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Theorem seq3f1olemqsumkj 10454

Description: Lemma for seq3f1o 10460. 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 9981	. . . . . . 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 5455	. . . . . . . . . . 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 5442	. . . . . . . . . 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	ffvelrnd 5632	. . . . . . . 8 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ (𝑀...𝑁))
10		elfzelz 9981	. . . . . . . 8 ⊢ ((^◡𝐽‘𝐾) ∈ (𝑀...𝑁) → (^◡𝐽‘𝐾) ∈ ℤ)
11	9, 10	syl 14	. . . . . . 7 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ ℤ)
12		peano2zm 9250	. . . . . . 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 10441	. . . . . . 7 ⊢ (𝜑 → 𝐾 < (^◡𝐽‘𝐾))
18		zltlem1 9269	. . . . . . . 8 ⊢ ((𝐾 ∈ ℤ ∧ (^◡𝐽‘𝐾) ∈ ℤ) → (𝐾 < (^◡𝐽‘𝐾) ↔ 𝐾 ≤ ((^◡𝐽‘𝐾) − 1)))
19	3, 11, 18	syl2anc 409	. . . . . . 7 ⊢ (𝜑 → (𝐾 < (^◡𝐽‘𝐾) ↔ 𝐾 ≤ ((^◡𝐽‘𝐾) − 1)))
20	17, 19	mpbid 146	. . . . . 6 ⊢ (𝜑 → 𝐾 ≤ ((^◡𝐽‘𝐾) − 1))
21		eluz2 9493	. . . . . 6 ⊢ (((^◡𝐽‘𝐾) − 1) ∈ (ℤ_≥‘𝐾) ↔ (𝐾 ∈ ℤ ∧ ((^◡𝐽‘𝐾) − 1) ∈ ℤ ∧ 𝐾 ≤ ((^◡𝐽‘𝐾) − 1)))
22	3, 13, 20, 21	syl3anbrc 1176	. . . . 5 ⊢ (𝜑 → ((^◡𝐽‘𝐾) − 1) ∈ (ℤ_≥‘𝐾))
23		1zzd 9239	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
24	1	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝐾 ∈ (𝑀...𝑁))
25	4	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
26		elfzel1 9980	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
27	1, 26	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℤ)
28	27	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑀 ∈ ℤ)
29		elfzel2 9979	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
30	1, 29	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℤ)
31	30	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑁 ∈ ℤ)
32		elfzelz 9981	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1)) → 𝑣 ∈ ℤ)
33	32	adantl 275	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑣 ∈ ℤ)
34	33	peano2zd 9337	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑣 + 1) ∈ ℤ)
35	28, 31, 34	3jca 1172	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑣 + 1) ∈ ℤ))
36	28	zred 9334	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑀 ∈ ℝ)
37	33	zred 9334	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑣 ∈ ℝ)
38	34	zred 9334	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑣 + 1) ∈ ℝ)
39	3	zred 9334	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ ℝ)
40	39	adantr 274	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝐾 ∈ ℝ)
41		elfzle1 9983	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝐾)
42	1, 41	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ≤ 𝐾)
43	42	adantr 274	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑀 ≤ 𝐾)
44		elfzle1 9983	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1)) → 𝐾 ≤ 𝑣)
45	44	adantl 275	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝐾 ≤ 𝑣)
46	36, 40, 37, 43, 45	letrd 8043	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑀 ≤ 𝑣)
47	37	lep1d 8847	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑣 ≤ (𝑣 + 1))
48	36, 37, 38, 46, 47	letrd 8043	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑀 ≤ (𝑣 + 1))
49	11	zred 9334	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ ℝ)
50	49	adantr 274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (^◡𝐽‘𝐾) ∈ ℝ)
51	31	zred 9334	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑁 ∈ ℝ)
52		elfzle2 9984	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1)) → 𝑣 ≤ ((^◡𝐽‘𝐾) − 1))
53	52	adantl 275	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑣 ≤ ((^◡𝐽‘𝐾) − 1))
54		1red 7935	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 1 ∈ ℝ)
55		leaddsub 8357	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ ℝ ∧ 1 ∈ ℝ ∧ (^◡𝐽‘𝐾) ∈ ℝ) → ((𝑣 + 1) ≤ (^◡𝐽‘𝐾) ↔ 𝑣 ≤ ((^◡𝐽‘𝐾) − 1)))
56	37, 54, 50, 55	syl3anc 1233	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → ((𝑣 + 1) ≤ (^◡𝐽‘𝐾) ↔ 𝑣 ≤ ((^◡𝐽‘𝐾) − 1)))
57	53, 56	mpbird 166	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑣 + 1) ≤ (^◡𝐽‘𝐾))
58		elfzle2 9984	. . . . . . . . . . . . . . 15 ⊢ ((^◡𝐽‘𝐾) ∈ (𝑀...𝑁) → (^◡𝐽‘𝐾) ≤ 𝑁)
59	9, 58	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (^◡𝐽‘𝐾) ≤ 𝑁)
60	59	adantr 274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (^◡𝐽‘𝐾) ≤ 𝑁)
61	38, 50, 51, 57, 60	letrd 8043	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑣 + 1) ≤ 𝑁)
62	48, 61	jca 304	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑀 ≤ (𝑣 + 1) ∧ (𝑣 + 1) ≤ 𝑁))
63		elfz2 9972	. . . . . . . . . . 11 ⊢ ((𝑣 + 1) ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑣 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑣 + 1) ∧ (𝑣 + 1) ≤ 𝑁)))
64	35, 62, 63	sylanbrc 415	. . . . . . . . . 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 10443	. . . . . . . . 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 274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (^◡𝐽‘𝐾) ∈ ℤ)
69	67, 68, 34	3jca 1172	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝐾 ∈ ℤ ∧ (^◡𝐽‘𝐾) ∈ ℤ ∧ (𝑣 + 1) ∈ ℤ))
70	40, 37, 38, 45, 47	letrd 8043	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝐾 ≤ (𝑣 + 1))
71	70, 57	jca 304	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝐾 ≤ (𝑣 + 1) ∧ (𝑣 + 1) ≤ (^◡𝐽‘𝐾)))
72		elfz2 9972	. . . . . . . . . . 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 415	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑣 + 1) ∈ (𝐾...(^◡𝐽‘𝐾)))
74	73	iftrued 3533	. . . . . . . . 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 2203	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑄‘(𝑣 + 1)) = if((𝑣 + 1) = 𝐾, 𝐾, (𝐽‘((𝑣 + 1) − 1))))
76		zleltp1 9267	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℤ ∧ 𝑣 ∈ ℤ) → (𝐾 ≤ 𝑣 ↔ 𝐾 < (𝑣 + 1)))
77	67, 33, 76	syl2anc 409	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝐾 ≤ 𝑣 ↔ 𝐾 < (𝑣 + 1)))
78	45, 77	mpbid 146	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝐾 < (𝑣 + 1))
79	40, 78	gtned 8032	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑣 + 1) ≠ 𝐾)
80	79	neneqd 2361	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → ¬ (𝑣 + 1) = 𝐾)
81	80	iffalsed 3536	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → if((𝑣 + 1) = 𝐾, 𝐾, (𝐽‘((𝑣 + 1) − 1))) = (𝐽‘((𝑣 + 1) − 1)))
82	33	zcnd 9335	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑣 ∈ ℂ)
83		pncan1 8296	. . . . . . . . . 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 5500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝐽‘((𝑣 + 1) − 1)) = (𝐽‘𝑣))
86	75, 81, 85	3eqtrd 2207	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑄‘(𝑣 + 1)) = (𝐽‘𝑣))
87	86	fveq2d 5500	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝐺‘(𝑄‘(𝑣 + 1))) = (𝐺‘(𝐽‘𝑣)))
88	1, 4, 65	iseqf1olemqf1o 10449	. . . . . . . 8 ⊢ (𝜑 → 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
89	88	adantr 274	. . . . . . 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 474	. . . . . . 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 10453	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (⦋𝑄 / 𝑓⦌𝑃‘(𝑣 + 1)) = (𝐺‘(𝑄‘(𝑣 + 1))))
94	28, 31, 33	3jca 1172	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑣 ∈ ℤ))
95	11, 23	zsubcld 9339	. . . . . . . . . . . 12 ⊢ (𝜑 → ((^◡𝐽‘𝐾) − 1) ∈ ℤ)
96	95	zred 9334	. . . . . . . . . . 11 ⊢ (𝜑 → ((^◡𝐽‘𝐾) − 1) ∈ ℝ)
97	96	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → ((^◡𝐽‘𝐾) − 1) ∈ ℝ)
98	50	lem1d 8849	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → ((^◡𝐽‘𝐾) − 1) ≤ (^◡𝐽‘𝐾))
99	97, 50, 51, 98, 60	letrd 8043	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → ((^◡𝐽‘𝐾) − 1) ≤ 𝑁)
100	37, 97, 51, 53, 99	letrd 8043	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑣 ≤ 𝑁)
101	46, 100	jca 304	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (𝑀 ≤ 𝑣 ∧ 𝑣 ≤ 𝑁))
102		elfz2 9972	. . . . . . . 8 ⊢ (𝑣 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑀 ≤ 𝑣 ∧ 𝑣 ≤ 𝑁)))
103	94, 101, 102	sylanbrc 415	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → 𝑣 ∈ (𝑀...𝑁))
104	103, 25, 103, 91, 92	iseqf1olemfvp 10453	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (⦋𝐽 / 𝑓⦌𝑃‘𝑣) = (𝐺‘(𝐽‘𝑣)))
105	87, 93, 104	3eqtr4rd 2214	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐾...((^◡𝐽‘𝐾) − 1))) → (⦋𝐽 / 𝑓⦌𝑃‘𝑣) = (⦋𝑄 / 𝑓⦌𝑃‘(𝑣 + 1)))
106		simpr 109	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → 𝑥 ∈ (ℤ_≥‘𝐾))
107		elfzuz 9977	. . . . . . . . 9 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ_≥‘𝑀))
108	1, 107	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘𝑀))
109	108	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → 𝐾 ∈ (ℤ_≥‘𝑀))
110		uztrn 9503	. . . . . . 7 ⊢ ((𝑥 ∈ (ℤ_≥‘𝐾) ∧ 𝐾 ∈ (ℤ_≥‘𝑀)) → 𝑥 ∈ (ℤ_≥‘𝑀))
111	106, 109, 110	syl2anc 409	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → 𝑥 ∈ (ℤ_≥‘𝑀))
112	1, 4, 65, 90, 92	iseqf1olemjpcl 10451	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (⦋𝐽 / 𝑓⦌𝑃‘𝑥) ∈ 𝑆)
113	111, 112	syldan 280	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → (⦋𝐽 / 𝑓⦌𝑃‘𝑥) ∈ 𝑆)
114		simpr 109	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝑥 ∈ (ℤ_≥‘(𝐾 + 1)))
115	3	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝐾 ∈ ℤ)
116	115	peano2zd 9337	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝐾 + 1))) → (𝐾 + 1) ∈ ℤ)
117	115	zred 9334	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝐾 ∈ ℝ)
118	117	lep1d 8847	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝐾 ≤ (𝐾 + 1))
119		eluz2 9493	. . . . . . . 8 ⊢ ((𝐾 + 1) ∈ (ℤ_≥‘𝐾) ↔ (𝐾 ∈ ℤ ∧ (𝐾 + 1) ∈ ℤ ∧ 𝐾 ≤ (𝐾 + 1)))
120	115, 116, 118, 119	syl3anbrc 1176	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝐾 + 1))) → (𝐾 + 1) ∈ (ℤ_≥‘𝐾))
121		uztrn 9503	. . . . . . 7 ⊢ ((𝑥 ∈ (ℤ_≥‘(𝐾 + 1)) ∧ (𝐾 + 1) ∈ (ℤ_≥‘𝐾)) → 𝑥 ∈ (ℤ_≥‘𝐾))
122	114, 120, 121	syl2anc 409	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝑥 ∈ (ℤ_≥‘𝐾))
123	1, 4, 65, 90, 92	iseqf1olemqpcl 10452	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (⦋𝑄 / 𝑓⦌𝑃‘𝑥) ∈ 𝑆)
124	111, 123	syldan 280	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → (⦋𝑄 / 𝑓⦌𝑃‘𝑥) ∈ 𝑆)
125	122, 124	syldan 280	. . . . 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 10429	. . . 4 ⊢ (𝜑 → (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘((^◡𝐽‘𝐾) − 1)) = (seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘(((^◡𝐽‘𝐾) − 1) + 1)))
128	11	zcnd 9335	. . . . . 6 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ ℂ)
129		npcan1 8297	. . . . . 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 5500	. . . 4 ⊢ (𝜑 → (seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘(((^◡𝐽‘𝐾) − 1) + 1)) = (seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)))
132	127, 131	eqtrd 2203	. . 3 ⊢ (𝜑 → (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘((^◡𝐽‘𝐾) − 1)) = (seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)))
133		f1ocnvfv2 5757	. . . . . 6 ⊢ ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐽‘(^◡𝐽‘𝐾)) = 𝐾)
134	4, 1, 133	syl2anc 409	. . . . 5 ⊢ (𝜑 → (𝐽‘(^◡𝐽‘𝐾)) = 𝐾)
135	134	fveq2d 5500	. . . 4 ⊢ (𝜑 → (𝐺‘(𝐽‘(^◡𝐽‘𝐾))) = (𝐺‘𝐾))
136	1, 4, 9, 90, 92	iseqf1olemfvp 10453	. . . 4 ⊢ (𝜑 → (⦋𝐽 / 𝑓⦌𝑃‘(^◡𝐽‘𝐾)) = (𝐺‘(𝐽‘(^◡𝐽‘𝐾))))
137	1, 88, 1, 90, 92	iseqf1olemfvp 10453	. . . . 5 ⊢ (𝜑 → (⦋𝑄 / 𝑓⦌𝑃‘𝐾) = (𝐺‘(𝑄‘𝐾)))
138	1, 4, 1, 65	iseqf1olemqval 10443	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝐾) = if(𝐾 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝐾 = 𝐾, 𝐾, (𝐽‘(𝐾 − 1))), (𝐽‘𝐾)))
139	14, 1, 4, 15	iseqf1olemkle 10440	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ≤ (^◡𝐽‘𝐾))
140		eluz2 9493	. . . . . . . . . 10 ⊢ ((^◡𝐽‘𝐾) ∈ (ℤ_≥‘𝐾) ↔ (𝐾 ∈ ℤ ∧ (^◡𝐽‘𝐾) ∈ ℤ ∧ 𝐾 ≤ (^◡𝐽‘𝐾)))
141	3, 11, 139, 140	syl3anbrc 1176	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ (ℤ_≥‘𝐾))
142		eluzfz1 9987	. . . . . . . . 9 ⊢ ((^◡𝐽‘𝐾) ∈ (ℤ_≥‘𝐾) → 𝐾 ∈ (𝐾...(^◡𝐽‘𝐾)))
143	141, 142	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (𝐾...(^◡𝐽‘𝐾)))
144	143	iftrued 3533	. . . . . . 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 2171	. . . . . . . 8 ⊢ (𝜑 → 𝐾 = 𝐾)
146	145	iftrued 3533	. . . . . . 7 ⊢ (𝜑 → if(𝐾 = 𝐾, 𝐾, (𝐽‘(𝐾 − 1))) = 𝐾)
147	138, 144, 146	3eqtrd 2207	. . . . . 6 ⊢ (𝜑 → (𝑄‘𝐾) = 𝐾)
148	147	fveq2d 5500	. . . . 5 ⊢ (𝜑 → (𝐺‘(𝑄‘𝐾)) = (𝐺‘𝐾))
149	137, 148	eqtrd 2203	. . . 4 ⊢ (𝜑 → (⦋𝑄 / 𝑓⦌𝑃‘𝐾) = (𝐺‘𝐾))
150	135, 136, 149	3eqtr4d 2213	. . 3 ⊢ (𝜑 → (⦋𝐽 / 𝑓⦌𝑃‘(^◡𝐽‘𝐾)) = (⦋𝑄 / 𝑓⦌𝑃‘𝐾))
151	132, 150	oveq12d 5871	. 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 9337	. . . 4 ⊢ (𝜑 → (𝐾 + 1) ∈ ℤ)
153		zltp1le 9266	. . . . . 6 ⊢ ((𝐾 ∈ ℤ ∧ (^◡𝐽‘𝐾) ∈ ℤ) → (𝐾 < (^◡𝐽‘𝐾) ↔ (𝐾 + 1) ≤ (^◡𝐽‘𝐾)))
154	3, 11, 153	syl2anc 409	. . . . 5 ⊢ (𝜑 → (𝐾 < (^◡𝐽‘𝐾) ↔ (𝐾 + 1) ≤ (^◡𝐽‘𝐾)))
155	17, 154	mpbid 146	. . . 4 ⊢ (𝜑 → (𝐾 + 1) ≤ (^◡𝐽‘𝐾))
156		eluz2 9493	. . . 4 ⊢ ((^◡𝐽‘𝐾) ∈ (ℤ_≥‘(𝐾 + 1)) ↔ ((𝐾 + 1) ∈ ℤ ∧ (^◡𝐽‘𝐾) ∈ ℤ ∧ (𝐾 + 1) ≤ (^◡𝐽‘𝐾)))
157	152, 11, 155, 156	syl3anbrc 1176	. . 3 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ (ℤ_≥‘(𝐾 + 1)))
158	3, 157, 113, 126	seq3m1 10424	. 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 10436	. . 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 5496	. . . . . . 7 ⊢ (𝑥 = (𝑄‘𝐾) → (𝐺‘𝑥) = (𝐺‘(𝑄‘𝐾)))
163	162	eleq1d 2239	. . . . . 6 ⊢ (𝑥 = (𝑄‘𝐾) → ((𝐺‘𝑥) ∈ 𝑆 ↔ (𝐺‘(𝑄‘𝐾)) ∈ 𝑆))
164	90	ralrimiva 2543	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ_≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆)
165	147, 108	eqeltrd 2247	. . . . . 6 ⊢ (𝜑 → (𝑄‘𝐾) ∈ (ℤ_≥‘𝑀))
166	163, 164, 165	rspcdva 2839	. . . . 5 ⊢ (𝜑 → (𝐺‘(𝑄‘𝐾)) ∈ 𝑆)
167	137, 166	eqeltrd 2247	. . . 4 ⊢ (𝜑 → (⦋𝑄 / 𝑓⦌𝑃‘𝐾) ∈ 𝑆)
168		eqid 2170	. . . . . 6 ⊢ (ℤ_≥‘(𝐾 + 1)) = (ℤ_≥‘(𝐾 + 1))
169	168, 152, 125, 126	seqf 10417	. . . . 5 ⊢ (𝜑 → seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃):(ℤ_≥‘(𝐾 + 1))⟶𝑆)
170	169, 157	ffvelrnd 5632	. . . 4 ⊢ (𝜑 → (seq(𝐾 + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘(^◡𝐽‘𝐾)) ∈ 𝑆)
171	161, 167, 170	caovcomd 6009	. . 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 2203	. 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 2213	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 103 ↔ wb 104 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 ∀wral 2448 ⦋csb 3049 ifcif 3526 class class class wbr 3989 ↦ cmpt 4050 ^◡ccnv 4610 ⟶wf 5194 –1-1-onto→wf1o 5197 ‘cfv 5198 (class class class)co 5853 ℂcc 7772 ℝcr 7773 1c1 7775 + caddc 7777 < clt 7954 ≤ cle 7955 − cmin 8090 ℤcz 9212 ℤ_≥cuz 9487 ...cfz 9965 ..^cfzo 10098 seqcseq 10401

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-1o 6395 df-er 6513 df-en 6719 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 df-fzo 10099 df-seqfrec 10402

This theorem is referenced by: seq3f1olemqsumk 10455

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