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Theorem seqf1o 14005

Description: Rearrange a sum via an arbitrary bijection on (𝑀...𝑁). (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)
seqf1o.6	⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
seqf1o.7	⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘𝑥) ∈ 𝐶)
seqf1o.8	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = (𝐺‘(𝐹‘𝑘)))

**Assertion**
Ref	Expression
seqf1o	⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

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

Proof of Theorem seqf1o Dummy variables 𝑓 𝑔 𝑠 𝑡 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqf1o.6	. . 3 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
2		seqf1o.7	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘𝑥) ∈ 𝐶)
3	2	fmpttd 7111	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)):(𝑀...𝑁)⟶𝐶)
4		seqf1o.4	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		oveq2 7413	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀))
6		f1oeq23 6821	. . . . . . . . . . 11 ⊢ (((𝑀...𝑥) = (𝑀...𝑀) ∧ (𝑀...𝑥) = (𝑀...𝑀)) → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)))
7	5, 5, 6	syl2anc 584	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)))
8	5	feq2d 6700	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝑔:(𝑀...𝑥)⟶𝐶 ↔ 𝑔:(𝑀...𝑀)⟶𝐶))
9	7, 8	anbi12d 631	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → ((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) ↔ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)))
10		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑀))
11		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑀))
12	10, 11	eqeq12d 2748	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → ((seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀)))
13	9, 12	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀))))
14	13	2albidv 1926	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (∀𝑔∀𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ∀𝑔∀𝑓((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀))))
15	14	imbi2d 340	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥))) ↔ (𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀)))))
16		oveq2 7413	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑘 → (𝑀...𝑥) = (𝑀...𝑘))
17		f1oeq23 6821	. . . . . . . . . . 11 ⊢ (((𝑀...𝑥) = (𝑀...𝑘) ∧ (𝑀...𝑥) = (𝑀...𝑘)) → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘)))
18	16, 16, 17	syl2anc 584	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘)))
19	16	feq2d 6700	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → (𝑔:(𝑀...𝑥)⟶𝐶 ↔ 𝑔:(𝑀...𝑘)⟶𝐶))
20	18, 19	anbi12d 631	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → ((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) ↔ (𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶)))
21		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘))
22		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑘))
23	21, 22	eqeq12d 2748	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → ((seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)))
24	20, 23	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))))
25	24	2albidv 1926	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (∀𝑔∀𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))))
26	25	imbi2d 340	. . . . . 6 ⊢ (𝑥 = 𝑘 → ((𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥))) ↔ (𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)))))
27		oveq2 7413	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → (𝑀...𝑥) = (𝑀...(𝑘 + 1)))
28		f1oeq23 6821	. . . . . . . . . . 11 ⊢ (((𝑀...𝑥) = (𝑀...(𝑘 + 1)) ∧ (𝑀...𝑥) = (𝑀...(𝑘 + 1))) → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1))))
29	27, 27, 28	syl2anc 584	. . . . . . . . . 10 ⊢ (𝑥 = (𝑘 + 1) → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1))))
30	27	feq2d 6700	. . . . . . . . . 10 ⊢ (𝑥 = (𝑘 + 1) → (𝑔:(𝑀...𝑥)⟶𝐶 ↔ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶))
31	29, 30	anbi12d 631	. . . . . . . . 9 ⊢ (𝑥 = (𝑘 + 1) → ((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) ↔ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)))
32		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = (𝑘 + 1) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , (𝑔 ∘ 𝑓))‘(𝑘 + 1)))
33		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = (𝑘 + 1) → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))
34	32, 33	eqeq12d 2748	. . . . . . . . 9 ⊢ (𝑥 = (𝑘 + 1) → ((seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔 ∘ 𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1))))
35	31, 34	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))))
36	35	2albidv 1926	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (∀𝑔∀𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ∀𝑔∀𝑓((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))))
37	36	imbi2d 340	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → ((𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥))) ↔ (𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1))))))
38		oveq2 7413	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁))
39		f1oeq23 6821	. . . . . . . . . . 11 ⊢ (((𝑀...𝑥) = (𝑀...𝑁) ∧ (𝑀...𝑥) = (𝑀...𝑁)) → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
40	38, 38, 39	syl2anc 584	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
41	38	feq2d 6700	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (𝑔:(𝑀...𝑥)⟶𝐶 ↔ 𝑔:(𝑀...𝑁)⟶𝐶))
42	40, 41	anbi12d 631	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → ((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) ↔ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶)))
43		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁))
44		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑁))
45	43, 44	eqeq12d 2748	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → ((seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)))
46	42, 45	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁))))
47	46	2albidv 1926	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (∀𝑔∀𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ∀𝑔∀𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁))))
48	47	imbi2d 340	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥))) ↔ (𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)))))
49		f1of 6830	. . . . . . . . . . . . 13 ⊢ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) → 𝑓:(𝑀...𝑀)⟶(𝑀...𝑀))
50	49	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → 𝑓:(𝑀...𝑀)⟶(𝑀...𝑀))
51		elfz3 13507	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (𝑀...𝑀))
52		fvco3 6987	. . . . . . . . . . . 12 ⊢ ((𝑓:(𝑀...𝑀)⟶(𝑀...𝑀) ∧ 𝑀 ∈ (𝑀...𝑀)) → ((𝑔 ∘ 𝑓)‘𝑀) = (𝑔‘(𝑓‘𝑀)))
53	50, 51, 52	syl2anr 597	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → ((𝑔 ∘ 𝑓)‘𝑀) = (𝑔‘(𝑓‘𝑀)))
54		ffvelcdm 7080	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(𝑀...𝑀)⟶(𝑀...𝑀) ∧ 𝑀 ∈ (𝑀...𝑀)) → (𝑓‘𝑀) ∈ (𝑀...𝑀))
55	49, 51, 54	syl2anr 597	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℤ ∧ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)) → (𝑓‘𝑀) ∈ (𝑀...𝑀))
56		fzsn 13539	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
57	56	eleq2d 2819	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℤ → ((𝑓‘𝑀) ∈ (𝑀...𝑀) ↔ (𝑓‘𝑀) ∈ {𝑀}))
58		elsni 4644	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑀) ∈ {𝑀} → (𝑓‘𝑀) = 𝑀)
59	57, 58	syl6bi 252	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℤ → ((𝑓‘𝑀) ∈ (𝑀...𝑀) → (𝑓‘𝑀) = 𝑀))
60	59	imp 407	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℤ ∧ (𝑓‘𝑀) ∈ (𝑀...𝑀)) → (𝑓‘𝑀) = 𝑀)
61	55, 60	syldan 591	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)) → (𝑓‘𝑀) = 𝑀)
62	61	adantrr 715	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (𝑓‘𝑀) = 𝑀)
63	62	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (𝑔‘(𝑓‘𝑀)) = (𝑔‘𝑀))
64	53, 63	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → ((𝑔 ∘ 𝑓)‘𝑀) = (𝑔‘𝑀))
65		seq1 13975	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑀) = ((𝑔 ∘ 𝑓)‘𝑀))
66	65	adantr 481	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑀) = ((𝑔 ∘ 𝑓)‘𝑀))
67		seq1 13975	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝑔)‘𝑀) = (𝑔‘𝑀))
68	67	adantr 481	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , 𝑔)‘𝑀) = (𝑔‘𝑀))
69	64, 66, 68	3eqtr4d 2782	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀))
70	69	ex 413	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → ((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀)))
71	70	alrimivv 1931	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → ∀𝑔∀𝑓((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀)))
72	71	a1d 25	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀))))
73		f1oeq1 6818	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑡 → (𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ↔ 𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘)))
74		feq1 6695	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑠 → (𝑔:(𝑀...𝑘)⟶𝐶 ↔ 𝑠:(𝑀...𝑘)⟶𝐶))
75	73, 74	bi2anan9r 638	. . . . . . . . . . 11 ⊢ ((𝑔 = 𝑠 ∧ 𝑓 = 𝑡) → ((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) ↔ (𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑠:(𝑀...𝑘)⟶𝐶)))
76		coeq1 5855	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑠 → (𝑔 ∘ 𝑓) = (𝑠 ∘ 𝑓))
77		coeq2 5856	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑡 → (𝑠 ∘ 𝑓) = (𝑠 ∘ 𝑡))
78	76, 77	sylan9eq 2792	. . . . . . . . . . . . . 14 ⊢ ((𝑔 = 𝑠 ∧ 𝑓 = 𝑡) → (𝑔 ∘ 𝑓) = (𝑠 ∘ 𝑡))
79	78	seqeq3d 13970	. . . . . . . . . . . . 13 ⊢ ((𝑔 = 𝑠 ∧ 𝑓 = 𝑡) → seq𝑀( + , (𝑔 ∘ 𝑓)) = seq𝑀( + , (𝑠 ∘ 𝑡)))
80	79	fveq1d 6890	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝑠 ∧ 𝑓 = 𝑡) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , (𝑠 ∘ 𝑡))‘𝑘))
81		simpl 483	. . . . . . . . . . . . . 14 ⊢ ((𝑔 = 𝑠 ∧ 𝑓 = 𝑡) → 𝑔 = 𝑠)
82	81	seqeq3d 13970	. . . . . . . . . . . . 13 ⊢ ((𝑔 = 𝑠 ∧ 𝑓 = 𝑡) → seq𝑀( + , 𝑔) = seq𝑀( + , 𝑠))
83	82	fveq1d 6890	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝑠 ∧ 𝑓 = 𝑡) → (seq𝑀( + , 𝑔)‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘))
84	80, 83	eqeq12d 2748	. . . . . . . . . . 11 ⊢ ((𝑔 = 𝑠 ∧ 𝑓 = 𝑡) → ((seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘) ↔ (seq𝑀( + , (𝑠 ∘ 𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘)))
85	75, 84	imbi12d 344	. . . . . . . . . 10 ⊢ ((𝑔 = 𝑠 ∧ 𝑓 = 𝑡) → (((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)) ↔ ((𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑠:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑠 ∘ 𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘))))
86	85	cbval2vw 2043	. . . . . . . . 9 ⊢ (∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)) ↔ ∀𝑠∀𝑡((𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑠:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑠 ∘ 𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘)))
87		simplll 773	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) ∧ ∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) → 𝜑)
88		seqf1o.1	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
89	87, 88	sylan 580	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) ∧ ∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
90		seqf1o.2	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
91	87, 90	sylan 580	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) ∧ ∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
92		seqf1o.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
93	87, 92	sylan 580	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) ∧ ∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
94		simpllr 774	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) ∧ ∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) → 𝑘 ∈ (ℤ_≥‘𝑀))
95		seqf1o.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ⊆ 𝑆)
96	87, 95	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) ∧ ∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) → 𝐶 ⊆ 𝑆)
97		simprl 769	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) ∧ ∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) → 𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)))
98		simprr 771	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) ∧ ∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) → 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)
99		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝑀...𝑘) ↦ (𝑓‘if(𝑤 < (^◡𝑓‘(𝑘 + 1)), 𝑤, (𝑤 + 1)))) = (𝑤 ∈ (𝑀...𝑘) ↦ (𝑓‘if(𝑤 < (^◡𝑓‘(𝑘 + 1)), 𝑤, (𝑤 + 1))))
100		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (^◡𝑓‘(𝑘 + 1)) = (^◡𝑓‘(𝑘 + 1))
101		simplr 767	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) ∧ ∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) → ∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)))
102	101, 86	sylib 217	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) ∧ ∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) → ∀𝑠∀𝑡((𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑠:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑠 ∘ 𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘)))
103	89, 91, 93, 94, 96, 97, 98, 99, 100, 102	seqf1olem2 14004	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) ∧ ∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))
104	103	exp31 420	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)) → ((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))))
105	86, 104	biimtrrid 242	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (∀𝑠∀𝑡((𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑠:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑠 ∘ 𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘)) → ((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))))
106	105	alrimdv 1932	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (∀𝑠∀𝑡((𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑠:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑠 ∘ 𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘)) → ∀𝑓((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))))
107	106	alrimdv 1932	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (∀𝑠∀𝑡((𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑠:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑠 ∘ 𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘)) → ∀𝑔∀𝑓((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))))
108	86, 107	biimtrid 241	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)) → ∀𝑔∀𝑓((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))))
109	108	expcom 414	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝜑 → (∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)) → ∀𝑔∀𝑓((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1))))))
110	109	a2d 29	. . . . . 6 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → ((𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) → (𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1))))))
111	15, 26, 37, 48, 72, 110	uzind4 12886	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁))))
112	4, 111	mpcom 38	. . . 4 ⊢ (𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)))
113		fvex 6901	. . . . . . 7 ⊢ (𝐺‘𝑥) ∈ V
114		eqid 2732	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥))
115	113, 114	fnmpti 6690	. . . . . 6 ⊢ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) Fn (𝑀...𝑁)
116		fzfi 13933	. . . . . 6 ⊢ (𝑀...𝑁) ∈ Fin
117		fnfi 9177	. . . . . 6 ⊢ (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) Fn (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∈ Fin)
118	115, 116, 117	mp2an 690	. . . . 5 ⊢ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∈ Fin
119		f1of 6830	. . . . . . 7 ⊢ (𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐹:(𝑀...𝑁)⟶(𝑀...𝑁))
120	1, 119	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶(𝑀...𝑁))
121		ovexd 7440	. . . . . 6 ⊢ (𝜑 → (𝑀...𝑁) ∈ V)
122		fex2 7920	. . . . . 6 ⊢ ((𝐹:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ V ∧ (𝑀...𝑁) ∈ V) → 𝐹 ∈ V)
123	120, 121, 121, 122	syl3anc 1371	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
124		f1oeq1 6818	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
125		feq1 6695	. . . . . . . 8 ⊢ (𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) → (𝑔:(𝑀...𝑁)⟶𝐶 ↔ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)):(𝑀...𝑁)⟶𝐶))
126	124, 125	bi2anan9r 638	. . . . . . 7 ⊢ ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∧ 𝑓 = 𝐹) → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) ↔ (𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)):(𝑀...𝑁)⟶𝐶)))
127		coeq1 5855	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) → (𝑔 ∘ 𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝑓))
128		coeq2 5856	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹))
129	127, 128	sylan9eq 2792	. . . . . . . . . 10 ⊢ ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∧ 𝑓 = 𝐹) → (𝑔 ∘ 𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹))
130	129	seqeq3d 13970	. . . . . . . . 9 ⊢ ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∧ 𝑓 = 𝐹) → seq𝑀( + , (𝑔 ∘ 𝑓)) = seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹)))
131	130	fveq1d 6890	. . . . . . . 8 ⊢ ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∧ 𝑓 = 𝐹) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹))‘𝑁))
132		simpl 483	. . . . . . . . . 10 ⊢ ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∧ 𝑓 = 𝐹) → 𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)))
133	132	seqeq3d 13970	. . . . . . . . 9 ⊢ ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∧ 𝑓 = 𝐹) → seq𝑀( + , 𝑔) = seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥))))
134	133	fveq1d 6890	. . . . . . . 8 ⊢ ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∧ 𝑓 = 𝐹) → (seq𝑀( + , 𝑔)‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)))‘𝑁))
135	131, 134	eqeq12d 2748	. . . . . . 7 ⊢ ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∧ 𝑓 = 𝐹) → ((seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁) ↔ (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)))‘𝑁)))
136	126, 135	imbi12d 344	. . . . . 6 ⊢ ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∧ 𝑓 = 𝐹) → (((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)) ↔ ((𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)):(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)))‘𝑁))))
137	136	spc2gv 3590	. . . . 5 ⊢ (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∈ Fin ∧ 𝐹 ∈ V) → (∀𝑔∀𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)) → ((𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)):(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)))‘𝑁))))
138	118, 123, 137	sylancr 587	. . . 4 ⊢ (𝜑 → (∀𝑔∀𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)) → ((𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)):(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)))‘𝑁))))
139	112, 138	mpd 15	. . 3 ⊢ (𝜑 → ((𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)):(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)))‘𝑁)))
140	1, 3, 139	mp2and 697	. 2 ⊢ (𝜑 → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)))‘𝑁))
141	120	ffvelcdmda 7083	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ (𝑀...𝑁))
142		fveq2 6888	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑘) → (𝐺‘𝑥) = (𝐺‘(𝐹‘𝑘)))
143		fvex 6901	. . . . . 6 ⊢ (𝐺‘(𝐹‘𝑘)) ∈ V
144	142, 114, 143	fvmpt 6995	. . . . 5 ⊢ ((𝐹‘𝑘) ∈ (𝑀...𝑁) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥))‘(𝐹‘𝑘)) = (𝐺‘(𝐹‘𝑘)))
145	141, 144	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥))‘(𝐹‘𝑘)) = (𝐺‘(𝐹‘𝑘)))
146		fvco3 6987	. . . . 5 ⊢ ((𝐹:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ 𝑘 ∈ (𝑀...𝑁)) → (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥))‘(𝐹‘𝑘)))
147	120, 146	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥))‘(𝐹‘𝑘)))
148		seqf1o.8	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = (𝐺‘(𝐹‘𝑘)))
149	145, 147, 148	3eqtr4d 2782	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹)‘𝑘) = (𝐻‘𝑘))
150	4, 149	seqfveq 13988	. 2 ⊢ (𝜑 → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , 𝐻)‘𝑁))
151		fveq2 6888	. . . . 5 ⊢ (𝑥 = 𝑘 → (𝐺‘𝑥) = (𝐺‘𝑘))
152		fvex 6901	. . . . 5 ⊢ (𝐺‘𝑘) ∈ V
153	151, 114, 152	fvmpt 6995	. . . 4 ⊢ (𝑘 ∈ (𝑀...𝑁) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥))‘𝑘) = (𝐺‘𝑘))
154	153	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥))‘𝑘) = (𝐺‘𝑘))
155	4, 154	seqfveq 13988	. 2 ⊢ (𝜑 → (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
156	140, 150, 155	3eqtr3d 2780	1 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 ∀wal 1539 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3947 ifcif 4527 {csn 4627 class class class wbr 5147 ↦ cmpt 5230 ^◡ccnv 5674 ∘ ccom 5679 Fn wfn 6535 ⟶wf 6536 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 1c1 11107 + caddc 11109 < clt 11244 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 seqcseq 13962

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963

This theorem is referenced by: summolem3 15656 prodmolem3 15873 eulerthlem2 16711 gsumval3eu 19766 gsumval3 19769

W3C validator