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Theorem seqf1o 14015

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 7090	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)):(𝑀...𝑁)⟶𝐶)
4		seqf1o.4	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		oveq2 7398	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀))
6		f1oeq23 6794	. . . . . . . . . . 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 6675	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝑔:(𝑀...𝑥)⟶𝐶 ↔ 𝑔:(𝑀...𝑀)⟶𝐶))
9	7, 8	anbi12d 632	. . . . . . . . 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 6861	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑀))
11		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑀))
12	10, 11	eqeq12d 2746	. . . . . . . . 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 1923	. . . . . . 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 7398	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑘 → (𝑀...𝑥) = (𝑀...𝑘))
17		f1oeq23 6794	. . . . . . . . . . 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 6675	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → (𝑔:(𝑀...𝑥)⟶𝐶 ↔ 𝑔:(𝑀...𝑘)⟶𝐶))
20	18, 19	anbi12d 632	. . . . . . . . 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 6861	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘))
22		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑘))
23	21, 22	eqeq12d 2746	. . . . . . . . 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 1923	. . . . . . 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 7398	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → (𝑀...𝑥) = (𝑀...(𝑘 + 1)))
28		f1oeq23 6794	. . . . . . . . . . 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 6675	. . . . . . . . . 10 ⊢ (𝑥 = (𝑘 + 1) → (𝑔:(𝑀...𝑥)⟶𝐶 ↔ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶))
31	29, 30	anbi12d 632	. . . . . . . . 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 6861	. . . . . . . . . 10 ⊢ (𝑥 = (𝑘 + 1) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , (𝑔 ∘ 𝑓))‘(𝑘 + 1)))
33		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑥 = (𝑘 + 1) → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))
34	32, 33	eqeq12d 2746	. . . . . . . . 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 1923	. . . . . . 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 7398	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁))
39		f1oeq23 6794	. . . . . . . . . . 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 6675	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (𝑔:(𝑀...𝑥)⟶𝐶 ↔ 𝑔:(𝑀...𝑁)⟶𝐶))
42	40, 41	anbi12d 632	. . . . . . . . 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 6861	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑥) = (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁))
44		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑁))
45	43, 44	eqeq12d 2746	. . . . . . . . 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 1923	. . . . . . 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 6803	. . . . . . . . . . . . 13 ⊢ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) → 𝑓:(𝑀...𝑀)⟶(𝑀...𝑀))
50	49	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → 𝑓:(𝑀...𝑀)⟶(𝑀...𝑀))
51		elfz3 13502	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (𝑀...𝑀))
52		fvco3 6963	. . . . . . . . . . . 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 7056	. . . . . . . . . . . . . . 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 13534	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
57	56	eleq2d 2815	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℤ → ((𝑓‘𝑀) ∈ (𝑀...𝑀) ↔ (𝑓‘𝑀) ∈ {𝑀}))
58		elsni 4609	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑀) ∈ {𝑀} → (𝑓‘𝑀) = 𝑀)
59	57, 58	biimtrdi 253	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℤ → ((𝑓‘𝑀) ∈ (𝑀...𝑀) → (𝑓‘𝑀) = 𝑀))
60	59	imp 406	. . . . . . . . . . . . . 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 717	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (𝑓‘𝑀) = 𝑀)
63	62	fveq2d 6865	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (𝑔‘(𝑓‘𝑀)) = (𝑔‘𝑀))
64	53, 63	eqtrd 2765	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → ((𝑔 ∘ 𝑓)‘𝑀) = (𝑔‘𝑀))
65		seq1 13986	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑀) = ((𝑔 ∘ 𝑓)‘𝑀))
66	65	adantr 480	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑀) = ((𝑔 ∘ 𝑓)‘𝑀))
67		seq1 13986	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝑔)‘𝑀) = (𝑔‘𝑀))
68	67	adantr 480	. . . . . . . . . 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 2775	. . . . . . . . 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 412	. . . . . . . 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 1928	. . . . . . 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 6791	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑡 → (𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ↔ 𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘)))
74		feq1 6669	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑠 → (𝑔:(𝑀...𝑘)⟶𝐶 ↔ 𝑠:(𝑀...𝑘)⟶𝐶))
75	73, 74	bi2anan9r 639	. . . . . . . . . . 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 5824	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑠 → (𝑔 ∘ 𝑓) = (𝑠 ∘ 𝑓))
77		coeq2 5825	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑡 → (𝑠 ∘ 𝑓) = (𝑠 ∘ 𝑡))
78	76, 77	sylan9eq 2785	. . . . . . . . . . . . . 14 ⊢ ((𝑔 = 𝑠 ∧ 𝑓 = 𝑡) → (𝑔 ∘ 𝑓) = (𝑠 ∘ 𝑡))
79	78	seqeq3d 13981	. . . . . . . . . . . . 13 ⊢ ((𝑔 = 𝑠 ∧ 𝑓 = 𝑡) → seq𝑀( + , (𝑔 ∘ 𝑓)) = seq𝑀( + , (𝑠 ∘ 𝑡)))
80	79	fveq1d 6863	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝑠 ∧ 𝑓 = 𝑡) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑘) = (seq𝑀( + , (𝑠 ∘ 𝑡))‘𝑘))
81		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑔 = 𝑠 ∧ 𝑓 = 𝑡) → 𝑔 = 𝑠)
82	81	seqeq3d 13981	. . . . . . . . . . . . 13 ⊢ ((𝑔 = 𝑠 ∧ 𝑓 = 𝑡) → seq𝑀( + , 𝑔) = seq𝑀( + , 𝑠))
83	82	fveq1d 6863	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝑠 ∧ 𝑓 = 𝑡) → (seq𝑀( + , 𝑔)‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘))
84	80, 83	eqeq12d 2746	. . . . . . . . . . 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 2040	. . . . . . . . 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 774	. . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . 14 ⊢ (^◡𝑓‘(𝑘 + 1)) = (^◡𝑓‘(𝑘 + 1))
101		simplr 768	. . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . 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 14014	. . . . . . . . . . . . 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 419	. . . . . . . . . . . 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 243	. . . . . . . . . . 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 1929	. . . . . . . . . 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 1929	. . . . . . . . 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 242	. . . . . . . 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 413	. . . . . . 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 12872	. . . . 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 6874	. . . . . . 7 ⊢ (𝐺‘𝑥) ∈ V
114		eqid 2730	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥))
115	113, 114	fnmpti 6664	. . . . . 6 ⊢ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) Fn (𝑀...𝑁)
116		fzfi 13944	. . . . . 6 ⊢ (𝑀...𝑁) ∈ Fin
117		fnfi 9148	. . . . . 6 ⊢ (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) Fn (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∈ Fin)
118	115, 116, 117	mp2an 692	. . . . 5 ⊢ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∈ Fin
119		f1of 6803	. . . . . . 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 7425	. . . . . 6 ⊢ (𝜑 → (𝑀...𝑁) ∈ V)
122		fex2 7915	. . . . . 6 ⊢ ((𝐹:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ V ∧ (𝑀...𝑁) ∈ V) → 𝐹 ∈ V)
123	120, 121, 121, 122	syl3anc 1373	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
124		f1oeq1 6791	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
125		feq1 6669	. . . . . . . 8 ⊢ (𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) → (𝑔:(𝑀...𝑁)⟶𝐶 ↔ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)):(𝑀...𝑁)⟶𝐶))
126	124, 125	bi2anan9r 639	. . . . . . 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 5824	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) → (𝑔 ∘ 𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝑓))
128		coeq2 5825	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹))
129	127, 128	sylan9eq 2785	. . . . . . . . . 10 ⊢ ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∧ 𝑓 = 𝐹) → (𝑔 ∘ 𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹))
130	129	seqeq3d 13981	. . . . . . . . 9 ⊢ ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∧ 𝑓 = 𝐹) → seq𝑀( + , (𝑔 ∘ 𝑓)) = seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹)))
131	130	fveq1d 6863	. . . . . . . 8 ⊢ ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∧ 𝑓 = 𝐹) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹))‘𝑁))
132		simpl 482	. . . . . . . . . 10 ⊢ ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∧ 𝑓 = 𝐹) → 𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)))
133	132	seqeq3d 13981	. . . . . . . . 9 ⊢ ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∧ 𝑓 = 𝐹) → seq𝑀( + , 𝑔) = seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥))))
134	133	fveq1d 6863	. . . . . . . 8 ⊢ ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∧ 𝑓 = 𝐹) → (seq𝑀( + , 𝑔)‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)))‘𝑁))
135	131, 134	eqeq12d 2746	. . . . . . 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 3569	. . . . 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 699	. 2 ⊢ (𝜑 → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)))‘𝑁))
141	120	ffvelcdmda 7059	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ (𝑀...𝑁))
142		fveq2 6861	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑘) → (𝐺‘𝑥) = (𝐺‘(𝐹‘𝑘)))
143		fvex 6874	. . . . . 6 ⊢ (𝐺‘(𝐹‘𝑘)) ∈ V
144	142, 114, 143	fvmpt 6971	. . . . 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 6963	. . . . 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 2775	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹)‘𝑘) = (𝐻‘𝑘))
150	4, 149	seqfveq 13998	. 2 ⊢ (𝜑 → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , 𝐻)‘𝑁))
151		fveq2 6861	. . . . 5 ⊢ (𝑥 = 𝑘 → (𝐺‘𝑥) = (𝐺‘𝑘))
152		fvex 6874	. . . . 5 ⊢ (𝐺‘𝑘) ∈ V
153	151, 114, 152	fvmpt 6971	. . . 4 ⊢ (𝑘 ∈ (𝑀...𝑁) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥))‘𝑘) = (𝐺‘𝑘))
154	153	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥))‘𝑘) = (𝐺‘𝑘))
155	4, 154	seqfveq 13998	. 2 ⊢ (𝜑 → (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺‘𝑥)))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
156	140, 150, 155	3eqtr3d 2773	1 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∀wal 1538 = wceq 1540 ∈ wcel 2109 Vcvv 3450 ⊆ wss 3917 ifcif 4491 {csn 4592 class class class wbr 5110 ↦ cmpt 5191 ^◡ccnv 5640 ∘ ccom 5645 Fn wfn 6509 ⟶wf 6510 –1-1-onto→wf1o 6513 ‘cfv 6514 (class class class)co 7390 Fincfn 8921 1c1 11076 + caddc 11078 < clt 11215 ℤcz 12536 ℤ_≥cuz 12800 ...cfz 13475 seqcseq 13973

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-fzo 13623 df-seq 13974

This theorem is referenced by: summolem3 15687 prodmolem3 15906 eulerthlem2 16759 gsumval3eu 19841 gsumval3 19844

W3C validator