MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqf1o Structured version   Visualization version   GIF version

Theorem seqf1o 12790
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 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seqf1o.2 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
seqf1o.3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
seqf1o.4 (𝜑𝑁 ∈ (ℤ𝑀))
seqf1o.5 (𝜑𝐶𝑆)
seqf1o.6 (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
seqf1o.7 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐺𝑥) ∈ 𝐶)
seqf1o.8 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐻𝑘) = (𝐺‘(𝐹𝑘)))
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.
StepHypRef Expression
1 seqf1o.6 . . 3 (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
2 seqf1o.7 . . . 4 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐺𝑥) ∈ 𝐶)
3 eqid 2621 . . . 4 (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))
42, 3fmptd 6346 . . 3 (𝜑 → (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)):(𝑀...𝑁)⟶𝐶)
5 seqf1o.4 . . . . 5 (𝜑𝑁 ∈ (ℤ𝑀))
6 oveq2 6618 . . . . . . . . . . 11 (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀))
7 f1oeq23 6092 . . . . . . . . . . 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)))
86, 6, 7syl2anc 692 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)))
96feq2d 5993 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑔:(𝑀...𝑥)⟶𝐶𝑔:(𝑀...𝑀)⟶𝐶))
108, 9anbi12d 746 . . . . . . . . 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)))
11 fveq2 6153 . . . . . . . . . 10 (𝑥 = 𝑀 → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘𝑀))
12 fveq2 6153 . . . . . . . . . 10 (𝑥 = 𝑀 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑀))
1311, 12eqeq12d 2636 . . . . . . . . 9 (𝑥 = 𝑀 → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀)))
1410, 13imbi12d 334 . . . . . . . 8 (𝑥 = 𝑀 → (((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀))))
15142albidv 1848 . . . . . . 7 (𝑥 = 𝑀 → (∀𝑔𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ∀𝑔𝑓((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀))))
1615imbi2d 330 . . . . . 6 (𝑥 = 𝑀 → ((𝜑 → ∀𝑔𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥))) ↔ (𝜑 → ∀𝑔𝑓((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀)))))
17 oveq2 6618 . . . . . . . . . . 11 (𝑥 = 𝑘 → (𝑀...𝑥) = (𝑀...𝑘))
18 f1oeq23 6092 . . . . . . . . . . 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)))
1917, 17, 18syl2anc 692 . . . . . . . . . 10 (𝑥 = 𝑘 → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘)))
2017feq2d 5993 . . . . . . . . . 10 (𝑥 = 𝑘 → (𝑔:(𝑀...𝑥)⟶𝐶𝑔:(𝑀...𝑘)⟶𝐶))
2119, 20anbi12d 746 . . . . . . . . 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)))
22 fveq2 6153 . . . . . . . . . 10 (𝑥 = 𝑘 → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘𝑘))
23 fveq2 6153 . . . . . . . . . 10 (𝑥 = 𝑘 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑘))
2422, 23eqeq12d 2636 . . . . . . . . 9 (𝑥 = 𝑘 → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)))
2521, 24imbi12d 334 . . . . . . . 8 (𝑥 = 𝑘 → (((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))))
26252albidv 1848 . . . . . . 7 (𝑥 = 𝑘 → (∀𝑔𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))))
2726imbi2d 330 . . . . . 6 (𝑥 = 𝑘 → ((𝜑 → ∀𝑔𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥))) ↔ (𝜑 → ∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)))))
28 oveq2 6618 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → (𝑀...𝑥) = (𝑀...(𝑘 + 1)))
29 f1oeq23 6092 . . . . . . . . . . 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))))
3028, 28, 29syl2anc 692 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1))))
3128feq2d 5993 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (𝑔:(𝑀...𝑥)⟶𝐶𝑔:(𝑀...(𝑘 + 1))⟶𝐶))
3230, 31anbi12d 746 . . . . . . . . 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)))
33 fveq2 6153 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)))
34 fveq2 6153 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))
3533, 34eqeq12d 2636 . . . . . . . . 9 (𝑥 = (𝑘 + 1) → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1))))
3632, 35imbi12d 334 . . . . . . . 8 (𝑥 = (𝑘 + 1) → (((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))))
37362albidv 1848 . . . . . . 7 (𝑥 = (𝑘 + 1) → (∀𝑔𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ∀𝑔𝑓((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))))
3837imbi2d 330 . . . . . 6 (𝑥 = (𝑘 + 1) → ((𝜑 → ∀𝑔𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥))) ↔ (𝜑 → ∀𝑔𝑓((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1))))))
39 oveq2 6618 . . . . . . . . . . 11 (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁))
40 f1oeq23 6092 . . . . . . . . . . 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)))
4139, 39, 40syl2anc 692 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
4239feq2d 5993 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝑔:(𝑀...𝑥)⟶𝐶𝑔:(𝑀...𝑁)⟶𝐶))
4341, 42anbi12d 746 . . . . . . . . 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)))
44 fveq2 6153 . . . . . . . . . 10 (𝑥 = 𝑁 → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘𝑁))
45 fveq2 6153 . . . . . . . . . 10 (𝑥 = 𝑁 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑁))
4644, 45eqeq12d 2636 . . . . . . . . 9 (𝑥 = 𝑁 → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)))
4743, 46imbi12d 334 . . . . . . . 8 (𝑥 = 𝑁 → (((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁))))
48472albidv 1848 . . . . . . 7 (𝑥 = 𝑁 → (∀𝑔𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥)) ↔ ∀𝑔𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁))))
4948imbi2d 330 . . . . . 6 (𝑥 = 𝑁 → ((𝜑 → ∀𝑔𝑓((𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ∧ 𝑔:(𝑀...𝑥)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥))) ↔ (𝜑 → ∀𝑔𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)))))
50 f1of 6099 . . . . . . . . . . . . 13 (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) → 𝑓:(𝑀...𝑀)⟶(𝑀...𝑀))
5150adantr 481 . . . . . . . . . . . 12 ((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → 𝑓:(𝑀...𝑀)⟶(𝑀...𝑀))
52 elfz3 12301 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ (𝑀...𝑀))
53 fvco3 6237 . . . . . . . . . . . 12 ((𝑓:(𝑀...𝑀)⟶(𝑀...𝑀) ∧ 𝑀 ∈ (𝑀...𝑀)) → ((𝑔𝑓)‘𝑀) = (𝑔‘(𝑓𝑀)))
5451, 52, 53syl2anr 495 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → ((𝑔𝑓)‘𝑀) = (𝑔‘(𝑓𝑀)))
55 ffvelrn 6318 . . . . . . . . . . . . . . 15 ((𝑓:(𝑀...𝑀)⟶(𝑀...𝑀) ∧ 𝑀 ∈ (𝑀...𝑀)) → (𝑓𝑀) ∈ (𝑀...𝑀))
5650, 52, 55syl2anr 495 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)) → (𝑓𝑀) ∈ (𝑀...𝑀))
57 fzsn 12333 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
5857eleq2d 2684 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℤ → ((𝑓𝑀) ∈ (𝑀...𝑀) ↔ (𝑓𝑀) ∈ {𝑀}))
59 elsni 4170 . . . . . . . . . . . . . . . 16 ((𝑓𝑀) ∈ {𝑀} → (𝑓𝑀) = 𝑀)
6058, 59syl6bi 243 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → ((𝑓𝑀) ∈ (𝑀...𝑀) → (𝑓𝑀) = 𝑀))
6160imp 445 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ (𝑓𝑀) ∈ (𝑀...𝑀)) → (𝑓𝑀) = 𝑀)
6256, 61syldan 487 . . . . . . . . . . . . 13 ((𝑀 ∈ ℤ ∧ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)) → (𝑓𝑀) = 𝑀)
6362adantrr 752 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (𝑓𝑀) = 𝑀)
6463fveq2d 6157 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (𝑔‘(𝑓𝑀)) = (𝑔𝑀))
6554, 64eqtrd 2655 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → ((𝑔𝑓)‘𝑀) = (𝑔𝑀))
66 seq1 12762 . . . . . . . . . . 11 (𝑀 ∈ ℤ → (seq𝑀( + , (𝑔𝑓))‘𝑀) = ((𝑔𝑓)‘𝑀))
6766adantr 481 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = ((𝑔𝑓)‘𝑀))
68 seq1 12762 . . . . . . . . . . 11 (𝑀 ∈ ℤ → (seq𝑀( + , 𝑔)‘𝑀) = (𝑔𝑀))
6968adantr 481 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , 𝑔)‘𝑀) = (𝑔𝑀))
7065, 67, 693eqtr4d 2665 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀))
7170ex 450 . . . . . . . 8 (𝑀 ∈ ℤ → ((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀)))
7271alrimivv 1853 . . . . . . 7 (𝑀 ∈ ℤ → ∀𝑔𝑓((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀)))
7372a1d 25 . . . . . 6 (𝑀 ∈ ℤ → (𝜑 → ∀𝑔𝑓((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀))))
74 f1oeq1 6089 . . . . . . . . . . . 12 (𝑓 = 𝑡 → (𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ↔ 𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘)))
75 feq1 5988 . . . . . . . . . . . 12 (𝑔 = 𝑠 → (𝑔:(𝑀...𝑘)⟶𝐶𝑠:(𝑀...𝑘)⟶𝐶))
7674, 75bi2anan9r 917 . . . . . . . . . . 11 ((𝑔 = 𝑠𝑓 = 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)))
77 coeq1 5244 . . . . . . . . . . . . . . 15 (𝑔 = 𝑠 → (𝑔𝑓) = (𝑠𝑓))
78 coeq2 5245 . . . . . . . . . . . . . . 15 (𝑓 = 𝑡 → (𝑠𝑓) = (𝑠𝑡))
7977, 78sylan9eq 2675 . . . . . . . . . . . . . 14 ((𝑔 = 𝑠𝑓 = 𝑡) → (𝑔𝑓) = (𝑠𝑡))
8079seqeq3d 12757 . . . . . . . . . . . . 13 ((𝑔 = 𝑠𝑓 = 𝑡) → seq𝑀( + , (𝑔𝑓)) = seq𝑀( + , (𝑠𝑡)))
8180fveq1d 6155 . . . . . . . . . . . 12 ((𝑔 = 𝑠𝑓 = 𝑡) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , (𝑠𝑡))‘𝑘))
82 simpl 473 . . . . . . . . . . . . . 14 ((𝑔 = 𝑠𝑓 = 𝑡) → 𝑔 = 𝑠)
8382seqeq3d 12757 . . . . . . . . . . . . 13 ((𝑔 = 𝑠𝑓 = 𝑡) → seq𝑀( + , 𝑔) = seq𝑀( + , 𝑠))
8483fveq1d 6155 . . . . . . . . . . . 12 ((𝑔 = 𝑠𝑓 = 𝑡) → (seq𝑀( + , 𝑔)‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘))
8581, 84eqeq12d 2636 . . . . . . . . . . 11 ((𝑔 = 𝑠𝑓 = 𝑡) → ((seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘) ↔ (seq𝑀( + , (𝑠𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘)))
8676, 85imbi12d 334 . . . . . . . . . 10 ((𝑔 = 𝑠𝑓 = 𝑡) → (((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)) ↔ ((𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑠:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑠𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘))))
8786cbval2v 2284 . . . . . . . . 9 (∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)) ↔ ∀𝑠𝑡((𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑠:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑠𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘)))
88 simplll 797 . . . . . . . . . . . . . . 15 ((((𝜑𝑘 ∈ (ℤ𝑀)) ∧ ∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) → 𝜑)
89 seqf1o.1 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
9088, 89sylan 488 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ (ℤ𝑀)) ∧ ∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
91 seqf1o.2 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
9288, 91sylan 488 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ (ℤ𝑀)) ∧ ∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
93 seqf1o.3 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
9488, 93sylan 488 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ (ℤ𝑀)) ∧ ∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
95 simpllr 798 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ (ℤ𝑀)) ∧ ∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) → 𝑘 ∈ (ℤ𝑀))
96 seqf1o.5 . . . . . . . . . . . . . . 15 (𝜑𝐶𝑆)
9788, 96syl 17 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ (ℤ𝑀)) ∧ ∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) → 𝐶𝑆)
98 simprl 793 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ (ℤ𝑀)) ∧ ∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘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)))
99 simprr 795 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ (ℤ𝑀)) ∧ ∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) → 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)
100 eqid 2621 . . . . . . . . . . . . . 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))))
101 eqid 2621 . . . . . . . . . . . . . 14 (𝑓‘(𝑘 + 1)) = (𝑓‘(𝑘 + 1))
102 simplr 791 . . . . . . . . . . . . . . 15 ((((𝜑𝑘 ∈ (ℤ𝑀)) ∧ ∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) → ∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)))
103102, 87sylib 208 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ (ℤ𝑀)) ∧ ∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) → ∀𝑠𝑡((𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑠:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑠𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘)))
10490, 92, 94, 95, 97, 98, 99, 100, 101, 103seqf1olem2 12789 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ (ℤ𝑀)) ∧ ∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) ∧ (𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶)) → (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))
105104exp31 629 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (ℤ𝑀)) → (∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)) → ((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))))
10687, 105syl5bir 233 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ℤ𝑀)) → (∀𝑠𝑡((𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑠:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑠𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘)) → ((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))))
107106alrimdv 1854 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝑀)) → (∀𝑠𝑡((𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑠:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑠𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘)) → ∀𝑓((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))))
108107alrimdv 1854 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝑀)) → (∀𝑠𝑡((𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑠:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑠𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘)) → ∀𝑔𝑓((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))))
10987, 108syl5bi 232 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑀)) → (∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)) → ∀𝑔𝑓((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))))
110109expcom 451 . . . . . . 7 (𝑘 ∈ (ℤ𝑀) → (𝜑 → (∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)) → ∀𝑔𝑓((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1))))))
111110a2d 29 . . . . . 6 (𝑘 ∈ (ℤ𝑀) → ((𝜑 → ∀𝑔𝑓((𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ∧ 𝑔:(𝑀...𝑘)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘))) → (𝜑 → ∀𝑔𝑓((𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)) ∧ 𝑔:(𝑀...(𝑘 + 1))⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1))))))
11216, 27, 38, 49, 73, 111uzind4 11698 . . . . 5 (𝑁 ∈ (ℤ𝑀) → (𝜑 → ∀𝑔𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁))))
1135, 112mpcom 38 . . . 4 (𝜑 → ∀𝑔𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)))
114 fvex 6163 . . . . . . 7 (𝐺𝑥) ∈ V
115114, 3fnmpti 5984 . . . . . 6 (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) Fn (𝑀...𝑁)
116 fzfi 12719 . . . . . 6 (𝑀...𝑁) ∈ Fin
117 fnfi 8190 . . . . . 6 (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) Fn (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∈ Fin)
118115, 116, 117mp2an 707 . . . . 5 (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∈ Fin
119 f1of 6099 . . . . . . 7 (𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐹:(𝑀...𝑁)⟶(𝑀...𝑁))
1201, 119syl 17 . . . . . 6 (𝜑𝐹:(𝑀...𝑁)⟶(𝑀...𝑁))
121 ovexd 6640 . . . . . 6 (𝜑 → (𝑀...𝑁) ∈ V)
122 fex2 7075 . . . . . 6 ((𝐹:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ V ∧ (𝑀...𝑁) ∈ V) → 𝐹 ∈ V)
123120, 121, 121, 122syl3anc 1323 . . . . 5 (𝜑𝐹 ∈ V)
124 f1oeq1 6089 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
125 feq1 5988 . . . . . . . 8 (𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) → (𝑔:(𝑀...𝑁)⟶𝐶 ↔ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)):(𝑀...𝑁)⟶𝐶))
126124, 125bi2anan9r 917 . . . . . . 7 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 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) ↦ (𝐺𝑥)):(𝑀...𝑁)⟶𝐶)))
127 coeq1 5244 . . . . . . . . . . 11 (𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) → (𝑔𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝑓))
128 coeq2 5245 . . . . . . . . . . 11 (𝑓 = 𝐹 → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))
129127, 128sylan9eq 2675 . . . . . . . . . 10 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → (𝑔𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))
130129seqeq3d 12757 . . . . . . . . 9 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → seq𝑀( + , (𝑔𝑓)) = seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹)))
131130fveq1d 6155 . . . . . . . 8 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁))
132 simpl 473 . . . . . . . . . 10 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → 𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))
133132seqeq3d 12757 . . . . . . . . 9 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → seq𝑀( + , 𝑔) = seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))))
134133fveq1d 6155 . . . . . . . 8 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → (seq𝑀( + , 𝑔)‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁))
135131, 134eqeq12d 2636 . . . . . . 7 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → ((seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁) ↔ (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁)))
136126, 135imbi12d 334 . . . . . 6 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → (((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)) ↔ ((𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)):(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁))))
137136spc2gv 3285 . . . . 5 (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∈ Fin ∧ 𝐹 ∈ V) → (∀𝑔𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)) → ((𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)):(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁))))
138118, 123, 137sylancr 694 . . . 4 (𝜑 → (∀𝑔𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)) → ((𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)):(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁))))
139113, 138mpd 15 . . 3 (𝜑 → ((𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)):(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁)))
1401, 4, 139mp2and 714 . 2 (𝜑 → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁))
141120ffvelrnda 6320 . . . . 5 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ (𝑀...𝑁))
142 fveq2 6153 . . . . . 6 (𝑥 = (𝐹𝑘) → (𝐺𝑥) = (𝐺‘(𝐹𝑘)))
143 fvex 6163 . . . . . 6 (𝐺‘(𝐹𝑘)) ∈ V
144142, 3, 143fvmpt 6244 . . . . 5 ((𝐹𝑘) ∈ (𝑀...𝑁) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))‘(𝐹𝑘)) = (𝐺‘(𝐹𝑘)))
145141, 144syl 17 . . . 4 ((𝜑𝑘 ∈ (𝑀...𝑁)) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))‘(𝐹𝑘)) = (𝐺‘(𝐹𝑘)))
146 fvco3 6237 . . . . 5 ((𝐹:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ 𝑘 ∈ (𝑀...𝑁)) → (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))‘(𝐹𝑘)))
147120, 146sylan 488 . . . 4 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))‘(𝐹𝑘)))
148 seqf1o.8 . . . 4 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐻𝑘) = (𝐺‘(𝐹𝑘)))
149145, 147, 1483eqtr4d 2665 . . 3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹)‘𝑘) = (𝐻𝑘))
1505, 149seqfveq 12773 . 2 (𝜑 → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , 𝐻)‘𝑁))
151 fveq2 6153 . . . . 5 (𝑥 = 𝑘 → (𝐺𝑥) = (𝐺𝑘))
152 fvex 6163 . . . . 5 (𝐺𝑘) ∈ V
153151, 3, 152fvmpt 6244 . . . 4 (𝑘 ∈ (𝑀...𝑁) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))‘𝑘) = (𝐺𝑘))
154153adantl 482 . . 3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))‘𝑘) = (𝐺𝑘))
1555, 154seqfveq 12773 . 2 (𝜑 → (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
156140, 150, 1553eqtr3d 2663 1 (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  Vcvv 3189  wss 3559  ifcif 4063  {csn 4153   class class class wbr 4618  cmpt 4678  ccnv 5078  ccom 5083   Fn wfn 5847  wf 5848  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  Fincfn 7907  1c1 9889   + caddc 9891   < clt 10026  cz 11329  cuz 11639  ...cfz 12276  seqcseq 12749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415  df-seq 12750
This theorem is referenced by:  summolem3  14386  prodmolem3  14599  eulerthlem2  15422  gsumval3eu  18237  gsumval3  18240
  Copyright terms: Public domain W3C validator