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Theorem seq3f1olemqsumkj 10620

Description: Lemma for seq3f1o 10626. 𝑄 gives the same sum as 𝐽 in the range (𝐾...(^◡𝐽‘𝐾)). (Contributed by Jim Kingdon, 29-Aug-2022.)

**Hypotheses**
Ref	Expression
iseqf1o.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqf1o.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
iseqf1o.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
iseqf1o.4	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
iseqf1o.6	⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
iseqf1o.7	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
iseqf1olemstep.k	⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))
iseqf1olemstep.j	⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
iseqf1olemstep.const	⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥)
iseqf1olemnk	⊢ (𝜑 → 𝐾 ≠ (^◡𝐽‘𝐾))
iseqf1olemqres.q	⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))
iseqf1olemqsumk.p	⊢ 𝑃 = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))

**Assertion**
Ref	Expression
seq3f1olemqsumkj	⊢ (𝜑 → (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)) = (seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)))

Distinct variable groups: 𝑢,𝐽 𝑢,𝐾,𝑥 𝑢,𝑀,𝑥 𝑢,𝑁 𝑥,𝐽 𝑥,𝑄 𝜑,𝑥 𝑥, + ,𝑦,𝑧 𝑓,𝐺,𝑥 𝑓,𝐽,𝑦,𝑧 𝑦,𝐾,𝑧 𝑓,𝑀 𝑓,𝑁,𝑥 𝑥,𝑃,𝑦,𝑧 𝑄,𝑓,𝑦,𝑧 𝑥,𝑆,𝑦,𝑧 𝜑,𝑢 𝜑,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑓) 𝑃(𝑢,𝑓) + (𝑢,𝑓) 𝑄(𝑢) 𝑆(𝑢,𝑓) 𝐹(𝑥,𝑦,𝑧,𝑢,𝑓) 𝐺(𝑦,𝑧,𝑢) 𝐾(𝑓) 𝑀(𝑦,𝑧) 𝑁(𝑦,𝑧)

Proof of Theorem seq3f1olemqsumkj Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqf1olemstep.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))
2		elfzelz 10117	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
3	1, 2	syl 14	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℤ)
4		iseqf1olemstep.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
5		f1ocnv 5520	. . . . . . . . . . 11 ⊢ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → ^◡𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
6	4, 5	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → ^◡𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
7		f1of 5507	. . . . . . . . . 10 ⊢ (^◡𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → ^◡𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
8	6, 7	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ^◡𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
9	8, 1	ffvelcdmd 5701	. . . . . . . 8 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ (𝑀...𝑁))
10		elfzelz 10117	. . . . . . . 8 ⊢ ((^◡𝐽‘𝐾) ∈ (𝑀...𝑁) → (^◡𝐽‘𝐾) ∈ ℤ)
11	9, 10	syl 14	. . . . . . 7 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ ℤ)
12		peano2zm 9381	. . . . . . 7 ⊢ ((^◡𝐽‘𝐾) ∈ ℤ → ((^◡𝐽‘𝐾) − 1) ∈ ℤ)
13	11, 12	syl 14	. . . . . 6 ⊢ (𝜑 → ((^◡𝐽‘𝐾) − 1) ∈ ℤ)
14		iseqf1o.4	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
15		iseqf1olemstep.const	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥)
16		iseqf1olemnk	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ≠ (^◡𝐽‘𝐾))
17	14, 1, 4, 15, 16	iseqf1olemklt 10607	. . . . . . 7 ⊢ (𝜑 → 𝐾 < (^◡𝐽‘𝐾))
18		zltlem1 9400	. . . . . . . 8 ⊢ ((𝐾 ∈ ℤ ∧ (^◡𝐽‘𝐾) ∈ ℤ) → (𝐾 < (^◡𝐽‘𝐾) ↔ 𝐾 ≤ ((^◡𝐽‘𝐾) − 1)))
19	3, 11, 18	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (𝐾 < (^◡𝐽‘𝐾) ↔ 𝐾 ≤ ((^◡𝐽‘𝐾) − 1)))
20	17, 19	mpbid 147	. . . . . 6 ⊢ (𝜑 → 𝐾 ≤ ((^◡𝐽‘𝐾) − 1))
21		eluz2 9624	. . . . . 6 ⊢ (((^◡𝐽‘𝐾) − 1) ∈ (ℤ_≥‘𝐾) ↔ (𝐾 ∈ ℤ ∧ ((^◡𝐽‘𝐾) − 1) ∈ ℤ ∧ 𝐾 ≤ ((^◡𝐽‘𝐾) − 1)))
22	3, 13, 20, 21	syl3anbrc 1183	. . . . 5 ⊢ (𝜑 → ((^◡𝐽‘𝐾) − 1) ∈ (ℤ_≥‘𝐾))
23		1zzd 9370	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
24	1	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝐾 ∈ (𝑀...𝑁))
25	4	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
26		elfzel1 10116	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
27	1, 26	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℤ)
28	27	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑀 ∈ ℤ)
29		elfzel2 10115	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
30	1, 29	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℤ)
31	30	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑁 ∈ ℤ)
32		elfzelz 10117	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1)) → 𝑣 ∈ ℤ)
33	32	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑣 ∈ ℤ)
34	33	peano2zd 9468	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑣 + 1) ∈ ℤ)
35	28, 31, 34	3jca 1179	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑣 + 1) ∈ ℤ))
36	28	zred 9465	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑀 ∈ ℝ)
37	33	zred 9465	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑣 ∈ ℝ)
38	34	zred 9465	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑣 + 1) ∈ ℝ)
39	3	zred 9465	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ ℝ)
40	39	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝐾 ∈ ℝ)
41		elfzle1 10119	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝐾)
42	1, 41	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ≤ 𝐾)
43	42	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑀 ≤ 𝐾)
44		elfzle1 10119	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1)) → 𝐾 ≤ 𝑣)
45	44	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝐾 ≤ 𝑣)
46	36, 40, 37, 43, 45	letrd 8167	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑀 ≤ 𝑣)
47	37	lep1d 8975	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑣 ≤ (𝑣 + 1))
48	36, 37, 38, 46, 47	letrd 8167	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑀 ≤ (𝑣 + 1))
49	11	zred 9465	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ ℝ)
50	49	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (^◡𝐽‘𝐾) ∈ ℝ)
51	31	zred 9465	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑁 ∈ ℝ)
52		elfzle2 10120	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1)) → 𝑣 ≤ ((^◡𝐽‘𝐾) − 1))
53	52	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑣 ≤ ((^◡𝐽‘𝐾) − 1))
54		1red 8058	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 1 ∈ ℝ)
55		leaddsub 8482	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ ℝ ∧ 1 ∈ ℝ ∧ (^◡𝐽‘𝐾) ∈ ℝ) → ((𝑣 + 1) ≤ (^◡𝐽‘𝐾) ↔ 𝑣 ≤ ((^◡𝐽‘𝐾) − 1)))
56	37, 54, 50, 55	syl3anc 1249	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → ((𝑣 + 1) ≤ (^◡𝐽‘𝐾) ↔ 𝑣 ≤ ((^◡𝐽‘𝐾) − 1)))
57	53, 56	mpbird 167	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑣 + 1) ≤ (^◡𝐽‘𝐾))
58		elfzle2 10120	. . . . . . . . . . . . . . 15 ⊢ ((^◡𝐽‘𝐾) ∈ (𝑀...𝑁) → (^◡𝐽‘𝐾) ≤ 𝑁)
59	9, 58	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (^◡𝐽‘𝐾) ≤ 𝑁)
60	59	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (^◡𝐽‘𝐾) ≤ 𝑁)
61	38, 50, 51, 57, 60	letrd 8167	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑣 + 1) ≤ 𝑁)
62	48, 61	jca 306	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑀 ≤ (𝑣 + 1) ∧ (𝑣 + 1) ≤ 𝑁))
63		elfz2 10107	. . . . . . . . . . 11 ⊢ ((𝑣 + 1) ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑣 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑣 + 1) ∧ (𝑣 + 1) ≤ 𝑁)))
64	35, 62, 63	sylanbrc 417	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑣 + 1) ∈ (𝑀...𝑁))
65		iseqf1olemqres.q	. . . . . . . . . 10 ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))
66	24, 25, 64, 65	iseqf1olemqval 10609	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑄‘(𝑣 + 1)) = if((𝑣 + 1) ∈ (𝐾...(^◡𝐽‘𝐾)), if((𝑣 + 1) = 𝐾, 𝐾, (𝐽‘((𝑣 + 1) − 1))), (𝐽‘(𝑣 + 1))))
67	24, 2	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝐾 ∈ ℤ)
68	11	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (^◡𝐽‘𝐾) ∈ ℤ)
69	67, 68, 34	3jca 1179	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝐾 ∈ ℤ ∧ (^◡𝐽‘𝐾) ∈ ℤ ∧ (𝑣 + 1) ∈ ℤ))
70	40, 37, 38, 45, 47	letrd 8167	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝐾 ≤ (𝑣 + 1))
71	70, 57	jca 306	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝐾 ≤ (𝑣 + 1) ∧ (𝑣 + 1) ≤ (^◡𝐽‘𝐾)))
72		elfz2 10107	. . . . . . . . . . 11 ⊢ ((𝑣 + 1) ∈ (𝐾...(^◡𝐽‘𝐾)) ↔ ((𝐾 ∈ ℤ ∧ (^◡𝐽‘𝐾) ∈ ℤ ∧ (𝑣 + 1) ∈ ℤ) ∧ (𝐾 ≤ (𝑣 + 1) ∧ (𝑣 + 1) ≤ (^◡𝐽‘𝐾))))
73	69, 71, 72	sylanbrc 417	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑣 + 1) ∈ (𝐾...(^◡𝐽‘𝐾)))
74	73	iftrued 3569	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → if((𝑣 + 1) ∈ (𝐾...(^◡𝐽‘𝐾)), if((𝑣 + 1) = 𝐾, 𝐾, (𝐽‘((𝑣 + 1) − 1))), (𝐽‘(𝑣 + 1))) = if((𝑣 + 1) = 𝐾, 𝐾, (𝐽‘((𝑣 + 1) − 1))))
75	66, 74	eqtrd 2229	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑄‘(𝑣 + 1)) = if((𝑣 + 1) = 𝐾, 𝐾, (𝐽‘((𝑣 + 1) − 1))))
76		zleltp1 9398	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℤ ∧ 𝑣 ∈ ℤ) → (𝐾 ≤ 𝑣 ↔ 𝐾 < (𝑣 + 1)))
77	67, 33, 76	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝐾 ≤ 𝑣 ↔ 𝐾 < (𝑣 + 1)))
78	45, 77	mpbid 147	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝐾 < (𝑣 + 1))
79	40, 78	gtned 8156	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑣 + 1) ≠ 𝐾)
80	79	neneqd 2388	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → ¬ (𝑣 + 1) = 𝐾)
81	80	iffalsed 3572	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → if((𝑣 + 1) = 𝐾, 𝐾, (𝐽‘((𝑣 + 1) − 1))) = (𝐽‘((𝑣 + 1) − 1)))
82	33	zcnd 9466	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑣 ∈ ℂ)
83		pncan1 8420	. . . . . . . . . 10 ⊢ (𝑣 ∈ ℂ → ((𝑣 + 1) − 1) = 𝑣)
84	82, 83	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → ((𝑣 + 1) − 1) = 𝑣)
85	84	fveq2d 5565	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝐽‘((𝑣 + 1) − 1)) = (𝐽‘𝑣))
86	75, 81, 85	3eqtrd 2233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑄‘(𝑣 + 1)) = (𝐽‘𝑣))
87	86	fveq2d 5565	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝐺‘(𝑄‘(𝑣 + 1))) = (𝐺‘(𝐽‘𝑣)))
88	1, 4, 65	iseqf1olemqf1o 10615	. . . . . . . 8 ⊢ (𝜑 → 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
89	88	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
90		iseqf1o.7	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
91	90	adantlr 477	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
92		iseqf1olemqsumk.p	. . . . . . 7 ⊢ 𝑃 = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))
93	24, 89, 64, 91, 92	iseqf1olemfvp 10619	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (⦋𝑄 / 𝑓⦌𝑃‘(𝑣 + 1)) = (𝐺‘(𝑄‘(𝑣 + 1))))
94	28, 31, 33	3jca 1179	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑣 ∈ ℤ))
95	11, 23	zsubcld 9470	. . . . . . . . . . . 12 ⊢ (𝜑 → ((^◡𝐽‘𝐾) − 1) ∈ ℤ)
96	95	zred 9465	. . . . . . . . . . 11 ⊢ (𝜑 → ((^◡𝐽‘𝐾) − 1) ∈ ℝ)
97	96	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → ((^◡𝐽‘𝐾) − 1) ∈ ℝ)
98	50	lem1d 8977	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → ((^◡𝐽‘𝐾) − 1) ≤ (^◡𝐽‘𝐾))
99	97, 50, 51, 98, 60	letrd 8167	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → ((^◡𝐽‘𝐾) − 1) ≤ 𝑁)
100	37, 97, 51, 53, 99	letrd 8167	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑣 ≤ 𝑁)
101	46, 100	jca 306	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑀 ≤ 𝑣 ∧ 𝑣 ≤ 𝑁))
102		elfz2 10107	. . . . . . . 8 ⊢ (𝑣 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑀 ≤ 𝑣 ∧ 𝑣 ≤ 𝑁)))
103	94, 101, 102	sylanbrc 417	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑣 ∈ (𝑀...𝑁))
104	103, 25, 103, 91, 92	iseqf1olemfvp 10619	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (⦋𝐽 / 𝑓⦌𝑃‘𝑣) = (𝐺‘(𝐽‘𝑣)))
105	87, 93, 104	3eqtr4rd 2240	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (⦋𝐽 / 𝑓⦌𝑃‘𝑣) = (⦋𝑄 / 𝑓⦌𝑃‘(𝑣 + 1)))
106		simpr 110	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → 𝑥 ∈ (ℤ_≥‘𝐾))
107		elfzuz 10113	. . . . . . . . 9 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ_≥‘𝑀))
108	1, 107	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘𝑀))
109	108	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → 𝐾 ∈ (ℤ_≥‘𝑀))
110		uztrn 9635	. . . . . . 7 ⊢ ((𝑥 ∈ (ℤ_≥‘𝐾) ∧ 𝐾 ∈ (ℤ_≥‘𝑀)) → 𝑥 ∈ (ℤ_≥‘𝑀))
111	106, 109, 110	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → 𝑥 ∈ (ℤ_≥‘𝑀))
112	1, 4, 65, 90, 92	iseqf1olemjpcl 10617	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (⦋𝐽 / 𝑓⦌𝑃‘𝑥) ∈ 𝑆)
113	111, 112	syldan 282	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → (⦋𝐽 / 𝑓⦌𝑃‘𝑥) ∈ 𝑆)
114		simpr 110	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝑥 ∈ (ℤ_≥‘(𝐾 + 1)))
115	3	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝐾 ∈ ℤ)
116	115	peano2zd 9468	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝐾 + 1))) → (𝐾 + 1) ∈ ℤ)
117	115	zred 9465	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝐾 ∈ ℝ)
118	117	lep1d 8975	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝐾 ≤ (𝐾 + 1))
119		eluz2 9624	. . . . . . . 8 ⊢ ((𝐾 + 1) ∈ (ℤ_≥‘𝐾) ↔ (𝐾 ∈ ℤ ∧ (𝐾 + 1) ∈ ℤ ∧ 𝐾 ≤ (𝐾 + 1)))
120	115, 116, 118, 119	syl3anbrc 1183	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝐾 + 1))) → (𝐾 + 1) ∈ (ℤ_≥‘𝐾))
121		uztrn 9635	. . . . . . 7 ⊢ ((𝑥 ∈ (ℤ_≥‘(𝐾 + 1)) ∧ (𝐾 + 1) ∈ (ℤ_≥‘𝐾)) → 𝑥 ∈ (ℤ_≥‘𝐾))
122	114, 120, 121	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝑥 ∈ (ℤ_≥‘𝐾))
123	1, 4, 65, 90, 92	iseqf1olemqpcl 10618	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (⦋𝑄 / 𝑓⦌𝑃‘𝑥) ∈ 𝑆)
124	111, 123	syldan 282	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → (⦋𝑄 / 𝑓⦌𝑃‘𝑥) ∈ 𝑆)
125	122, 124	syldan 282	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝐾 + 1))) → (⦋𝑄 / 𝑓⦌𝑃‘𝑥) ∈ 𝑆)
126		iseqf1o.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
127	22, 23, 105, 113, 125, 126	seq3shft2 10590	. . . 4 ⊢ (𝜑 → (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘((^◡𝐽‘𝐾) − 1)) = (seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘(((^◡𝐽‘𝐾) − 1) + 1)))
128	11	zcnd 9466	. . . . . 6 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ ℂ)
129		npcan1 8421	. . . . . 6 ⊢ ((^◡𝐽‘𝐾) ∈ ℂ → (((^◡𝐽‘𝐾) − 1) + 1) = (^◡𝐽‘𝐾))
130	128, 129	syl 14	. . . . 5 ⊢ (𝜑 → (((^◡𝐽‘𝐾) − 1) + 1) = (^◡𝐽‘𝐾))
131	130	fveq2d 5565	. . . 4 ⊢ (𝜑 → (seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘(((^◡𝐽‘𝐾) − 1) + 1)) = (seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)))
132	127, 131	eqtrd 2229	. . 3 ⊢ (𝜑 → (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘((^◡𝐽‘𝐾) − 1)) = (seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)))
133		f1ocnvfv2 5828	. . . . . 6 ⊢ ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐽‘(^◡𝐽‘𝐾)) = 𝐾)
134	4, 1, 133	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝐽‘(^◡𝐽‘𝐾)) = 𝐾)
135	134	fveq2d 5565	. . . 4 ⊢ (𝜑 → (𝐺‘(𝐽‘(^◡𝐽‘𝐾))) = (𝐺‘𝐾))
136	1, 4, 9, 90, 92	iseqf1olemfvp 10619	. . . 4 ⊢ (𝜑 → (⦋𝐽 / 𝑓⦌𝑃‘(^◡𝐽‘𝐾)) = (𝐺‘(𝐽‘(^◡𝐽‘𝐾))))
137	1, 88, 1, 90, 92	iseqf1olemfvp 10619	. . . . 5 ⊢ (𝜑 → (⦋𝑄 / 𝑓⦌𝑃‘𝐾) = (𝐺‘(𝑄‘𝐾)))
138	1, 4, 1, 65	iseqf1olemqval 10609	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝐾) = if(𝐾 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝐾 = 𝐾, 𝐾, (𝐽‘(𝐾 − 1))), (𝐽‘𝐾)))
139	14, 1, 4, 15	iseqf1olemkle 10606	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ≤ (^◡𝐽‘𝐾))
140		eluz2 9624	. . . . . . . . . 10 ⊢ ((^◡𝐽‘𝐾) ∈ (ℤ_≥‘𝐾) ↔ (𝐾 ∈ ℤ ∧ (^◡𝐽‘𝐾) ∈ ℤ ∧ 𝐾 ≤ (^◡𝐽‘𝐾)))
141	3, 11, 139, 140	syl3anbrc 1183	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ (ℤ_≥‘𝐾))
142		eluzfz1 10123	. . . . . . . . 9 ⊢ ((^◡𝐽‘𝐾) ∈ (ℤ_≥‘𝐾) → 𝐾 ∈ (𝐾...(^◡𝐽‘𝐾)))
143	141, 142	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (𝐾...(^◡𝐽‘𝐾)))
144	143	iftrued 3569	. . . . . . 7 ⊢ (𝜑 → if(𝐾 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝐾 = 𝐾, 𝐾, (𝐽‘(𝐾 − 1))), (𝐽‘𝐾)) = if(𝐾 = 𝐾, 𝐾, (𝐽‘(𝐾 − 1))))
145		eqidd 2197	. . . . . . . 8 ⊢ (𝜑 → 𝐾 = 𝐾)
146	145	iftrued 3569	. . . . . . 7 ⊢ (𝜑 → if(𝐾 = 𝐾, 𝐾, (𝐽‘(𝐾 − 1))) = 𝐾)
147	138, 144, 146	3eqtrd 2233	. . . . . 6 ⊢ (𝜑 → (𝑄‘𝐾) = 𝐾)
148	147	fveq2d 5565	. . . . 5 ⊢ (𝜑 → (𝐺‘(𝑄‘𝐾)) = (𝐺‘𝐾))
149	137, 148	eqtrd 2229	. . . 4 ⊢ (𝜑 → (⦋𝑄 / 𝑓⦌𝑃‘𝐾) = (𝐺‘𝐾))
150	135, 136, 149	3eqtr4d 2239	. . 3 ⊢ (𝜑 → (⦋𝐽 / 𝑓⦌𝑃‘(^◡𝐽‘𝐾)) = (⦋𝑄 / 𝑓⦌𝑃‘𝐾))
151	132, 150	oveq12d 5943	. 2 ⊢ (𝜑 → ((seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘((^◡𝐽‘𝐾) − 1)) + (⦋𝐽 / 𝑓⦌𝑃‘(^◡𝐽‘𝐾))) = ((seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)) + (⦋𝑄 / 𝑓⦌𝑃‘𝐾)))
152	3	peano2zd 9468	. . . 4 ⊢ (𝜑 → (𝐾 + 1) ∈ ℤ)
153		zltp1le 9397	. . . . . 6 ⊢ ((𝐾 ∈ ℤ ∧ (^◡𝐽‘𝐾) ∈ ℤ) → (𝐾 < (^◡𝐽‘𝐾) ↔ (𝐾 + 1) ≤ (^◡𝐽‘𝐾)))
154	3, 11, 153	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝐾 < (^◡𝐽‘𝐾) ↔ (𝐾 + 1) ≤ (^◡𝐽‘𝐾)))
155	17, 154	mpbid 147	. . . 4 ⊢ (𝜑 → (𝐾 + 1) ≤ (^◡𝐽‘𝐾))
156		eluz2 9624	. . . 4 ⊢ ((^◡𝐽‘𝐾) ∈ (ℤ_≥‘(𝐾 + 1)) ↔ ((𝐾 + 1) ∈ ℤ ∧ (^◡𝐽‘𝐾) ∈ ℤ ∧ (𝐾 + 1) ≤ (^◡𝐽‘𝐾)))
157	152, 11, 155, 156	syl3anbrc 1183	. . 3 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ (ℤ_≥‘(𝐾 + 1)))
158	3, 157, 113, 126	seq3m1 10582	. 2 ⊢ (𝜑 → (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)) = ((seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘((^◡𝐽‘𝐾) − 1)) + (⦋𝐽 / 𝑓⦌𝑃‘(^◡𝐽‘𝐾))))
159		iseqf1o.3	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
160	126, 159, 157, 3, 124	seq3-1p 10599	. . 3 ⊢ (𝜑 → (seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)) = ((⦋𝑄 / 𝑓⦌𝑃‘𝐾) + (seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾))))
161		iseqf1o.2	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
162		fveq2 5561	. . . . . . 7 ⊢ (𝑥 = (𝑄‘𝐾) → (𝐺‘𝑥) = (𝐺‘(𝑄‘𝐾)))
163	162	eleq1d 2265	. . . . . 6 ⊢ (𝑥 = (𝑄‘𝐾) → ((𝐺‘𝑥) ∈ 𝑆 ↔ (𝐺‘(𝑄‘𝐾)) ∈ 𝑆))
164	90	ralrimiva 2570	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ_≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆)
165	147, 108	eqeltrd 2273	. . . . . 6 ⊢ (𝜑 → (𝑄‘𝐾) ∈ (ℤ_≥‘𝑀))
166	163, 164, 165	rspcdva 2873	. . . . 5 ⊢ (𝜑 → (𝐺‘(𝑄‘𝐾)) ∈ 𝑆)
167	137, 166	eqeltrd 2273	. . . 4 ⊢ (𝜑 → (⦋𝑄 / 𝑓⦌𝑃‘𝐾) ∈ 𝑆)
168		eqid 2196	. . . . . 6 ⊢ (ℤ_≥‘(𝐾 + 1)) = (ℤ_≥‘(𝐾 + 1))
169	168, 152, 125, 126	seqf 10573	. . . . 5 ⊢ (𝜑 → seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃):(ℤ_≥‘(𝐾 + 1))⟶𝑆)
170	169, 157	ffvelcdmd 5701	. . . 4 ⊢ (𝜑 → (seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)) ∈ 𝑆)
171	161, 167, 170	caovcomd 6084	. . 3 ⊢ (𝜑 → ((⦋𝑄 / 𝑓⦌𝑃‘𝐾) + (seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾))) = ((seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)) + (⦋𝑄 / 𝑓⦌𝑃‘𝐾)))
172	160, 171	eqtrd 2229	. 2 ⊢ (𝜑 → (seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)) = ((seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)) + (⦋𝑄 / 𝑓⦌𝑃‘𝐾)))
173	151, 158, 172	3eqtr4d 2239	1 ⊢ (𝜑 → (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)) = (seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ wcel 2167 ≠ wne 2367 ∀wral 2475 ⦋csb 3084 ifcif 3562 class class class wbr 4034 ↦ cmpt 4095 ^◡ccnv 4663 ⟶wf 5255 –1-1-onto→wf1o 5258 ‘cfv 5259 (class class class)co 5925 ℂcc 7894 ℝcr 7895 1c1 7897 + caddc 7899 < clt 8078 ≤ cle 8079 − cmin 8214 ℤcz 9343 ℤ_≥cuz 9618 ...cfz 10100 ..^cfzo 10234 seqcseq 10556

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-addcom 7996 ax-addass 7998 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-0id 8004 ax-rnegex 8005 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-1o 6483 df-er 6601 df-en 6809 df-fin 6811 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-inn 9008 df-n0 9267 df-z 9344 df-uz 9619 df-fz 10101 df-fzo 10235 df-seqfrec 10557

This theorem is referenced by: seq3f1olemqsumk 10621

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