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Theorem seq3f1olemqsum 10781

Description: Lemma for seq3f1o 10785. 𝑄 gives the same sum as 𝐽. (Contributed by Jim Kingdon, 21-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
seq3f1olemqsum	⊢ (𝜑 → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁))

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

Proof of Theorem seq3f1olemqsum Dummy variables 𝑏 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqf1olemstep.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))
2		elfzel1 10264	. . . . . . . 8 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
3	1, 2	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	3	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → 𝑀 ∈ ℤ)
5		elfzelz 10265	. . . . . . . . 9 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
6	1, 5	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℤ)
7	6	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → 𝐾 ∈ ℤ)
8		peano2zm 9522	. . . . . . 7 ⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈ ℤ)
9	7, 8	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → (𝐾 − 1) ∈ ℤ)
10		simpr 110	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → 𝑀 < 𝐾)
11		zltlem1 9542	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 < 𝐾 ↔ 𝑀 ≤ (𝐾 − 1)))
12	4, 7, 11	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → (𝑀 < 𝐾 ↔ 𝑀 ≤ (𝐾 − 1)))
13	10, 12	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → 𝑀 ≤ (𝐾 − 1))
14		eluz2 9766	. . . . . 6 ⊢ ((𝐾 − 1) ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝐾 − 1) ∈ ℤ ∧ 𝑀 ≤ (𝐾 − 1)))
15	4, 9, 13, 14	syl3anbrc 1207	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → (𝐾 − 1) ∈ (ℤ_≥‘𝑀))
16	3	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝑀 ∈ ℤ)
17		elfzel2 10263	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
18	1, 17	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
19	18	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝑁 ∈ ℤ)
20		elfzelz 10265	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (𝑀...(𝐾 − 1)) → 𝑏 ∈ ℤ)
21	20	adantl 277	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝑏 ∈ ℤ)
22		elfzle1 10267	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (𝑀...(𝐾 − 1)) → 𝑀 ≤ 𝑏)
23	22	adantl 277	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝑀 ≤ 𝑏)
24	21	zred 9607	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝑏 ∈ ℝ)
25	6	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝐾 ∈ ℤ)
26	25	zred 9607	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝐾 ∈ ℝ)
27	19	zred 9607	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝑁 ∈ ℝ)
28		peano2rem 8451	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℝ → (𝐾 − 1) ∈ ℝ)
29	26, 28	syl 14	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (𝐾 − 1) ∈ ℝ)
30		elfzle2 10268	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝑀...(𝐾 − 1)) → 𝑏 ≤ (𝐾 − 1))
31	30	adantl 277	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝑏 ≤ (𝐾 − 1))
32	26	lem1d 9118	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (𝐾 − 1) ≤ 𝐾)
33	24, 29, 26, 31, 32	letrd 8308	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝑏 ≤ 𝐾)
34		elfzle2 10268	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ≤ 𝑁)
35	1, 34	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ≤ 𝑁)
36	35	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝐾 ≤ 𝑁)
37	24, 26, 27, 33, 36	letrd 8308	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝑏 ≤ 𝑁)
38		elfz4 10258	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑀 ≤ 𝑏 ∧ 𝑏 ≤ 𝑁)) → 𝑏 ∈ (𝑀...𝑁))
39	16, 19, 21, 23, 37, 38	syl32anc 1281	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝑏 ∈ (𝑀...𝑁))
40		elfzel1 10264	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (𝐾...(^◡𝐽‘𝐾)) → 𝐾 ∈ ℤ)
41	40	zred 9607	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝐾...(^◡𝐽‘𝐾)) → 𝐾 ∈ ℝ)
42		elfzelz 10265	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (𝐾...(^◡𝐽‘𝐾)) → 𝑏 ∈ ℤ)
43	42	zred 9607	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝐾...(^◡𝐽‘𝐾)) → 𝑏 ∈ ℝ)
44		elfzle1 10267	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝐾...(^◡𝐽‘𝐾)) → 𝐾 ≤ 𝑏)
45	41, 43, 44	lensymd 8306	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (𝐾...(^◡𝐽‘𝐾)) → ¬ 𝑏 < 𝐾)
46		zltlem1 9542	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑏 < 𝐾 ↔ 𝑏 ≤ (𝐾 − 1)))
47	21, 25, 46	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (𝑏 < 𝐾 ↔ 𝑏 ≤ (𝐾 − 1)))
48	31, 47	mpbird 167	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝑏 < 𝐾)
49	45, 48	nsyl3 631	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → ¬ 𝑏 ∈ (𝐾...(^◡𝐽‘𝐾)))
50	49	iffalsed 3616	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → if(𝑏 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑏 = 𝐾, 𝐾, (𝐽‘(𝑏 − 1))), (𝐽‘𝑏)) = (𝐽‘𝑏))
51		iseqf1olemstep.j	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
52		f1of 5586	. . . . . . . . . . . . 13 ⊢ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
53	51, 52	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
54	53	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
55	54, 39	ffvelcdmd 5786	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (𝐽‘𝑏) ∈ (𝑀...𝑁))
56	50, 55	eqeltrd 2307	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → if(𝑏 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑏 = 𝐾, 𝐾, (𝐽‘(𝑏 − 1))), (𝐽‘𝑏)) ∈ (𝑀...𝑁))
57		eleq1w 2291	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑏 → (𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)) ↔ 𝑏 ∈ (𝐾...(^◡𝐽‘𝐾))))
58		eqeq1 2237	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑏 → (𝑢 = 𝐾 ↔ 𝑏 = 𝐾))
59		fvoveq1 6046	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑏 → (𝐽‘(𝑢 − 1)) = (𝐽‘(𝑏 − 1)))
60	58, 59	ifbieq2d 3631	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑏 → if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))) = if(𝑏 = 𝐾, 𝐾, (𝐽‘(𝑏 − 1))))
61		fveq2 5642	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑏 → (𝐽‘𝑢) = (𝐽‘𝑏))
62	57, 60, 61	ifbieq12d 3633	. . . . . . . . . 10 ⊢ (𝑢 = 𝑏 → if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)) = if(𝑏 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑏 = 𝐾, 𝐾, (𝐽‘(𝑏 − 1))), (𝐽‘𝑏)))
63		iseqf1olemqres.q	. . . . . . . . . 10 ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))
64	62, 63	fvmptg 5725	. . . . . . . . 9 ⊢ ((𝑏 ∈ (𝑀...𝑁) ∧ if(𝑏 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑏 = 𝐾, 𝐾, (𝐽‘(𝑏 − 1))), (𝐽‘𝑏)) ∈ (𝑀...𝑁)) → (𝑄‘𝑏) = if(𝑏 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑏 = 𝐾, 𝐾, (𝐽‘(𝑏 − 1))), (𝐽‘𝑏)))
65	39, 56, 64	syl2anc 411	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (𝑄‘𝑏) = if(𝑏 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑏 = 𝐾, 𝐾, (𝐽‘(𝑏 − 1))), (𝐽‘𝑏)))
66	65, 50	eqtrd 2263	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (𝑄‘𝑏) = (𝐽‘𝑏))
67	66	fveq2d 5646	. . . . . 6 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (𝐺‘(𝑄‘𝑏)) = (𝐺‘(𝐽‘𝑏)))
68		iseqf1olemqsumk.p	. . . . . . . . . . 11 ⊢ 𝑃 = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))
69	68	csbeq2i 3153	. . . . . . . . . 10 ⊢ ⦋𝑄 / 𝑓⦌𝑃 = ⦋𝑄 / 𝑓⦌(𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))
70	3, 18	fzfigd 10699	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
71		mptexg 5884	. . . . . . . . . . . . 13 ⊢ ((𝑀...𝑁) ∈ Fin → (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) ∈ V)
72	70, 71	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) ∈ V)
73	63, 72	eqeltrid 2317	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ V)
74		nfcvd 2374	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ V → Ⅎ𝑓(𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑄‘𝑥)), (𝐺‘𝑀))))
75		fveq1 5641	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑄 → (𝑓‘𝑥) = (𝑄‘𝑥))
76	75	fveq2d 5646	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑄 → (𝐺‘(𝑓‘𝑥)) = (𝐺‘(𝑄‘𝑥)))
77	76	ifeq1d 3624	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑄 → if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)) = if(𝑥 ≤ 𝑁, (𝐺‘(𝑄‘𝑥)), (𝐺‘𝑀)))
78	77	mpteq2dv 4181	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑄 → (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀))) = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑄‘𝑥)), (𝐺‘𝑀))))
79	74, 78	csbiegf 3170	. . . . . . . . . . 11 ⊢ (𝑄 ∈ V → ⦋𝑄 / 𝑓⦌(𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀))) = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑄‘𝑥)), (𝐺‘𝑀))))
80	73, 79	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → ⦋𝑄 / 𝑓⦌(𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀))) = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑄‘𝑥)), (𝐺‘𝑀))))
81	69, 80	eqtrid 2275	. . . . . . . . 9 ⊢ (𝜑 → ⦋𝑄 / 𝑓⦌𝑃 = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑄‘𝑥)), (𝐺‘𝑀))))
82	81	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → ⦋𝑄 / 𝑓⦌𝑃 = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑄‘𝑥)), (𝐺‘𝑀))))
83		breq1 4092	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → (𝑥 ≤ 𝑁 ↔ 𝑏 ≤ 𝑁))
84		2fveq3 5647	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → (𝐺‘(𝑄‘𝑥)) = (𝐺‘(𝑄‘𝑏)))
85	83, 84	ifbieq1d 3629	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → if(𝑥 ≤ 𝑁, (𝐺‘(𝑄‘𝑥)), (𝐺‘𝑀)) = if(𝑏 ≤ 𝑁, (𝐺‘(𝑄‘𝑏)), (𝐺‘𝑀)))
86	85	adantl 277	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) ∧ 𝑥 = 𝑏) → if(𝑥 ≤ 𝑁, (𝐺‘(𝑄‘𝑥)), (𝐺‘𝑀)) = if(𝑏 ≤ 𝑁, (𝐺‘(𝑄‘𝑏)), (𝐺‘𝑀)))
87		elfzuz 10261	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝑀...(𝐾 − 1)) → 𝑏 ∈ (ℤ_≥‘𝑀))
88	87	adantl 277	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝑏 ∈ (ℤ_≥‘𝑀))
89	37	iftrued 3613	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → if(𝑏 ≤ 𝑁, (𝐺‘(𝑄‘𝑏)), (𝐺‘𝑀)) = (𝐺‘(𝑄‘𝑏)))
90		fveq2 5642	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑄‘𝑏) → (𝐺‘𝑎) = (𝐺‘(𝑄‘𝑏)))
91	90	eleq1d 2299	. . . . . . . . . 10 ⊢ (𝑎 = (𝑄‘𝑏) → ((𝐺‘𝑎) ∈ 𝑆 ↔ (𝐺‘(𝑄‘𝑏)) ∈ 𝑆))
92		iseqf1o.7	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
93	92	ralrimiva 2604	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ_≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆)
94		fveq2 5642	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (𝐺‘𝑥) = (𝐺‘𝑎))
95	94	eleq1d 2299	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → ((𝐺‘𝑥) ∈ 𝑆 ↔ (𝐺‘𝑎) ∈ 𝑆))
96	95	cbvralv 2766	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (ℤ_≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆 ↔ ∀𝑎 ∈ (ℤ_≥‘𝑀)(𝐺‘𝑎) ∈ 𝑆)
97	93, 96	sylib 122	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑎 ∈ (ℤ_≥‘𝑀)(𝐺‘𝑎) ∈ 𝑆)
98	97	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → ∀𝑎 ∈ (ℤ_≥‘𝑀)(𝐺‘𝑎) ∈ 𝑆)
99	1, 51, 63	iseqf1olemqf 10772	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄:(𝑀...𝑁)⟶(𝑀...𝑁))
100	99	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → 𝑄:(𝑀...𝑁)⟶(𝑀...𝑁))
101	100, 39	ffvelcdmd 5786	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (𝑄‘𝑏) ∈ (𝑀...𝑁))
102		elfzuz 10261	. . . . . . . . . . 11 ⊢ ((𝑄‘𝑏) ∈ (𝑀...𝑁) → (𝑄‘𝑏) ∈ (ℤ_≥‘𝑀))
103	101, 102	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (𝑄‘𝑏) ∈ (ℤ_≥‘𝑀))
104	91, 98, 103	rspcdva 2914	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (𝐺‘(𝑄‘𝑏)) ∈ 𝑆)
105	89, 104	eqeltrd 2307	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → if(𝑏 ≤ 𝑁, (𝐺‘(𝑄‘𝑏)), (𝐺‘𝑀)) ∈ 𝑆)
106	82, 86, 88, 105	fvmptd 5730	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (⦋𝑄 / 𝑓⦌𝑃‘𝑏) = if(𝑏 ≤ 𝑁, (𝐺‘(𝑄‘𝑏)), (𝐺‘𝑀)))
107	106, 89	eqtrd 2263	. . . . . 6 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (⦋𝑄 / 𝑓⦌𝑃‘𝑏) = (𝐺‘(𝑄‘𝑏)))
108	68	csbeq2i 3153	. . . . . . . . . 10 ⊢ ⦋𝐽 / 𝑓⦌𝑃 = ⦋𝐽 / 𝑓⦌(𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))
109		fex 5888	. . . . . . . . . . . 12 ⊢ ((𝐽:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → 𝐽 ∈ V)
110	53, 70, 109	syl2anc 411	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ V)
111		nfcvd 2374	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ V → Ⅎ𝑓(𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝐽‘𝑥)), (𝐺‘𝑀))))
112		fveq1 5641	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝐽 → (𝑓‘𝑥) = (𝐽‘𝑥))
113	112	fveq2d 5646	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐽 → (𝐺‘(𝑓‘𝑥)) = (𝐺‘(𝐽‘𝑥)))
114	113	ifeq1d 3624	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐽 → if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)) = if(𝑥 ≤ 𝑁, (𝐺‘(𝐽‘𝑥)), (𝐺‘𝑀)))
115	114	mpteq2dv 4181	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐽 → (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀))) = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝐽‘𝑥)), (𝐺‘𝑀))))
116	111, 115	csbiegf 3170	. . . . . . . . . . 11 ⊢ (𝐽 ∈ V → ⦋𝐽 / 𝑓⦌(𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀))) = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝐽‘𝑥)), (𝐺‘𝑀))))
117	110, 116	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → ⦋𝐽 / 𝑓⦌(𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀))) = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝐽‘𝑥)), (𝐺‘𝑀))))
118	108, 117	eqtrid 2275	. . . . . . . . 9 ⊢ (𝜑 → ⦋𝐽 / 𝑓⦌𝑃 = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝐽‘𝑥)), (𝐺‘𝑀))))
119	118	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → ⦋𝐽 / 𝑓⦌𝑃 = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝐽‘𝑥)), (𝐺‘𝑀))))
120		2fveq3 5647	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → (𝐺‘(𝐽‘𝑥)) = (𝐺‘(𝐽‘𝑏)))
121	83, 120	ifbieq1d 3629	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → if(𝑥 ≤ 𝑁, (𝐺‘(𝐽‘𝑥)), (𝐺‘𝑀)) = if(𝑏 ≤ 𝑁, (𝐺‘(𝐽‘𝑏)), (𝐺‘𝑀)))
122	121	adantl 277	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) ∧ 𝑥 = 𝑏) → if(𝑥 ≤ 𝑁, (𝐺‘(𝐽‘𝑥)), (𝐺‘𝑀)) = if(𝑏 ≤ 𝑁, (𝐺‘(𝐽‘𝑏)), (𝐺‘𝑀)))
123	37	iftrued 3613	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → if(𝑏 ≤ 𝑁, (𝐺‘(𝐽‘𝑏)), (𝐺‘𝑀)) = (𝐺‘(𝐽‘𝑏)))
124		fveq2 5642	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐽‘𝑏) → (𝐺‘𝑎) = (𝐺‘(𝐽‘𝑏)))
125	124	eleq1d 2299	. . . . . . . . . 10 ⊢ (𝑎 = (𝐽‘𝑏) → ((𝐺‘𝑎) ∈ 𝑆 ↔ (𝐺‘(𝐽‘𝑏)) ∈ 𝑆))
126		elfzuz 10261	. . . . . . . . . . 11 ⊢ ((𝐽‘𝑏) ∈ (𝑀...𝑁) → (𝐽‘𝑏) ∈ (ℤ_≥‘𝑀))
127	55, 126	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (𝐽‘𝑏) ∈ (ℤ_≥‘𝑀))
128	125, 98, 127	rspcdva 2914	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (𝐺‘(𝐽‘𝑏)) ∈ 𝑆)
129	123, 128	eqeltrd 2307	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → if(𝑏 ≤ 𝑁, (𝐺‘(𝐽‘𝑏)), (𝐺‘𝑀)) ∈ 𝑆)
130	119, 122, 88, 129	fvmptd 5730	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (⦋𝐽 / 𝑓⦌𝑃‘𝑏) = if(𝑏 ≤ 𝑁, (𝐺‘(𝐽‘𝑏)), (𝐺‘𝑀)))
131	130, 123	eqtrd 2263	. . . . . 6 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (⦋𝐽 / 𝑓⦌𝑃‘𝑏) = (𝐺‘(𝐽‘𝑏)))
132	67, 107, 131	3eqtr4rd 2274	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑏 ∈ (𝑀...(𝐾 − 1))) → (⦋𝐽 / 𝑓⦌𝑃‘𝑏) = (⦋𝑄 / 𝑓⦌𝑃‘𝑏))
133	1	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → 𝐾 ∈ (𝑀...𝑁))
134	51	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
135	92	adantlr 477	. . . . . 6 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
136	133, 134, 63, 135, 68	iseqf1olemjpcl 10776	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (⦋𝐽 / 𝑓⦌𝑃‘𝑥) ∈ 𝑆)
137	133, 134, 63, 135, 68	iseqf1olemqpcl 10777	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (⦋𝑄 / 𝑓⦌𝑃‘𝑥) ∈ 𝑆)
138		iseqf1o.1	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
139	138	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
140	15, 132, 136, 137, 139	seq3fveq 10747	. . . 4 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘(𝐾 − 1)) = (seq𝑀( + , ⦋𝑄 / 𝑓⦌𝑃)‘(𝐾 − 1)))
141		iseqf1o.2	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
142		iseqf1o.3	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
143		iseqf1o.4	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
144		iseqf1o.6	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
145		iseqf1olemstep.const	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥)
146		iseqf1olemnk	. . . . . . 7 ⊢ (𝜑 → 𝐾 ≠ (^◡𝐽‘𝐾))
147	138, 141, 142, 143, 144, 92, 1, 51, 145, 146, 63, 68	seq3f1olemqsumk 10780	. . . . . 6 ⊢ (𝜑 → (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁))
148	147	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁))
149	7	zcnd 9608	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → 𝐾 ∈ ℂ)
150		npcan1 8562	. . . . . . . 8 ⊢ (𝐾 ∈ ℂ → ((𝐾 − 1) + 1) = 𝐾)
151	149, 150	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → ((𝐾 − 1) + 1) = 𝐾)
152	151	seqeq1d 10721	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → seq((𝐾 − 1) + 1)( + , ⦋𝐽 / 𝑓⦌𝑃) = seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃))
153	152	fveq1d 5644	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → (seq((𝐾 − 1) + 1)( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁))
154	151	seqeq1d 10721	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → seq((𝐾 − 1) + 1)( + , ⦋𝑄 / 𝑓⦌𝑃) = seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃))
155	154	fveq1d 5644	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → (seq((𝐾 − 1) + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁) = (seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁))
156	148, 153, 155	3eqtr4d 2273	. . . 4 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → (seq((𝐾 − 1) + 1)( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq((𝐾 − 1) + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁))
157	140, 156	oveq12d 6041	. . 3 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → ((seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘(𝐾 − 1)) + (seq((𝐾 − 1) + 1)( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁)) = ((seq𝑀( + , ⦋𝑄 / 𝑓⦌𝑃)‘(𝐾 − 1)) + (seq((𝐾 − 1) + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁)))
158	142	adantlr 477	. . . 4 ⊢ (((𝜑 ∧ 𝑀 < 𝐾) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
159		elfzuz3 10262	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘𝐾))
160	1, 159	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝐾))
161	160	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → 𝑁 ∈ (ℤ_≥‘𝐾))
162	151	fveq2d 5646	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → (ℤ_≥‘((𝐾 − 1) + 1)) = (ℤ_≥‘𝐾))
163	161, 162	eleqtrrd 2310	. . . 4 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → 𝑁 ∈ (ℤ_≥‘((𝐾 − 1) + 1)))
164	139, 158, 163, 15, 136	seq3split 10756	. . 3 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = ((seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘(𝐾 − 1)) + (seq((𝐾 − 1) + 1)( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁)))
165	139, 158, 163, 15, 137	seq3split 10756	. . 3 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → (seq𝑀( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁) = ((seq𝑀( + , ⦋𝑄 / 𝑓⦌𝑃)‘(𝐾 − 1)) + (seq((𝐾 − 1) + 1)( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁)))
166	157, 164, 165	3eqtr4d 2273	. 2 ⊢ ((𝜑 ∧ 𝑀 < 𝐾) → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁))
167	147	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝑀 = 𝐾) → (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁))
168		seqeq1 10718	. . . . . 6 ⊢ (𝑀 = 𝐾 → seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃) = seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃))
169	168	fveq1d 5644	. . . . 5 ⊢ (𝑀 = 𝐾 → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁))
170		seqeq1 10718	. . . . . 6 ⊢ (𝑀 = 𝐾 → seq𝑀( + , ⦋𝑄 / 𝑓⦌𝑃) = seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃))
171	170	fveq1d 5644	. . . . 5 ⊢ (𝑀 = 𝐾 → (seq𝑀( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁) = (seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁))
172	169, 171	eqeq12d 2245	. . . 4 ⊢ (𝑀 = 𝐾 → ((seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁) ↔ (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁)))
173	172	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝑀 = 𝐾) → ((seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁) ↔ (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁)))
174	167, 173	mpbird 167	. 2 ⊢ ((𝜑 ∧ 𝑀 = 𝐾) → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁))
175		elfzle1 10267	. . . 4 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝐾)
176	1, 175	syl 14	. . 3 ⊢ (𝜑 → 𝑀 ≤ 𝐾)
177		zleloe 9531	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ≤ 𝐾 ↔ (𝑀 < 𝐾 ∨ 𝑀 = 𝐾)))
178	3, 6, 177	syl2anc 411	. . 3 ⊢ (𝜑 → (𝑀 ≤ 𝐾 ↔ (𝑀 < 𝐾 ∨ 𝑀 = 𝐾)))
179	176, 178	mpbid 147	. 2 ⊢ (𝜑 → (𝑀 < 𝐾 ∨ 𝑀 = 𝐾))
180	166, 174, 179	mpjaodan 805	1 ⊢ (𝜑 → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 715 ∧ w3a 1004 = wceq 1397 ∈ wcel 2201 ≠ wne 2401 ∀wral 2509 Vcvv 2801 ⦋csb 3126 ifcif 3604 class class class wbr 4089 ↦ cmpt 4151 ^◡ccnv 4726 ⟶wf 5324 –1-1-onto→wf1o 5327 ‘cfv 5328 (class class class)co 6023 Fincfn 6914 ℂcc 8035 ℝcr 8036 1c1 8038 + caddc 8040 < clt 8219 ≤ cle 8220 − cmin 8355 ℤcz 9484 ℤ_≥cuz 9760 ...cfz 10248 ..^cfzo 10382 seqcseq 10715

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2203 ax-14 2204 ax-ext 2212 ax-coll 4205 ax-sep 4208 ax-nul 4216 ax-pow 4266 ax-pr 4301 ax-un 4532 ax-setind 4637 ax-iinf 4688 ax-cnex 8128 ax-resscn 8129 ax-1cn 8130 ax-1re 8131 ax-icn 8132 ax-addcl 8133 ax-addrcl 8134 ax-mulcl 8135 ax-addcom 8137 ax-addass 8139 ax-distr 8141 ax-i2m1 8142 ax-0lt1 8143 ax-0id 8145 ax-rnegex 8146 ax-cnre 8148 ax-pre-ltirr 8149 ax-pre-ltwlin 8150 ax-pre-lttrn 8151 ax-pre-apti 8152 ax-pre-ltadd 8153

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1810 df-eu 2081 df-mo 2082 df-clab 2217 df-cleq 2223 df-clel 2226 df-nfc 2362 df-ne 2402 df-nel 2497 df-ral 2514 df-rex 2515 df-reu 2516 df-rab 2518 df-v 2803 df-sbc 3031 df-csb 3127 df-dif 3201 df-un 3203 df-in 3205 df-ss 3212 df-nul 3494 df-if 3605 df-pw 3655 df-sn 3676 df-pr 3677 df-op 3679 df-uni 3895 df-int 3930 df-iun 3973 df-br 4090 df-opab 4152 df-mpt 4153 df-tr 4189 df-id 4392 df-iord 4465 df-on 4467 df-ilim 4468 df-suc 4470 df-iom 4691 df-xp 4733 df-rel 4734 df-cnv 4735 df-co 4736 df-dm 4737 df-rn 4738 df-res 4739 df-ima 4740 df-iota 5288 df-fun 5330 df-fn 5331 df-f 5332 df-f1 5333 df-fo 5334 df-f1o 5335 df-fv 5336 df-riota 5976 df-ov 6026 df-oprab 6027 df-mpo 6028 df-1st 6308 df-2nd 6309 df-recs 6476 df-frec 6562 df-1o 6587 df-er 6707 df-en 6915 df-fin 6917 df-pnf 8221 df-mnf 8222 df-xr 8223 df-ltxr 8224 df-le 8225 df-sub 8357 df-neg 8358 df-inn 9149 df-n0 9408 df-z 9485 df-uz 9761 df-fz 10249 df-fzo 10383 df-seqfrec 10716

This theorem is referenced by: seq3f1olemstep 10782

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