 4 ⊢ (𝜑 → 1 ≤ (𝑁 + (1 − 𝑀)))
96		eluz2 9927	. . . 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 5553	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝑆):(1...(𝑁 + (1 − 𝑀)))⟶𝐵)
99	86, 87, 69, 97, 98	gzsumval2 13714	. 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 4130 ↦ cmpt 4192 ∘ ccom 4778 ⟶wf 5373 –1-1-onto→wf1o 5376 ‘cfv 5377 (class class class)co 6085 Fincfn 7022 ℂcc 8177 ℝcr 8178 1c1 8180 + caddc 8182 ≤ cle 8361 − cmin 8497 ℤcz 9644 ℤ_≥cuz 9921 ...cfz 10411 seqcseq 10884 Basecbs 13352 +_gcplusg 13431 Σ_gz cgzsu 13611 CMndccmn 14087

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 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-addcom 8279 ax-addass 8281 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-0id 8287 ax-rnegex 8288 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295

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 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-1o 6687 df-er 6807 df-en 7023 df-fin 7025 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-inn 9305 df-2 9363 df-n0 9564 df-z 9645 df-uz 9922 df-fz 10412 df-seqfrec 10885 df-ndx 13355 df-slot 13356 df-base 13358 df-plusg 13444 df-0g 13612 df-gzsum 13613

This theorem is used by: gzsumgsum 14155

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