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Theorem seqf1o 13410
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)
32fmpttd 6878 . . 3 (𝜑 → (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)):(𝑀...𝑁)⟶𝐶)
4 seqf1o.4 . . . . 5 (𝜑𝑁 ∈ (ℤ𝑀))
5 oveq2 7163 . . . . . . . . . . 11 (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀))
6 f1oeq23 6606 . . . . . . . . . . 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)))
75, 5, 6syl2anc 586 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)))
85feq2d 6499 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑔:(𝑀...𝑥)⟶𝐶𝑔:(𝑀...𝑀)⟶𝐶))
97, 8anbi12d 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 6669 . . . . . . . . . 10 (𝑥 = 𝑀 → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘𝑀))
11 fveq2 6669 . . . . . . . . . 10 (𝑥 = 𝑀 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑀))
1210, 11eqeq12d 2837 . . . . . . . . 9 (𝑥 = 𝑀 → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀)))
139, 12imbi12d 347 . . . . . . . 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))))
14132albidv 1920 . . . . . . 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))))
1514imbi2d 343 . . . . . 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)))))
16 oveq2 7163 . . . . . . . . . . 11 (𝑥 = 𝑘 → (𝑀...𝑥) = (𝑀...𝑘))
17 f1oeq23 6606 . . . . . . . . . . 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)))
1816, 16, 17syl2anc 586 . . . . . . . . . 10 (𝑥 = 𝑘 → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘)))
1916feq2d 6499 . . . . . . . . . 10 (𝑥 = 𝑘 → (𝑔:(𝑀...𝑥)⟶𝐶𝑔:(𝑀...𝑘)⟶𝐶))
2018, 19anbi12d 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 6669 . . . . . . . . . 10 (𝑥 = 𝑘 → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘𝑘))
22 fveq2 6669 . . . . . . . . . 10 (𝑥 = 𝑘 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑘))
2321, 22eqeq12d 2837 . . . . . . . . 9 (𝑥 = 𝑘 → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)))
2420, 23imbi12d 347 . . . . . . . 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))))
25242albidv 1920 . . . . . . 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))))
2625imbi2d 343 . . . . . 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)))))
27 oveq2 7163 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → (𝑀...𝑥) = (𝑀...(𝑘 + 1)))
28 f1oeq23 6606 . . . . . . . . . . 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))))
2927, 27, 28syl2anc 586 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1))))
3027feq2d 6499 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (𝑔:(𝑀...𝑥)⟶𝐶𝑔:(𝑀...(𝑘 + 1))⟶𝐶))
3129, 30anbi12d 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 6669 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)))
33 fveq2 6669 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))
3432, 33eqeq12d 2837 . . . . . . . . 9 (𝑥 = (𝑘 + 1) → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1))))
3531, 34imbi12d 347 . . . . . . . 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)))))
36352albidv 1920 . . . . . . 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)))))
3736imbi2d 343 . . . . . 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))))))
38 oveq2 7163 . . . . . . . . . . 11 (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁))
39 f1oeq23 6606 . . . . . . . . . . 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)))
4038, 38, 39syl2anc 586 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
4138feq2d 6499 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝑔:(𝑀...𝑥)⟶𝐶𝑔:(𝑀...𝑁)⟶𝐶))
4240, 41anbi12d 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 6669 . . . . . . . . . 10 (𝑥 = 𝑁 → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘𝑁))
44 fveq2 6669 . . . . . . . . . 10 (𝑥 = 𝑁 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑁))
4543, 44eqeq12d 2837 . . . . . . . . 9 (𝑥 = 𝑁 → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)))
4642, 45imbi12d 347 . . . . . . . 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))))
47462albidv 1920 . . . . . . 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))))
4847imbi2d 343 . . . . . 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)))))
49 f1of 6614 . . . . . . . . . . . . 13 (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) → 𝑓:(𝑀...𝑀)⟶(𝑀...𝑀))
5049adantr 483 . . . . . . . . . . . 12 ((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → 𝑓:(𝑀...𝑀)⟶(𝑀...𝑀))
51 elfz3 12916 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ (𝑀...𝑀))
52 fvco3 6759 . . . . . . . . . . . 12 ((𝑓:(𝑀...𝑀)⟶(𝑀...𝑀) ∧ 𝑀 ∈ (𝑀...𝑀)) → ((𝑔𝑓)‘𝑀) = (𝑔‘(𝑓𝑀)))
5350, 51, 52syl2anr 598 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → ((𝑔𝑓)‘𝑀) = (𝑔‘(𝑓𝑀)))
54 ffvelrn 6848 . . . . . . . . . . . . . . 15 ((𝑓:(𝑀...𝑀)⟶(𝑀...𝑀) ∧ 𝑀 ∈ (𝑀...𝑀)) → (𝑓𝑀) ∈ (𝑀...𝑀))
5549, 51, 54syl2anr 598 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)) → (𝑓𝑀) ∈ (𝑀...𝑀))
56 fzsn 12948 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
5756eleq2d 2898 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℤ → ((𝑓𝑀) ∈ (𝑀...𝑀) ↔ (𝑓𝑀) ∈ {𝑀}))
58 elsni 4583 . . . . . . . . . . . . . . . 16 ((𝑓𝑀) ∈ {𝑀} → (𝑓𝑀) = 𝑀)
5957, 58syl6bi 255 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → ((𝑓𝑀) ∈ (𝑀...𝑀) → (𝑓𝑀) = 𝑀))
6059imp 409 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ (𝑓𝑀) ∈ (𝑀...𝑀)) → (𝑓𝑀) = 𝑀)
6155, 60syldan 593 . . . . . . . . . . . . 13 ((𝑀 ∈ ℤ ∧ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)) → (𝑓𝑀) = 𝑀)
6261adantrr 715 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (𝑓𝑀) = 𝑀)
6362fveq2d 6673 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (𝑔‘(𝑓𝑀)) = (𝑔𝑀))
6453, 63eqtrd 2856 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → ((𝑔𝑓)‘𝑀) = (𝑔𝑀))
65 seq1 13381 . . . . . . . . . . 11 (𝑀 ∈ ℤ → (seq𝑀( + , (𝑔𝑓))‘𝑀) = ((𝑔𝑓)‘𝑀))
6665adantr 483 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = ((𝑔𝑓)‘𝑀))
67 seq1 13381 . . . . . . . . . . 11 (𝑀 ∈ ℤ → (seq𝑀( + , 𝑔)‘𝑀) = (𝑔𝑀))
6867adantr 483 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , 𝑔)‘𝑀) = (𝑔𝑀))
6964, 66, 683eqtr4d 2866 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀))
7069ex 415 . . . . . . . 8 (𝑀 ∈ ℤ → ((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀)))
7170alrimivv 1925 . . . . . . 7 (𝑀 ∈ ℤ → ∀𝑔𝑓((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀)))
7271a1d 25 . . . . . 6 (𝑀 ∈ ℤ → (𝜑 → ∀𝑔𝑓((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀))))
73 f1oeq1 6603 . . . . . . . . . . . 12 (𝑓 = 𝑡 → (𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ↔ 𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘)))
74 feq1 6494 . . . . . . . . . . . 12 (𝑔 = 𝑠 → (𝑔:(𝑀...𝑘)⟶𝐶𝑠:(𝑀...𝑘)⟶𝐶))
7573, 74bi2anan9r 638 . . . . . . . . . . 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)))
76 coeq1 5727 . . . . . . . . . . . . . . 15 (𝑔 = 𝑠 → (𝑔𝑓) = (𝑠𝑓))
77 coeq2 5728 . . . . . . . . . . . . . . 15 (𝑓 = 𝑡 → (𝑠𝑓) = (𝑠𝑡))
7876, 77sylan9eq 2876 . . . . . . . . . . . . . 14 ((𝑔 = 𝑠𝑓 = 𝑡) → (𝑔𝑓) = (𝑠𝑡))
7978seqeq3d 13376 . . . . . . . . . . . . 13 ((𝑔 = 𝑠𝑓 = 𝑡) → seq𝑀( + , (𝑔𝑓)) = seq𝑀( + , (𝑠𝑡)))
8079fveq1d 6671 . . . . . . . . . . . 12 ((𝑔 = 𝑠𝑓 = 𝑡) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , (𝑠𝑡))‘𝑘))
81 simpl 485 . . . . . . . . . . . . . 14 ((𝑔 = 𝑠𝑓 = 𝑡) → 𝑔 = 𝑠)
8281seqeq3d 13376 . . . . . . . . . . . . 13 ((𝑔 = 𝑠𝑓 = 𝑡) → seq𝑀( + , 𝑔) = seq𝑀( + , 𝑠))
8382fveq1d 6671 . . . . . . . . . . . 12 ((𝑔 = 𝑠𝑓 = 𝑡) → (seq𝑀( + , 𝑔)‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘))
8480, 83eqeq12d 2837 . . . . . . . . . . 11 ((𝑔 = 𝑠𝑓 = 𝑡) → ((seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘) ↔ (seq𝑀( + , (𝑠𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘)))
8575, 84imbi12d 347 . . . . . . . . . 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))))
8685cbval2vw 2043 . . . . . . . . 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)))
87 simplll 773 . . . . . . . . . . . . . . 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)
88 seqf1o.1 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
8987, 88sylan 582 . . . . . . . . . . . . . 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)
90 seqf1o.2 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
9187, 90sylan 582 . . . . . . . . . . . . . 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))
92 seqf1o.3 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
9387, 92sylan 582 . . . . . . . . . . . . . 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)))
94 simpllr 774 . . . . . . . . . . . . . 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))
95 seqf1o.5 . . . . . . . . . . . . . . 15 (𝜑𝐶𝑆)
9687, 95syl 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)
97 simprl 769 . . . . . . . . . . . . . 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)))
98 simprr 771 . . . . . . . . . . . . . 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)
99 eqid 2821 . . . . . . . . . . . . . 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 2821 . . . . . . . . . . . . . 14 (𝑓‘(𝑘 + 1)) = (𝑓‘(𝑘 + 1))
101 simplr 767 . . . . . . . . . . . . . . 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)))
102101, 86sylib 220 . . . . . . . . . . . . . 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)))
10389, 91, 93, 94, 96, 97, 98, 99, 100, 102seqf1olem2 13409 . . . . . . . . . . . . 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)))
104103exp31 422 . . . . . . . . . . . 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)))))
10586, 104syl5bir 245 . . . . . . . . . . 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)))))
106105alrimdv 1926 . . . . . . . . . 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)))))
107106alrimdv 1926 . . . . . . . . 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)))))
10886, 107syl5bi 244 . . . . . . . 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)))))
109108expcom 416 . . . . . . 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))))))
110109a2d 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))))))
11115, 26, 37, 48, 72, 110uzind4 12305 . . . . 5 (𝑁 ∈ (ℤ𝑀) → (𝜑 → ∀𝑔𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁))))
1124, 111mpcom 38 . . . 4 (𝜑 → ∀𝑔𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)))
113 fvex 6682 . . . . . . 7 (𝐺𝑥) ∈ V
114 eqid 2821 . . . . . . 7 (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))
115113, 114fnmpti 6490 . . . . . 6 (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) Fn (𝑀...𝑁)
116 fzfi 13339 . . . . . 6 (𝑀...𝑁) ∈ Fin
117 fnfi 8795 . . . . . 6 (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) Fn (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∈ Fin)
118115, 116, 117mp2an 690 . . . . 5 (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∈ Fin
119 f1of 6614 . . . . . . 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 7190 . . . . . 6 (𝜑 → (𝑀...𝑁) ∈ V)
122 fex2 7637 . . . . . 6 ((𝐹:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ V ∧ (𝑀...𝑁) ∈ V) → 𝐹 ∈ V)
123120, 121, 121, 122syl3anc 1367 . . . . 5 (𝜑𝐹 ∈ V)
124 f1oeq1 6603 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
125 feq1 6494 . . . . . . . 8 (𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) → (𝑔:(𝑀...𝑁)⟶𝐶 ↔ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)):(𝑀...𝑁)⟶𝐶))
126124, 125bi2anan9r 638 . . . . . . 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 5727 . . . . . . . . . . 11 (𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) → (𝑔𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝑓))
128 coeq2 5728 . . . . . . . . . . 11 (𝑓 = 𝐹 → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))
129127, 128sylan9eq 2876 . . . . . . . . . 10 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → (𝑔𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))
130129seqeq3d 13376 . . . . . . . . 9 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → seq𝑀( + , (𝑔𝑓)) = seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹)))
131130fveq1d 6671 . . . . . . . 8 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁))
132 simpl 485 . . . . . . . . . 10 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → 𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))
133132seqeq3d 13376 . . . . . . . . 9 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → seq𝑀( + , 𝑔) = seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))))
134133fveq1d 6671 . . . . . . . 8 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → (seq𝑀( + , 𝑔)‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁))
135131, 134eqeq12d 2837 . . . . . . 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 347 . . . . . 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 3600 . . . . 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 589 . . . 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))))
139112, 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, 3, 139mp2and 697 . 2 (𝜑 → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁))
141120ffvelrnda 6850 . . . . 5 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ (𝑀...𝑁))
142 fveq2 6669 . . . . . 6 (𝑥 = (𝐹𝑘) → (𝐺𝑥) = (𝐺‘(𝐹𝑘)))
143 fvex 6682 . . . . . 6 (𝐺‘(𝐹𝑘)) ∈ V
144142, 114, 143fvmpt 6767 . . . . 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 6759 . . . . 5 ((𝐹:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ 𝑘 ∈ (𝑀...𝑁)) → (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))‘(𝐹𝑘)))
147120, 146sylan 582 . . . 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 2866 . . 3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹)‘𝑘) = (𝐻𝑘))
1504, 149seqfveq 13393 . 2 (𝜑 → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , 𝐻)‘𝑁))
151 fveq2 6669 . . . . 5 (𝑥 = 𝑘 → (𝐺𝑥) = (𝐺𝑘))
152 fvex 6682 . . . . 5 (𝐺𝑘) ∈ V
153151, 114, 152fvmpt 6767 . . . 4 (𝑘 ∈ (𝑀...𝑁) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))‘𝑘) = (𝐺𝑘))
154153adantl 484 . . 3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))‘𝑘) = (𝐺𝑘))
1554, 154seqfveq 13393 . 2 (𝜑 → (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
156140, 150, 1553eqtr3d 2864 1 (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083  wal 1531   = wceq 1533  wcel 2110  Vcvv 3494  wss 3935  ifcif 4466  {csn 4566   class class class wbr 5065  cmpt 5145  ccnv 5553  ccom 5558   Fn wfn 6349  wf 6350  1-1-ontowf1o 6353  cfv 6354  (class class class)co 7155  Fincfn 8508  1c1 10537   + caddc 10539   < clt 10674  cz 11980  cuz 12242  ...cfz 12891  seqcseq 13368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-n0 11897  df-z 11981  df-uz 12243  df-fz 12892  df-fzo 13033  df-seq 13369
This theorem is referenced by:  summolem3  15070  prodmolem3  15286  eulerthlem2  16118  gsumval3eu  19023  gsumval3  19026
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