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Theorem seqf1olem2 14008

Description: Lemma for seqf1o 14009. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)

**Hypotheses**
Ref	Expression
seqf1o.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seqf1o.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
seqf1o.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
seqf1o.4	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
seqf1o.5	⊢ (𝜑 → 𝐶 ⊆ 𝑆)
seqf1olem.5	⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
seqf1olem.6	⊢ (𝜑 → 𝐺:(𝑀...(𝑁 + 1))⟶𝐶)
seqf1olem.7	⊢ 𝐽 = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))
seqf1olem.8	⊢ 𝐾 = (^◡𝐹‘(𝑁 + 1))
seqf1olem.9	⊢ (𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)))

**Assertion**
Ref	Expression
seqf1olem2	⊢ (𝜑 → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = (seq𝑀( + , 𝐺)‘(𝑁 + 1)))

Distinct variable groups: 𝑓,𝑔,𝑘,𝑥,𝑦,𝑧,𝐹 𝑓,𝐺,𝑔,𝑘,𝑥,𝑦,𝑧 𝑓,𝑀,𝑔,𝑘,𝑥,𝑦,𝑧 + ,𝑓,𝑔,𝑘,𝑥,𝑦,𝑧 𝑓,𝐽,𝑔,𝑥,𝑦,𝑧 𝑓,𝑁,𝑔,𝑘,𝑥,𝑦,𝑧 𝑘,𝐾,𝑥,𝑦,𝑧 𝜑,𝑓,𝑔,𝑘,𝑥,𝑦,𝑧 𝑆,𝑘,𝑥,𝑦,𝑧 𝐶,𝑓,𝑔,𝑘,𝑥,𝑦,𝑧 Allowed substitution hints: 𝑆(𝑓,𝑔) 𝐽(𝑘) 𝐾(𝑓,𝑔)

**Proof of Theorem seqf1olem2**
Step	Hyp	Ref	Expression
1		seqf1olem.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:(𝑀...(𝑁 + 1))⟶𝐶)
2	1	ffnd 6719	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 Fn (𝑀...(𝑁 + 1)))
3		fzssp1 13544	. . . . . . . . 9 ⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))
4		fnssres 6674	. . . . . . . . 9 ⊢ ((𝐺 Fn (𝑀...(𝑁 + 1)) ∧ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) → (𝐺 ↾ (𝑀...𝑁)) Fn (𝑀...𝑁))
5	2, 3, 4	sylancl 587	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)) Fn (𝑀...𝑁))
6		fzfid 13938	. . . . . . . 8 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
7		fnfi 9181	. . . . . . . 8 ⊢ (((𝐺 ↾ (𝑀...𝑁)) Fn (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → (𝐺 ↾ (𝑀...𝑁)) ∈ Fin)
8	5, 6, 7	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)) ∈ Fin)
9	8	elexd 3495	. . . . . 6 ⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)) ∈ V)
10		seqf1o.1	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
11		seqf1o.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
12		seqf1o.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
13		seqf1o.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
14		seqf1o.5	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ⊆ 𝑆)
15		seqf1olem.5	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
16		seqf1olem.7	. . . . . . . . 9 ⊢ 𝐽 = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))
17		seqf1olem.8	. . . . . . . . 9 ⊢ 𝐾 = (^◡𝐹‘(𝑁 + 1))
18	10, 11, 12, 13, 14, 15, 1, 16, 17	seqf1olem1 14007	. . . . . . . 8 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
19		f1of 6834	. . . . . . . 8 ⊢ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
20	18, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
21		fex2 7924	. . . . . . 7 ⊢ ((𝐽:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin ∧ (𝑀...𝑁) ∈ Fin) → 𝐽 ∈ V)
22	20, 6, 6, 21	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ V)
23	9, 22	jca 513	. . . . 5 ⊢ (𝜑 → ((𝐺 ↾ (𝑀...𝑁)) ∈ V ∧ 𝐽 ∈ V))
24		seqf1olem.9	. . . . 5 ⊢ (𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)))
25		fssres 6758	. . . . . . 7 ⊢ ((𝐺:(𝑀...(𝑁 + 1))⟶𝐶 ∧ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶)
26	1, 3, 25	sylancl 587	. . . . . 6 ⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶)
27	18, 26	jca 513	. . . . 5 ⊢ (𝜑 → (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶))
28		f1oeq1 6822	. . . . . . . 8 ⊢ (𝑓 = 𝐽 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
29		feq1 6699	. . . . . . . 8 ⊢ (𝑔 = (𝐺 ↾ (𝑀...𝑁)) → (𝑔:(𝑀...𝑁)⟶𝐶 ↔ (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶))
30	28, 29	bi2anan9r 639	. . . . . . 7 ⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) ↔ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶)))
31		coeq1 5858	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐺 ↾ (𝑀...𝑁)) → (𝑔 ∘ 𝑓) = ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝑓))
32		coeq2 5859	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐽 → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝑓) = ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))
33	31, 32	sylan9eq 2793	. . . . . . . . . 10 ⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → (𝑔 ∘ 𝑓) = ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))
34	33	seqeq3d 13974	. . . . . . . . 9 ⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → seq𝑀( + , (𝑔 ∘ 𝑓)) = seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)))
35	34	fveq1d 6894	. . . . . . . 8 ⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁))
36		simpl 484	. . . . . . . . . 10 ⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → 𝑔 = (𝐺 ↾ (𝑀...𝑁)))
37	36	seqeq3d 13974	. . . . . . . . 9 ⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → seq𝑀( + , 𝑔) = seq𝑀( + , (𝐺 ↾ (𝑀...𝑁))))
38	37	fveq1d 6894	. . . . . . . 8 ⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → (seq𝑀( + , 𝑔)‘𝑁) = (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁))
39	35, 38	eqeq12d 2749	. . . . . . 7 ⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → ((seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁) ↔ (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁)))
40	30, 39	imbi12d 345	. . . . . 6 ⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → (((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)) ↔ ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁))))
41	40	spc2gv 3591	. . . . 5 ⊢ (((𝐺 ↾ (𝑀...𝑁)) ∈ V ∧ 𝐽 ∈ V) → (∀𝑔∀𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)) → ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁))))
42	23, 24, 27, 41	syl3c 66	. . . 4 ⊢ (𝜑 → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁))
43		fvres 6911	. . . . . 6 ⊢ (𝑥 ∈ (𝑀...𝑁) → ((𝐺 ↾ (𝑀...𝑁))‘𝑥) = (𝐺‘𝑥))
44	43	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘𝑥) = (𝐺‘𝑥))
45	13, 44	seqfveq 13992	. . . 4 ⊢ (𝜑 → (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
46	42, 45	eqtrd 2773	. . 3 ⊢ (𝜑 → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
47	46	oveq1d 7424	. 2 ⊢ (𝜑 → ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘(𝑁 + 1))))
48	10	adantlr 714	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
49	12	adantlr 714	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
50		elfzuz3 13498	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘𝐾))
51	50	adantl 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝑁 ∈ (ℤ_≥‘𝐾))
52		eluzp1p1 12850	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝐾) → (𝑁 + 1) ∈ (ℤ_≥‘(𝐾 + 1)))
53	51, 52	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝑁 + 1) ∈ (ℤ_≥‘(𝐾 + 1)))
54		elfzuz 13497	. . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ_≥‘𝑀))
55	54	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐾 ∈ (ℤ_≥‘𝑀))
56		f1of 6834	. . . . . . . . . 10 ⊢ (𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
57	15, 56	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
58		fco 6742	. . . . . . . . 9 ⊢ ((𝐺:(𝑀...(𝑁 + 1))⟶𝐶 ∧ 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝐶)
59	1, 57, 58	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝐶)
60	59, 14	fssd 6736	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝑆)
61	60	ffvelcdmda 7087	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆)
62	61	adantlr 714	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆)
63	48, 49, 53, 55, 62	seqsplit 14001	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))))
64		elfzp12 13580	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁))))
65	64	biimpa 478	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)))
66	13, 65	sylan 581	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)))
67		seqeq1 13969	. . . . . . . . . . 11 ⊢ (𝐾 = 𝑀 → seq𝐾( + , (𝐺 ∘ 𝐹)) = seq𝑀( + , (𝐺 ∘ 𝐹)))
68	67	eqcomd 2739	. . . . . . . . . 10 ⊢ (𝐾 = 𝑀 → seq𝑀( + , (𝐺 ∘ 𝐹)) = seq𝐾( + , (𝐺 ∘ 𝐹)))
69	68	fveq1d 6894	. . . . . . . . 9 ⊢ (𝐾 = 𝑀 → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) = (seq𝐾( + , (𝐺 ∘ 𝐹))‘𝐾))
70		f1ocnv 6846	. . . . . . . . . . . . 13 ⊢ (𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → ^◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
71		f1of 6834	. . . . . . . . . . . . 13 ⊢ (^◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → ^◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
72	15, 70, 71	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ^◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
73		peano2uz 12885	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 + 1) ∈ (ℤ_≥‘𝑀))
74		eluzfz2 13509	. . . . . . . . . . . . 13 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘𝑀) → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1)))
75	13, 73, 74	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1)))
76	72, 75	ffvelcdmd 7088	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝐹‘(𝑁 + 1)) ∈ (𝑀...(𝑁 + 1)))
77	17, 76	eqeltrid 2838	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (𝑀...(𝑁 + 1)))
78		elfzelz 13501	. . . . . . . . . 10 ⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → 𝐾 ∈ ℤ)
79		seq1 13979	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (seq𝐾( + , (𝐺 ∘ 𝐹))‘𝐾) = ((𝐺 ∘ 𝐹)‘𝐾))
80	77, 78, 79	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (seq𝐾( + , (𝐺 ∘ 𝐹))‘𝐾) = ((𝐺 ∘ 𝐹)‘𝐾))
81	69, 80	sylan9eqr 2795	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) = ((𝐺 ∘ 𝐹)‘𝐾))
82	81	oveq1d 7424	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))))
83		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → 𝐾 = 𝑀)
84		eluzfz1 13508	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
85	13, 84	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
86	85	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → 𝑀 ∈ (𝑀...𝑁))
87	83, 86	eqeltrd 2834	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → 𝐾 ∈ (𝑀...𝑁))
88	11	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
89	14	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐶 ⊆ 𝑆)
90	59	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝐶)
91	77	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐾 ∈ (𝑀...(𝑁 + 1)))
92		peano2uz 12885	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (𝐾 + 1) ∈ (ℤ_≥‘𝑀))
93		fzss1 13540	. . . . . . . . . . 11 ⊢ ((𝐾 + 1) ∈ (ℤ_≥‘𝑀) → ((𝐾 + 1)...(𝑁 + 1)) ⊆ (𝑀...(𝑁 + 1)))
94	55, 92, 93	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → ((𝐾 + 1)...(𝑁 + 1)) ⊆ (𝑀...(𝑁 + 1)))
95	48, 88, 49, 53, 89, 90, 91, 94	seqf1olem2a 14006	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) + ((𝐺 ∘ 𝐹)‘𝐾)))
96		1zzd 12593	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 1 ∈ ℤ)
97		elfzuz 13497	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → 𝐾 ∈ (ℤ_≥‘𝑀))
98		fzss1 13540	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁))
99	77, 97, 98	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐾...𝑁) ⊆ (𝑀...𝑁))
100	99	sselda 3983	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝑥 ∈ (𝑀...𝑁))
101	20	ffvelcdmda 7087	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐽‘𝑥) ∈ (𝑀...𝑁))
102	100, 101	syldan 592	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐽‘𝑥) ∈ (𝑀...𝑁))
103	102	fvresd 6912	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = (𝐺‘(𝐽‘𝑥)))
104		breq1 5152	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑥 → (𝑘 < 𝐾 ↔ 𝑥 < 𝐾))
105		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑥 → 𝑘 = 𝑥)
106		oveq1 7416	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑥 → (𝑘 + 1) = (𝑥 + 1))
107	104, 105, 106	ifbieq12d 4557	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑥 → if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)) = if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)))
108	107	fveq2d 6896	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))))
109		fvex 6905	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) ∈ V
110	108, 16, 109	fvmpt 6999	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝑀...𝑁) → (𝐽‘𝑥) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))))
111	100, 110	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐽‘𝑥) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))))
112	77, 78	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐾 ∈ ℤ)
113	112	zred 12666	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 ∈ ℝ)
114	113	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝐾 ∈ ℝ)
115		elfzelz 13501	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝐾...𝑁) → 𝑥 ∈ ℤ)
116	115	adantl 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝑥 ∈ ℤ)
117	116	zred 12666	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝑥 ∈ ℝ)
118		elfzle1 13504	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝐾...𝑁) → 𝐾 ≤ 𝑥)
119	118	adantl 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝐾 ≤ 𝑥)
120	114, 117, 119	lensymd 11365	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ¬ 𝑥 < 𝐾)
121		iffalse 4538	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑥 < 𝐾 → if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)) = (𝑥 + 1))
122	121	fveq2d 6896	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 < 𝐾 → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) = (𝐹‘(𝑥 + 1)))
123	120, 122	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) = (𝐹‘(𝑥 + 1)))
124	111, 123	eqtrd 2773	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐽‘𝑥) = (𝐹‘(𝑥 + 1)))
125	124	fveq2d 6896	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐺‘(𝐽‘𝑥)) = (𝐺‘(𝐹‘(𝑥 + 1))))
126	103, 125	eqtrd 2773	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = (𝐺‘(𝐹‘(𝑥 + 1))))
127		fvco3 6991	. . . . . . . . . . . . . . 15 ⊢ ((𝐽:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)))
128	20, 127	sylan 581	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)))
129	100, 128	syldan 592	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)))
130		fzp1elp1 13554	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑀...𝑁) → (𝑥 + 1) ∈ (𝑀...(𝑁 + 1)))
131	100, 130	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝑥 + 1) ∈ (𝑀...(𝑁 + 1)))
132		fvco3 6991	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)) ∧ (𝑥 + 1) ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘(𝑥 + 1)) = (𝐺‘(𝐹‘(𝑥 + 1))))
133	57, 132	sylan 581	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 + 1) ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘(𝑥 + 1)) = (𝐺‘(𝐹‘(𝑥 + 1))))
134	131, 133	syldan 592	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ((𝐺 ∘ 𝐹)‘(𝑥 + 1)) = (𝐺‘(𝐹‘(𝑥 + 1))))
135	126, 129, 134	3eqtr4d 2783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ∘ 𝐹)‘(𝑥 + 1)))
136	135	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ∘ 𝐹)‘(𝑥 + 1)))
137	51, 96, 136	seqshft2 13994	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)))
138		fvco3 6991	. . . . . . . . . . . . 13 ⊢ ((𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)) ∧ 𝐾 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝐾) = (𝐺‘(𝐹‘𝐾)))
139	57, 77, 138	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝐾) = (𝐺‘(𝐹‘𝐾)))
140	17	fveq2i 6895	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝐾) = (𝐹‘(^◡𝐹‘(𝑁 + 1)))
141		f1ocnvfv2 7275	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) → (𝐹‘(^◡𝐹‘(𝑁 + 1))) = (𝑁 + 1))
142	15, 75, 141	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘(^◡𝐹‘(𝑁 + 1))) = (𝑁 + 1))
143	140, 142	eqtrid 2785	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘𝐾) = (𝑁 + 1))
144	143	fveq2d 6896	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺‘(𝐹‘𝐾)) = (𝐺‘(𝑁 + 1)))
145	139, 144	eqtr2d 2774	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺‘(𝑁 + 1)) = ((𝐺 ∘ 𝐹)‘𝐾))
146	145	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐺‘(𝑁 + 1)) = ((𝐺 ∘ 𝐹)‘𝐾))
147	137, 146	oveq12d 7427	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))) = ((seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) + ((𝐺 ∘ 𝐹)‘𝐾)))
148	95, 147	eqtr4d 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))
149	87, 148	syldan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))
150	83	seqeq1d 13972	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)) = seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)))
151	150	fveq1d 6894	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁))
152	151	oveq1d 7424	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))
153	82, 149, 152	3eqtrd 2777	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))
154		eluzel2 12827	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
155	13, 154	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
156		elfzuz 13497	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ((𝑀 + 1)...𝑁) → 𝐾 ∈ (ℤ_≥‘(𝑀 + 1)))
157		eluzp1m1 12848	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ (ℤ_≥‘(𝑀 + 1))) → (𝐾 − 1) ∈ (ℤ_≥‘𝑀))
158	155, 156, 157	syl2an 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (𝐾 − 1) ∈ (ℤ_≥‘𝑀))
159		eluzelz 12832	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
160	13, 159	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℤ)
161	160	zcnd 12667	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℂ)
162		ax-1cn 11168	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
163		pncan 11466	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
164	161, 162, 163	sylancl 587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
165		peano2zm 12605	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈ ℤ)
166	77, 78, 165	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐾 − 1) ∈ ℤ)
167		elfzuz3 13498	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → (𝑁 + 1) ∈ (ℤ_≥‘𝐾))
168	77, 167	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘𝐾))
169	112	zcnd 12667	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐾 ∈ ℂ)
170		npcan 11469	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 − 1) + 1) = 𝐾)
171	169, 162, 170	sylancl 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾)
172	171	fveq2d 6896	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℤ_≥‘((𝐾 − 1) + 1)) = (ℤ_≥‘𝐾))
173	168, 172	eleqtrrd 2837	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘((𝐾 − 1) + 1)))
174		eluzp1m1 12848	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 − 1) ∈ ℤ ∧ (𝑁 + 1) ∈ (ℤ_≥‘((𝐾 − 1) + 1))) → ((𝑁 + 1) − 1) ∈ (ℤ_≥‘(𝐾 − 1)))
175	166, 173, 174	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 + 1) − 1) ∈ (ℤ_≥‘(𝐾 − 1)))
176	164, 175	eqeltrrd 2835	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝐾 − 1)))
177		fzss2 13541	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘(𝐾 − 1)) → (𝑀...(𝐾 − 1)) ⊆ (𝑀...𝑁))
178	176, 177	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀...(𝐾 − 1)) ⊆ (𝑀...𝑁))
179	178	sselda 3983	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → 𝑥 ∈ (𝑀...𝑁))
180	179, 101	syldan 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐽‘𝑥) ∈ (𝑀...𝑁))
181	180	fvresd 6912	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = (𝐺‘(𝐽‘𝑥)))
182	179, 110	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐽‘𝑥) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))))
183		elfzm11 13572	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑥 ∈ (𝑀...(𝐾 − 1)) ↔ (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥 ∧ 𝑥 < 𝐾)))
184	155, 112, 183	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑥 ∈ (𝑀...(𝐾 − 1)) ↔ (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥 ∧ 𝑥 < 𝐾)))
185	184	biimpa 478	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥 ∧ 𝑥 < 𝐾))
186	185	simp3d 1145	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → 𝑥 < 𝐾)
187		iftrue 4535	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 < 𝐾 → if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)) = 𝑥)
188	187	fveq2d 6896	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 < 𝐾 → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) = (𝐹‘𝑥))
189	186, 188	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) = (𝐹‘𝑥))
190	182, 189	eqtrd 2773	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐽‘𝑥) = (𝐹‘𝑥))
191	190	fveq2d 6896	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐺‘(𝐽‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
192	181, 191	eqtr2d 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐺‘(𝐹‘𝑥)) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)))
193		peano2uz 12885	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘(𝐾 − 1)) → (𝑁 + 1) ∈ (ℤ_≥‘(𝐾 − 1)))
194		fzss2 13541	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘(𝐾 − 1)) → (𝑀...(𝐾 − 1)) ⊆ (𝑀...(𝑁 + 1)))
195	176, 193, 194	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀...(𝐾 − 1)) ⊆ (𝑀...(𝑁 + 1)))
196	195	sselda 3983	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → 𝑥 ∈ (𝑀...(𝑁 + 1)))
197		fvco3 6991	. . . . . . . . . . . . . 14 ⊢ ((𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
198	57, 197	sylan 581	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
199	196, 198	syldan 592	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
200	179, 128	syldan 592	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)))
201	192, 199, 200	3eqtr4d 2783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥))
202	201	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥))
203	158, 202	seqfveq 13992	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)))
204		fzp1ss 13552	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁))
205	13, 154, 204	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁))
206	205	sselda 3983	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝐾 ∈ (𝑀...𝑁))
207	206, 148	syldan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))
208	203, 207	oveq12d 7427	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))))
209	196, 61	syldan 592	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆)
210	209	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆)
211	10	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
212	158, 210, 211	seqcl 13988	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) ∈ 𝑆)
213	59, 77	ffvelcdmd 7088	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝐶)
214	14, 213	sseldd 3984	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝑆)
215	214	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝑆)
216	94	sselda 3983	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ ((𝐾 + 1)...(𝑁 + 1))) → 𝑥 ∈ (𝑀...(𝑁 + 1)))
217	216, 62	syldan 592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ ((𝐾 + 1)...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆)
218	53, 217, 48	seqcl 13988	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) ∈ 𝑆)
219	206, 218	syldan 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) ∈ 𝑆)
220	212, 215, 219	3jca 1129	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) ∈ 𝑆 ∧ ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝑆 ∧ (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) ∈ 𝑆))
221	12	caovassg 7605	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) ∈ 𝑆 ∧ ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝑆 ∧ (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) ∈ 𝑆)) → (((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + ((𝐺 ∘ 𝐹)‘𝐾)) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)))))
222	220, 221	syldan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + ((𝐺 ∘ 𝐹)‘𝐾)) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + (((𝐺 ∘ 𝐹)‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)))))
223	1, 14	fssd 6736	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺:(𝑀...(𝑁 + 1))⟶𝑆)
224		fssres 6758	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:(𝑀...(𝑁 + 1))⟶𝑆 ∧ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝑆)
225	223, 3, 224	sylancl 587	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝑆)
226		fco 6742	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝑆 ∧ 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽):(𝑀...𝑁)⟶𝑆)
227	225, 20, 226	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽):(𝑀...𝑁)⟶𝑆)
228	227	ffvelcdmda 7087	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆)
229	179, 228	syldan 592	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆)
230	229	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆)
231	158, 230, 211	seqcl 13988	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) ∈ 𝑆)
232		elfzuz3 13498	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ((𝑀 + 1)...𝑁) → 𝑁 ∈ (ℤ_≥‘𝐾))
233	232	adantl 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝑁 ∈ (ℤ_≥‘𝐾))
234	100, 228	syldan 592	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆)
235	234	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆)
236	233, 235, 211	seqcl 13988	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) ∈ 𝑆)
237	223, 75	ffvelcdmd 7088	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺‘(𝑁 + 1)) ∈ 𝑆)
238	237	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (𝐺‘(𝑁 + 1)) ∈ 𝑆)
239	231, 236, 238	3jca 1129	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) ∈ 𝑆 ∧ (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) ∈ 𝑆 ∧ (𝐺‘(𝑁 + 1)) ∈ 𝑆))
240	12	caovassg 7605	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) ∈ 𝑆 ∧ (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) ∈ 𝑆 ∧ (𝐺‘(𝑁 + 1)) ∈ 𝑆)) → (((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) + (𝐺‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))))
241	239, 240	syldan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) + (𝐺‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + ((seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1)))))
242	208, 222, 241	3eqtr4d 2783	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + ((𝐺 ∘ 𝐹)‘𝐾)) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = (((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) + (𝐺‘(𝑁 + 1))))
243		seqm1 13985	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ (ℤ_≥‘(𝑀 + 1))) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + ((𝐺 ∘ 𝐹)‘𝐾)))
244	155, 156, 243	syl2an 597	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + ((𝐺 ∘ 𝐹)‘𝐾)))
245	244	oveq1d 7424	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = (((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + ((𝐺 ∘ 𝐹)‘𝐾)) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))))
246	12	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
247		elfzelz 13501	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ ((𝑀 + 1)...𝑁) → 𝐾 ∈ ℤ)
248	247	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝐾 ∈ ℤ)
249	248	zcnd 12667	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝐾 ∈ ℂ)
250	249, 162, 170	sylancl 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((𝐾 − 1) + 1) = 𝐾)
251	250	fveq2d 6896	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (ℤ_≥‘((𝐾 − 1) + 1)) = (ℤ_≥‘𝐾))
252	233, 251	eleqtrrd 2837	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝑁 ∈ (ℤ_≥‘((𝐾 − 1) + 1)))
253	228	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆)
254	211, 246, 252, 158, 253	seqsplit 14001	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq((𝐾 − 1) + 1)( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)))
255	250	seqeq1d 13972	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → seq((𝐾 − 1) + 1)( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)) = seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)))
256	255	fveq1d 6894	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq((𝐾 − 1) + 1)( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁))
257	256	oveq2d 7425	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq((𝐾 − 1) + 1)( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)))
258	254, 257	eqtrd 2773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)))
259	258	oveq1d 7424	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))) = (((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) + (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁)) + (𝐺‘(𝑁 + 1))))
260	242, 245, 259	3eqtr4d 2783	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))
261	153, 260	jaodan 957	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁))) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))
262	66, 261	syldan 592	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))
263	63, 262	eqtrd 2773	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))
264	13	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → 𝑁 ∈ (ℤ_≥‘𝑀))
265		seqp1 13981	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝑁) + ((𝐺 ∘ 𝐹)‘(𝑁 + 1))))
266	264, 265	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝑁) + ((𝐺 ∘ 𝐹)‘(𝑁 + 1))))
267	110	adantl 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐽‘𝑥) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))))
268		elfzelz 13501	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ)
269	268	zred 12666	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℝ)
270	269	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ)
271	160	zred 12666	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ)
272	271	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℝ)
273		peano2re 11387	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
274	272, 273	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑁 + 1) ∈ ℝ)
275		elfzle2 13505	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ≤ 𝑁)
276	275	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ≤ 𝑁)
277	272	ltp1d 12144	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 < (𝑁 + 1))
278	270, 272, 274, 276, 277	lelttrd 11372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 < (𝑁 + 1))
279	278	adantlr 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 < (𝑁 + 1))
280		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 = (𝑁 + 1))
281	279, 280	breqtrrd 5177	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 < 𝐾)
282	281, 188	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) = (𝐹‘𝑥))
283	267, 282	eqtrd 2773	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐽‘𝑥) = (𝐹‘𝑥))
284	283	fveq2d 6896	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐹‘𝑥)))
285	269	adantl 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ)
286	285, 281	gtned 11349	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 ≠ 𝑥)
287	57	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
288		fzelp1 13553	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ (𝑀...(𝑁 + 1)))
289	288	adantl 483	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ (𝑀...(𝑁 + 1)))
290	287, 289	ffvelcdmd 7088	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)))
291	13	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 ∈ (ℤ_≥‘𝑀))
292		elfzp1 13551	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘𝑥) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑥) = (𝑁 + 1))))
293	291, 292	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘𝑥) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑥) = (𝑁 + 1))))
294	290, 293	mpbid 231	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑥) = (𝑁 + 1)))
295	294	ord 863	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (¬ (𝐹‘𝑥) ∈ (𝑀...𝑁) → (𝐹‘𝑥) = (𝑁 + 1)))
296	15	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
297		f1ocnvfv 7276	. . . . . . . . . . . . . 14 ⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐹‘𝑥) = (𝑁 + 1) → (^◡𝐹‘(𝑁 + 1)) = 𝑥))
298	296, 289, 297	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) = (𝑁 + 1) → (^◡𝐹‘(𝑁 + 1)) = 𝑥))
299	17	eqeq1i 2738	. . . . . . . . . . . . 13 ⊢ (𝐾 = 𝑥 ↔ (^◡𝐹‘(𝑁 + 1)) = 𝑥)
300	298, 299	imbitrrdi 251	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) = (𝑁 + 1) → 𝐾 = 𝑥))
301	295, 300	syld 47	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (¬ (𝐹‘𝑥) ∈ (𝑀...𝑁) → 𝐾 = 𝑥))
302	301	necon1ad 2958	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐾 ≠ 𝑥 → (𝐹‘𝑥) ∈ (𝑀...𝑁)))
303	286, 302	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ (𝑀...𝑁))
304	303	fvresd 6912	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
305	284, 304	eqtr2d 2774	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘(𝐹‘𝑥)) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)))
306	57, 288, 197	syl2an 597	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
307	306	adantlr 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
308	128	adantlr 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)))
309	305, 307, 308	3eqtr4d 2783	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ∘ 𝐹)‘𝑥) = (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥))
310	264, 309	seqfveq 13992	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝑁) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁))
311		fvco3 6991	. . . . . . . 8 ⊢ ((𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)) ∧ (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘(𝑁 + 1)) = (𝐺‘(𝐹‘(𝑁 + 1))))
312	57, 75, 311	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘(𝑁 + 1)) = (𝐺‘(𝐹‘(𝑁 + 1))))
313	312	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → ((𝐺 ∘ 𝐹)‘(𝑁 + 1)) = (𝐺‘(𝐹‘(𝑁 + 1))))
314		simpr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → 𝐾 = (𝑁 + 1))
315	17, 314	eqtr3id 2787	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (^◡𝐹‘(𝑁 + 1)) = (𝑁 + 1))
316	315	fveq2d 6896	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐹‘(^◡𝐹‘(𝑁 + 1))) = (𝐹‘(𝑁 + 1)))
317	142	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐹‘(^◡𝐹‘(𝑁 + 1))) = (𝑁 + 1))
318	316, 317	eqtr3d 2775	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐹‘(𝑁 + 1)) = (𝑁 + 1))
319	318	fveq2d 6896	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐺‘(𝐹‘(𝑁 + 1))) = (𝐺‘(𝑁 + 1)))
320	313, 319	eqtrd 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → ((𝐺 ∘ 𝐹)‘(𝑁 + 1)) = (𝐺‘(𝑁 + 1)))
321	310, 320	oveq12d 7427	. . . 4 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝑁) + ((𝐺 ∘ 𝐹)‘(𝑁 + 1))) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))
322	266, 321	eqtrd 2773	. . 3 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))
323		elfzp1 13551	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝐾 ∈ (𝑀...(𝑁 + 1)) ↔ (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1))))
324	13, 323	syl 17	. . . 4 ⊢ (𝜑 → (𝐾 ∈ (𝑀...(𝑁 + 1)) ↔ (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1))))
325	77, 324	mpbid 231	. . 3 ⊢ (𝜑 → (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1)))
326	263, 322, 325	mpjaodan 958	. 2 ⊢ (𝜑 → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))
327		seqp1 13981	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (seq𝑀( + , 𝐺)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘(𝑁 + 1))))
328	13, 327	syl 17	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐺)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘(𝑁 + 1))))
329	47, 326, 328	3eqtr4d 2783	1 ⊢ (𝜑 → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = (seq𝑀( + , 𝐺)‘(𝑁 + 1)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 ∀wal 1540 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ⊆ wss 3949 ifcif 4529 class class class wbr 5149 ↦ cmpt 5232 ^◡ccnv 5676 ↾ cres 5679 ∘ ccom 5681 Fn wfn 6539 ⟶wf 6540 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 ℂcc 11108 ℝcr 11109 1c1 11111 + caddc 11113 < clt 11248 ≤ cle 11249 − cmin 11444 ℤcz 12558 ℤ_≥cuz 12822 ...cfz 13484 seqcseq 13966

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-seq 13967

This theorem is referenced by: seqf1o 14009

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