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Theorem seqf1olem2 13763

Description: Lemma for seqf1o 13764. (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 6601	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 Fn (𝑀...(𝑁 + 1)))
3		fzssp1 13299	. . . . . . . . 9 ⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))
4		fnssres 6555	. . . . . . . . 9 ⊢ ((𝐺 Fn (𝑀...(𝑁 + 1)) ∧ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) → (𝐺 ↾ (𝑀...𝑁)) Fn (𝑀...𝑁))
5	2, 3, 4	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)) Fn (𝑀...𝑁))
6		fzfid 13693	. . . . . . . 8 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
7		fnfi 8964	. . . . . . . 8 ⊢ (((𝐺 ↾ (𝑀...𝑁)) Fn (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → (𝐺 ↾ (𝑀...𝑁)) ∈ Fin)
8	5, 6, 7	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)) ∈ Fin)
9	8	elexd 3452	. . . . . 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 13762	. . . . . . . 8 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
19		f1of 6716	. . . . . . . 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 7780	. . . . . . 7 ⊢ ((𝐽:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin ∧ (𝑀...𝑁) ∈ Fin) → 𝐽 ∈ V)
22	20, 6, 6, 21	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ V)
23	9, 22	jca 512	. . . . 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 6640	. . . . . . 7 ⊢ ((𝐺:(𝑀...(𝑁 + 1))⟶𝐶 ∧ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶)
26	1, 3, 25	sylancl 586	. . . . . 6 ⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶)
27	18, 26	jca 512	. . . . 5 ⊢ (𝜑 → (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶))
28		f1oeq1 6704	. . . . . . . 8 ⊢ (𝑓 = 𝐽 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
29		feq1 6581	. . . . . . . 8 ⊢ (𝑔 = (𝐺 ↾ (𝑀...𝑁)) → (𝑔:(𝑀...𝑁)⟶𝐶 ↔ (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝐶))
30	28, 29	bi2anan9r 637	. . . . . . 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 5766	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐺 ↾ (𝑀...𝑁)) → (𝑔 ∘ 𝑓) = ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝑓))
32		coeq2 5767	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐽 → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝑓) = ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))
33	31, 32	sylan9eq 2798	. . . . . . . . . 10 ⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → (𝑔 ∘ 𝑓) = ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))
34	33	seqeq3d 13729	. . . . . . . . 9 ⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → seq𝑀( + , (𝑔 ∘ 𝑓)) = seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)))
35	34	fveq1d 6776	. . . . . . . 8 ⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁))
36		simpl 483	. . . . . . . . . 10 ⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → 𝑔 = (𝐺 ↾ (𝑀...𝑁)))
37	36	seqeq3d 13729	. . . . . . . . 9 ⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → seq𝑀( + , 𝑔) = seq𝑀( + , (𝐺 ↾ (𝑀...𝑁))))
38	37	fveq1d 6776	. . . . . . . 8 ⊢ ((𝑔 = (𝐺 ↾ (𝑀...𝑁)) ∧ 𝑓 = 𝐽) → (seq𝑀( + , 𝑔)‘𝑁) = (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁))
39	35, 38	eqeq12d 2754	. . . . . . 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 3539	. . . . 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 6793	. . . . . 6 ⊢ (𝑥 ∈ (𝑀...𝑁) → ((𝐺 ↾ (𝑀...𝑁))‘𝑥) = (𝐺‘𝑥))
44	43	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘𝑥) = (𝐺‘𝑥))
45	13, 44	seqfveq 13747	. . . 4 ⊢ (𝜑 → (seq𝑀( + , (𝐺 ↾ (𝑀...𝑁)))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
46	42, 45	eqtrd 2778	. . 3 ⊢ (𝜑 → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
47	46	oveq1d 7290	. 2 ⊢ (𝜑 → ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘(𝑁 + 1))))
48	10	adantlr 712	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
49	12	adantlr 712	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
50		elfzuz3 13253	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘𝐾))
51	50	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝑁 ∈ (ℤ_≥‘𝐾))
52		eluzp1p1 12610	. . . . . 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 13252	. . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ_≥‘𝑀))
55	54	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐾 ∈ (ℤ_≥‘𝑀))
56		f1of 6716	. . . . . . . . . 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 6624	. . . . . . . . 9 ⊢ ((𝐺:(𝑀...(𝑁 + 1))⟶𝐶 ∧ 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝐶)
59	1, 57, 58	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝐶)
60	59, 14	fssd 6618	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝑆)
61	60	ffvelrnda 6961	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆)
62	61	adantlr 712	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆)
63	48, 49, 53, 55, 62	seqsplit 13756	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) + (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1))))
64		elfzp12 13335	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁))))
65	64	biimpa 477	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)))
66	13, 65	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)))
67		seqeq1 13724	. . . . . . . . . . 11 ⊢ (𝐾 = 𝑀 → seq𝐾( + , (𝐺 ∘ 𝐹)) = seq𝑀( + , (𝐺 ∘ 𝐹)))
68	67	eqcomd 2744	. . . . . . . . . 10 ⊢ (𝐾 = 𝑀 → seq𝑀( + , (𝐺 ∘ 𝐹)) = seq𝐾( + , (𝐺 ∘ 𝐹)))
69	68	fveq1d 6776	. . . . . . . . 9 ⊢ (𝐾 = 𝑀 → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) = (seq𝐾( + , (𝐺 ∘ 𝐹))‘𝐾))
70		f1ocnv 6728	. . . . . . . . . . . . 13 ⊢ (𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → ^◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
71		f1of 6716	. . . . . . . . . . . . 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 12641	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 + 1) ∈ (ℤ_≥‘𝑀))
74		eluzfz2 13264	. . . . . . . . . . . . 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	ffvelrnd 6962	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝐹‘(𝑁 + 1)) ∈ (𝑀...(𝑁 + 1)))
77	17, 76	eqeltrid 2843	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (𝑀...(𝑁 + 1)))
78		elfzelz 13256	. . . . . . . . . 10 ⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → 𝐾 ∈ ℤ)
79		seq1 13734	. . . . . . . . . 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 2800	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) = ((𝐺 ∘ 𝐹)‘𝐾))
82	81	oveq1d 7290	. . . . . . 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 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → 𝐾 = 𝑀)
84		eluzfz1 13263	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
85	13, 84	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
86	85	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → 𝑀 ∈ (𝑀...𝑁))
87	83, 86	eqeltrd 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → 𝐾 ∈ (𝑀...𝑁))
88	11	adantlr 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
89	14	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐶 ⊆ 𝑆)
90	59	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐺 ∘ 𝐹):(𝑀...(𝑁 + 1))⟶𝐶)
91	77	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐾 ∈ (𝑀...(𝑁 + 1)))
92		peano2uz 12641	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (𝐾 + 1) ∈ (ℤ_≥‘𝑀))
93		fzss1 13295	. . . . . . . . . . 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 13761	. . . . . . . . 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 12351	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → 1 ∈ ℤ)
97		elfzuz 13252	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → 𝐾 ∈ (ℤ_≥‘𝑀))
98		fzss1 13295	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁))
99	77, 97, 98	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐾...𝑁) ⊆ (𝑀...𝑁))
100	99	sselda 3921	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝑥 ∈ (𝑀...𝑁))
101	20	ffvelrnda 6961	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐽‘𝑥) ∈ (𝑀...𝑁))
102	100, 101	syldan 591	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐽‘𝑥) ∈ (𝑀...𝑁))
103	102	fvresd 6794	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = (𝐺‘(𝐽‘𝑥)))
104		breq1 5077	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑥 → (𝑘 < 𝐾 ↔ 𝑥 < 𝐾))
105		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑥 → 𝑘 = 𝑥)
106		oveq1 7282	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑥 → (𝑘 + 1) = (𝑥 + 1))
107	104, 105, 106	ifbieq12d 4487	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑥 → if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)) = if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)))
108	107	fveq2d 6778	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))))
109		fvex 6787	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))) ∈ V
110	108, 16, 109	fvmpt 6875	. . . . . . . . . . . . . . . . 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 12426	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 ∈ ℝ)
114	113	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝐾 ∈ ℝ)
115		elfzelz 13256	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝐾...𝑁) → 𝑥 ∈ ℤ)
116	115	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝑥 ∈ ℤ)
117	116	zred 12426	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝑥 ∈ ℝ)
118		elfzle1 13259	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝐾...𝑁) → 𝐾 ≤ 𝑥)
119	118	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → 𝐾 ≤ 𝑥)
120	114, 117, 119	lensymd 11126	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ¬ 𝑥 < 𝐾)
121		iffalse 4468	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑥 < 𝐾 → if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)) = (𝑥 + 1))
122	121	fveq2d 6778	. . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐽‘𝑥) = (𝐹‘(𝑥 + 1)))
125	124	fveq2d 6778	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐺‘(𝐽‘𝑥)) = (𝐺‘(𝐹‘(𝑥 + 1))))
126	103, 125	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = (𝐺‘(𝐹‘(𝑥 + 1))))
127		fvco3 6867	. . . . . . . . . . . . . . 15 ⊢ ((𝐽:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)))
128	20, 127	sylan 580	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)))
129	100, 128	syldan 591	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)))
130		fzp1elp1 13309	. . . . . . . . . . . . . . 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 6867	. . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 + 1) ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘(𝑥 + 1)) = (𝐺‘(𝐹‘(𝑥 + 1))))
134	131, 133	syldan 591	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → ((𝐺 ∘ 𝐹)‘(𝑥 + 1)) = (𝐺‘(𝐹‘(𝑥 + 1))))
135	126, 129, 134	3eqtr4d 2788	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ∘ 𝐹)‘(𝑥 + 1)))
136	135	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ∘ 𝐹)‘(𝑥 + 1)))
137	51, 96, 136	seqshft2 13749	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)))
138		fvco3 6867	. . . . . . . . . . . . 13 ⊢ ((𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)) ∧ 𝐾 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝐾) = (𝐺‘(𝐹‘𝐾)))
139	57, 77, 138	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝐾) = (𝐺‘(𝐹‘𝐾)))
140	17	fveq2i 6777	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝐾) = (𝐹‘(^◡𝐹‘(𝑁 + 1)))
141		f1ocnvfv2 7149	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘(^◡𝐹‘(𝑁 + 1))) = (𝑁 + 1))
143	140, 142	eqtrid 2790	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘𝐾) = (𝑁 + 1))
144	143	fveq2d 6778	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺‘(𝐹‘𝐾)) = (𝐺‘(𝑁 + 1)))
145	139, 144	eqtr2d 2779	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺‘(𝑁 + 1)) = ((𝐺 ∘ 𝐹)‘𝐾))
146	145	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐺‘(𝑁 + 1)) = ((𝐺 ∘ 𝐹)‘𝐾))
147	137, 146	oveq12d 7293	. . . . . . . . 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 2781	. . . . . . . 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 591	. . . . . . 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 13727	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)) = seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)))
151	150	fveq1d 6776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = 𝑀) → (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁))
152	151	oveq1d 7290	. . . . . . 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 2782	. . . . . 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 12587	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
155	13, 154	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
156		elfzuz 13252	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ((𝑀 + 1)...𝑁) → 𝐾 ∈ (ℤ_≥‘(𝑀 + 1)))
157		eluzp1m1 12608	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ (ℤ_≥‘(𝑀 + 1))) → (𝐾 − 1) ∈ (ℤ_≥‘𝑀))
158	155, 156, 157	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (𝐾 − 1) ∈ (ℤ_≥‘𝑀))
159		eluzelz 12592	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
160	13, 159	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℤ)
161	160	zcnd 12427	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℂ)
162		ax-1cn 10929	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
163		pncan 11227	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
164	161, 162, 163	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
165		peano2zm 12363	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈ ℤ)
166	77, 78, 165	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐾 − 1) ∈ ℤ)
167		elfzuz3 13253	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → (𝑁 + 1) ∈ (ℤ_≥‘𝐾))
168	77, 167	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘𝐾))
169	112	zcnd 12427	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐾 ∈ ℂ)
170		npcan 11230	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 − 1) + 1) = 𝐾)
171	169, 162, 170	sylancl 586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾)
172	171	fveq2d 6778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℤ_≥‘((𝐾 − 1) + 1)) = (ℤ_≥‘𝐾))
173	168, 172	eleqtrrd 2842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘((𝐾 − 1) + 1)))
174		eluzp1m1 12608	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 − 1) ∈ ℤ ∧ (𝑁 + 1) ∈ (ℤ_≥‘((𝐾 − 1) + 1))) → ((𝑁 + 1) − 1) ∈ (ℤ_≥‘(𝐾 − 1)))
175	166, 173, 174	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 + 1) − 1) ∈ (ℤ_≥‘(𝐾 − 1)))
176	164, 175	eqeltrrd 2840	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝐾 − 1)))
177		fzss2 13296	. . . . . . . . . . . . . . . . 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 3921	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → 𝑥 ∈ (𝑀...𝑁))
180	179, 101	syldan 591	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐽‘𝑥) ∈ (𝑀...𝑁))
181	180	fvresd 6794	. . . . . . . . . . . . 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 13327	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑥 ∈ (𝑀...(𝐾 − 1)) ↔ (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥 ∧ 𝑥 < 𝐾)))
184	155, 112, 183	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑥 ∈ (𝑀...(𝐾 − 1)) ↔ (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥 ∧ 𝑥 < 𝐾)))
185	184	biimpa 477	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝑥 ∈ ℤ ∧ 𝑀 ≤ 𝑥 ∧ 𝑥 < 𝐾))
186	185	simp3d 1143	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → 𝑥 < 𝐾)
187		iftrue 4465	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 < 𝐾 → if(𝑥 < 𝐾, 𝑥, (𝑥 + 1)) = 𝑥)
188	187	fveq2d 6778	. . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐽‘𝑥) = (𝐹‘𝑥))
191	190	fveq2d 6778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐺‘(𝐽‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
192	181, 191	eqtr2d 2779	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐺‘(𝐹‘𝑥)) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)))
193		peano2uz 12641	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘(𝐾 − 1)) → (𝑁 + 1) ∈ (ℤ_≥‘(𝐾 − 1)))
194		fzss2 13296	. . . . . . . . . . . . . . 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 3921	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → 𝑥 ∈ (𝑀...(𝑁 + 1)))
197		fvco3 6867	. . . . . . . . . . . . . 14 ⊢ ((𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
198	57, 197	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
199	196, 198	syldan 591	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
200	179, 128	syldan 591	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)))
201	192, 199, 200	3eqtr4d 2788	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥))
202	201	adantlr 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) = (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥))
203	158, 202	seqfveq 13747	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)))
204		fzp1ss 13307	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁))
205	13, 154, 204	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁))
206	205	sselda 3921	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝐾 ∈ (𝑀...𝑁))
207	206, 148	syldan 591	. . . . . . . . 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 7293	. . . . . . . 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 591	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆)
210	209	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆)
211	10	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
212	158, 210, 211	seqcl 13743	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) ∈ 𝑆)
213	59, 77	ffvelrnd 6962	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝐶)
214	14, 213	sseldd 3922	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝑆)
215	214	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((𝐺 ∘ 𝐹)‘𝐾) ∈ 𝑆)
216	94	sselda 3921	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ ((𝐾 + 1)...(𝑁 + 1))) → 𝑥 ∈ (𝑀...(𝑁 + 1)))
217	216, 62	syldan 591	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) ∧ 𝑥 ∈ ((𝐾 + 1)...(𝑁 + 1))) → ((𝐺 ∘ 𝐹)‘𝑥) ∈ 𝑆)
218	53, 217, 48	seqcl 13743	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) ∈ 𝑆)
219	206, 218	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq(𝐾 + 1)( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) ∈ 𝑆)
220	212, 215, 219	3jca 1127	. . . . . . . . 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 7470	. . . . . . . . 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 591	. . . . . . . 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 6618	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺:(𝑀...(𝑁 + 1))⟶𝑆)
224		fssres 6640	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:(𝑀...(𝑁 + 1))⟶𝑆 ∧ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝑆)
225	223, 3, 224	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝑆)
226		fco 6624	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ↾ (𝑀...𝑁)):(𝑀...𝑁)⟶𝑆 ∧ 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽):(𝑀...𝑁)⟶𝑆)
227	225, 20, 226	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽):(𝑀...𝑁)⟶𝑆)
228	227	ffvelrnda 6961	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆)
229	179, 228	syldan 591	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆)
230	229	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆)
231	158, 230, 211	seqcl 13743	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘(𝐾 − 1)) ∈ 𝑆)
232		elfzuz3 13253	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ((𝑀 + 1)...𝑁) → 𝑁 ∈ (ℤ_≥‘𝐾))
233	232	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝑁 ∈ (ℤ_≥‘𝐾))
234	100, 228	syldan 591	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆)
235	234	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝐾...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆)
236	233, 235, 211	seqcl 13743	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) ∈ 𝑆)
237	223, 75	ffvelrnd 6962	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺‘(𝑁 + 1)) ∈ 𝑆)
238	237	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (𝐺‘(𝑁 + 1)) ∈ 𝑆)
239	231, 236, 238	3jca 1127	. . . . . . . . 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 7470	. . . . . . . . 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 591	. . . . . . . 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 2788	. . . . . . 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 13740	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ (ℤ_≥‘(𝑀 + 1))) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + ((𝐺 ∘ 𝐹)‘𝐾)))
244	155, 156, 243	syl2an 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝐾) = ((seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝐾 − 1)) + ((𝐺 ∘ 𝐹)‘𝐾)))
245	244	oveq1d 7290	. . . . . . 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 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
247		elfzelz 13256	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ ((𝑀 + 1)...𝑁) → 𝐾 ∈ ℤ)
248	247	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝐾 ∈ ℤ)
249	248	zcnd 12427	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝐾 ∈ ℂ)
250	249, 162, 170	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → ((𝐾 − 1) + 1) = 𝐾)
251	250	fveq2d 6778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (ℤ_≥‘((𝐾 − 1) + 1)) = (ℤ_≥‘𝐾))
252	233, 251	eleqtrrd 2842	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → 𝑁 ∈ (ℤ_≥‘((𝐾 − 1) + 1)))
253	228	adantlr 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) ∈ 𝑆)
254	211, 246, 252, 158, 253	seqsplit 13756	. . . . . . . . 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 13727	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → seq((𝐾 − 1) + 1)( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)) = seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)))
256	255	fveq1d 6776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (seq((𝐾 − 1) + 1)( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) = (seq𝐾( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁))
257	256	oveq2d 7291	. . . . . . . . 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 2778	. . . . . . . 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 7290	. . . . . . 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 2788	. . . . . 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 955	. . . . 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 591	. . . 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 2778	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝑀...𝑁)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))
264	13	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → 𝑁 ∈ (ℤ_≥‘𝑀))
265		seqp1 13736	. . . . 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 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐽‘𝑥) = (𝐹‘if(𝑥 < 𝐾, 𝑥, (𝑥 + 1))))
268		elfzelz 13256	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ)
269	268	zred 12426	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℝ)
270	269	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ)
271	160	zred 12426	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ)
272	271	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℝ)
273		peano2re 11148	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
274	272, 273	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑁 + 1) ∈ ℝ)
275		elfzle2 13260	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ≤ 𝑁)
276	275	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ≤ 𝑁)
277	272	ltp1d 11905	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 < (𝑁 + 1))
278	270, 272, 274, 276, 277	lelttrd 11133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 < (𝑁 + 1))
279	278	adantlr 712	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 < (𝑁 + 1))
280		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 = (𝑁 + 1))
281	279, 280	breqtrrd 5102	. . . . . . . . . . 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 2778	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐽‘𝑥) = (𝐹‘𝑥))
284	283	fveq2d 6778	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐹‘𝑥)))
285	269	adantl 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ)
286	285, 281	gtned 11110	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 ≠ 𝑥)
287	57	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
288		fzelp1 13308	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ (𝑀...(𝑁 + 1)))
289	288	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ (𝑀...(𝑁 + 1)))
290	287, 289	ffvelrnd 6962	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)))
291	13	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 ∈ (ℤ_≥‘𝑀))
292		elfzp1 13306	. . . . . . . . . . . . . . 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 861	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (¬ (𝐹‘𝑥) ∈ (𝑀...𝑁) → (𝐹‘𝑥) = (𝑁 + 1)))
296	15	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
297		f1ocnvfv 7150	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) = (𝑁 + 1) → (^◡𝐹‘(𝑁 + 1)) = 𝑥))
299	17	eqeq1i 2743	. . . . . . . . . . . . 13 ⊢ (𝐾 = 𝑥 ↔ (^◡𝐹‘(𝑁 + 1)) = 𝑥)
300	298, 299	syl6ibr 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 2960	. . . . . . . . . 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 6794	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ↾ (𝑀...𝑁))‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
305	284, 304	eqtr2d 2779	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘(𝐹‘𝑥)) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)))
306	57, 288, 197	syl2an 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
307	306	adantlr 712	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
308	128	adantlr 712	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥) = ((𝐺 ↾ (𝑀...𝑁))‘(𝐽‘𝑥)))
309	305, 307, 308	3eqtr4d 2788	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 = (𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐺 ∘ 𝐹)‘𝑥) = (((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽)‘𝑥))
310	264, 309	seqfveq 13747	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘𝑁) = (seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁))
311		fvco3 6867	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘(𝑁 + 1)) = (𝐺‘(𝐹‘(𝑁 + 1))))
313	312	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → ((𝐺 ∘ 𝐹)‘(𝑁 + 1)) = (𝐺‘(𝐹‘(𝑁 + 1))))
314		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → 𝐾 = (𝑁 + 1))
315	17, 314	eqtr3id 2792	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (^◡𝐹‘(𝑁 + 1)) = (𝑁 + 1))
316	315	fveq2d 6778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐹‘(^◡𝐹‘(𝑁 + 1))) = (𝐹‘(𝑁 + 1)))
317	142	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐹‘(^◡𝐹‘(𝑁 + 1))) = (𝑁 + 1))
318	316, 317	eqtr3d 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐹‘(𝑁 + 1)) = (𝑁 + 1))
319	318	fveq2d 6778	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (𝐺‘(𝐹‘(𝑁 + 1))) = (𝐺‘(𝑁 + 1)))
320	313, 319	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → ((𝐺 ∘ 𝐹)‘(𝑁 + 1)) = (𝐺‘(𝑁 + 1)))
321	310, 320	oveq12d 7293	. . . 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 2778	. . 3 ⊢ ((𝜑 ∧ 𝐾 = (𝑁 + 1)) → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))
323		elfzp1 13306	. . . . 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 956	. 2 ⊢ (𝜑 → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = ((seq𝑀( + , ((𝐺 ↾ (𝑀...𝑁)) ∘ 𝐽))‘𝑁) + (𝐺‘(𝑁 + 1))))
327		seqp1 13736	. . 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 2788	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 396 ∨ wo 844 ∧ w3a 1086 ∀wal 1537 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 Vcvv 3432 ⊆ wss 3887 ifcif 4459 class class class wbr 5074 ↦ cmpt 5157 ^◡ccnv 5588 ↾ cres 5591 ∘ ccom 5593 Fn wfn 6428 ⟶wf 6429 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 ℂcc 10869 ℝcr 10870 1c1 10872 + caddc 10874 < clt 11009 ≤ cle 11010 − cmin 11205 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 seqcseq 13721

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722

This theorem is referenced by: seqf1o 13764

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