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Theorem seqf1o 13048
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 6574 . . 3 (𝜑 → (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)):(𝑀...𝑁)⟶𝐶)
4 seqf1o.4 . . . . 5 (𝜑𝑁 ∈ (ℤ𝑀))
5 oveq2 6849 . . . . . . . . . . 11 (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀))
6 f1oeq23 6312 . . . . . . . . . . 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 579 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)))
85feq2d 6208 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑔:(𝑀...𝑥)⟶𝐶𝑔:(𝑀...𝑀)⟶𝐶))
97, 8anbi12d 624 . . . . . . . . 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 6374 . . . . . . . . . 10 (𝑥 = 𝑀 → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘𝑀))
11 fveq2 6374 . . . . . . . . . 10 (𝑥 = 𝑀 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑀))
1210, 11eqeq12d 2779 . . . . . . . . 9 (𝑥 = 𝑀 → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀)))
139, 12imbi12d 335 . . . . . . . 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 2018 . . . . . . 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 331 . . . . . 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 6849 . . . . . . . . . . 11 (𝑥 = 𝑘 → (𝑀...𝑥) = (𝑀...𝑘))
17 f1oeq23 6312 . . . . . . . . . . 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 579 . . . . . . . . . 10 (𝑥 = 𝑘 → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘)))
1916feq2d 6208 . . . . . . . . . 10 (𝑥 = 𝑘 → (𝑔:(𝑀...𝑥)⟶𝐶𝑔:(𝑀...𝑘)⟶𝐶))
2018, 19anbi12d 624 . . . . . . . . 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 6374 . . . . . . . . . 10 (𝑥 = 𝑘 → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘𝑘))
22 fveq2 6374 . . . . . . . . . 10 (𝑥 = 𝑘 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑘))
2321, 22eqeq12d 2779 . . . . . . . . 9 (𝑥 = 𝑘 → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)))
2420, 23imbi12d 335 . . . . . . . 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 2018 . . . . . . 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 331 . . . . . 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 6849 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → (𝑀...𝑥) = (𝑀...(𝑘 + 1)))
28 f1oeq23 6312 . . . . . . . . . . 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 579 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1))))
3027feq2d 6208 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (𝑔:(𝑀...𝑥)⟶𝐶𝑔:(𝑀...(𝑘 + 1))⟶𝐶))
3129, 30anbi12d 624 . . . . . . . . 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 6374 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)))
33 fveq2 6374 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))
3432, 33eqeq12d 2779 . . . . . . . . 9 (𝑥 = (𝑘 + 1) → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1))))
3531, 34imbi12d 335 . . . . . . . 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 2018 . . . . . . 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 331 . . . . . 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 6849 . . . . . . . . . . 11 (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁))
39 f1oeq23 6312 . . . . . . . . . . 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 579 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
4138feq2d 6208 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝑔:(𝑀...𝑥)⟶𝐶𝑔:(𝑀...𝑁)⟶𝐶))
4240, 41anbi12d 624 . . . . . . . . 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 6374 . . . . . . . . . 10 (𝑥 = 𝑁 → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘𝑁))
44 fveq2 6374 . . . . . . . . . 10 (𝑥 = 𝑁 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑁))
4543, 44eqeq12d 2779 . . . . . . . . 9 (𝑥 = 𝑁 → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)))
4642, 45imbi12d 335 . . . . . . . 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 2018 . . . . . . 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 331 . . . . . 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 6319 . . . . . . . . . . . . 13 (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) → 𝑓:(𝑀...𝑀)⟶(𝑀...𝑀))
5049adantr 472 . . . . . . . . . . . 12 ((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → 𝑓:(𝑀...𝑀)⟶(𝑀...𝑀))
51 elfz3 12557 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ (𝑀...𝑀))
52 fvco3 6463 . . . . . . . . . . . 12 ((𝑓:(𝑀...𝑀)⟶(𝑀...𝑀) ∧ 𝑀 ∈ (𝑀...𝑀)) → ((𝑔𝑓)‘𝑀) = (𝑔‘(𝑓𝑀)))
5350, 51, 52syl2anr 590 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → ((𝑔𝑓)‘𝑀) = (𝑔‘(𝑓𝑀)))
54 ffvelrn 6546 . . . . . . . . . . . . . . 15 ((𝑓:(𝑀...𝑀)⟶(𝑀...𝑀) ∧ 𝑀 ∈ (𝑀...𝑀)) → (𝑓𝑀) ∈ (𝑀...𝑀))
5549, 51, 54syl2anr 590 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)) → (𝑓𝑀) ∈ (𝑀...𝑀))
56 fzsn 12589 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
5756eleq2d 2829 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℤ → ((𝑓𝑀) ∈ (𝑀...𝑀) ↔ (𝑓𝑀) ∈ {𝑀}))
58 elsni 4350 . . . . . . . . . . . . . . . 16 ((𝑓𝑀) ∈ {𝑀} → (𝑓𝑀) = 𝑀)
5957, 58syl6bi 244 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → ((𝑓𝑀) ∈ (𝑀...𝑀) → (𝑓𝑀) = 𝑀))
6059imp 395 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ (𝑓𝑀) ∈ (𝑀...𝑀)) → (𝑓𝑀) = 𝑀)
6155, 60syldan 585 . . . . . . . . . . . . 13 ((𝑀 ∈ ℤ ∧ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)) → (𝑓𝑀) = 𝑀)
6261adantrr 708 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (𝑓𝑀) = 𝑀)
6362fveq2d 6378 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (𝑔‘(𝑓𝑀)) = (𝑔𝑀))
6453, 63eqtrd 2798 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → ((𝑔𝑓)‘𝑀) = (𝑔𝑀))
65 seq1 13020 . . . . . . . . . . 11 (𝑀 ∈ ℤ → (seq𝑀( + , (𝑔𝑓))‘𝑀) = ((𝑔𝑓)‘𝑀))
6665adantr 472 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = ((𝑔𝑓)‘𝑀))
67 seq1 13020 . . . . . . . . . . 11 (𝑀 ∈ ℤ → (seq𝑀( + , 𝑔)‘𝑀) = (𝑔𝑀))
6867adantr 472 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , 𝑔)‘𝑀) = (𝑔𝑀))
6964, 66, 683eqtr4d 2808 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀))
7069ex 401 . . . . . . . 8 (𝑀 ∈ ℤ → ((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀)))
7170alrimivv 2023 . . . . . . 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 6309 . . . . . . . . . . . 12 (𝑓 = 𝑡 → (𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ↔ 𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘)))
74 feq1 6203 . . . . . . . . . . . 12 (𝑔 = 𝑠 → (𝑔:(𝑀...𝑘)⟶𝐶𝑠:(𝑀...𝑘)⟶𝐶))
7573, 74bi2anan9r 630 . . . . . . . . . . 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 5447 . . . . . . . . . . . . . . 15 (𝑔 = 𝑠 → (𝑔𝑓) = (𝑠𝑓))
77 coeq2 5448 . . . . . . . . . . . . . . 15 (𝑓 = 𝑡 → (𝑠𝑓) = (𝑠𝑡))
7876, 77sylan9eq 2818 . . . . . . . . . . . . . 14 ((𝑔 = 𝑠𝑓 = 𝑡) → (𝑔𝑓) = (𝑠𝑡))
7978seqeq3d 13015 . . . . . . . . . . . . 13 ((𝑔 = 𝑠𝑓 = 𝑡) → seq𝑀( + , (𝑔𝑓)) = seq𝑀( + , (𝑠𝑡)))
8079fveq1d 6376 . . . . . . . . . . . 12 ((𝑔 = 𝑠𝑓 = 𝑡) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , (𝑠𝑡))‘𝑘))
81 simpl 474 . . . . . . . . . . . . . 14 ((𝑔 = 𝑠𝑓 = 𝑡) → 𝑔 = 𝑠)
8281seqeq3d 13015 . . . . . . . . . . . . 13 ((𝑔 = 𝑠𝑓 = 𝑡) → seq𝑀( + , 𝑔) = seq𝑀( + , 𝑠))
8382fveq1d 6376 . . . . . . . . . . . 12 ((𝑔 = 𝑠𝑓 = 𝑡) → (seq𝑀( + , 𝑔)‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘))
8480, 83eqeq12d 2779 . . . . . . . . . . 11 ((𝑔 = 𝑠𝑓 = 𝑡) → ((seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘) ↔ (seq𝑀( + , (𝑠𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘)))
8575, 84imbi12d 335 . . . . . . . . . 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))))
8685cbval2v 2383 . . . . . . . . 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 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)) → 𝜑)
88 seqf1o.1 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
8987, 88sylan 575 . . . . . . . . . . . . . 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 575 . . . . . . . . . . . . . 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 575 . . . . . . . . . . . . . 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 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)) → 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 787 . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . . 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 2764 . . . . . . . . . . . . . 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 2764 . . . . . . . . . . . . . 14 (𝑓‘(𝑘 + 1)) = (𝑓‘(𝑘 + 1))
101 simplr 785 . . . . . . . . . . . . . . 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 209 . . . . . . . . . . . . . 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 13047 . . . . . . . . . . . . 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 410 . . . . . . . . . . . 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 234 . . . . . . . . . . 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 2024 . . . . . . . . . 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 2024 . . . . . . . . 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 233 . . . . . . . 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 402 . . . . . . 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 11945 . . . . 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 6387 . . . . . . 7 (𝐺𝑥) ∈ V
114 eqid 2764 . . . . . . 7 (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))
115113, 114fnmpti 6199 . . . . . 6 (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) Fn (𝑀...𝑁)
116 fzfi 12978 . . . . . 6 (𝑀...𝑁) ∈ Fin
117 fnfi 8444 . . . . . 6 (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) Fn (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∈ Fin)
118115, 116, 117mp2an 683 . . . . 5 (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∈ Fin
119 f1of 6319 . . . . . . 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 6875 . . . . . 6 (𝜑 → (𝑀...𝑁) ∈ V)
122 fex2 7318 . . . . . 6 ((𝐹:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ V ∧ (𝑀...𝑁) ∈ V) → 𝐹 ∈ V)
123120, 121, 121, 122syl3anc 1490 . . . . 5 (𝜑𝐹 ∈ V)
124 f1oeq1 6309 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
125 feq1 6203 . . . . . . . 8 (𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) → (𝑔:(𝑀...𝑁)⟶𝐶 ↔ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)):(𝑀...𝑁)⟶𝐶))
126124, 125bi2anan9r 630 . . . . . . 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 5447 . . . . . . . . . . 11 (𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) → (𝑔𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝑓))
128 coeq2 5448 . . . . . . . . . . 11 (𝑓 = 𝐹 → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))
129127, 128sylan9eq 2818 . . . . . . . . . 10 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → (𝑔𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))
130129seqeq3d 13015 . . . . . . . . 9 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → seq𝑀( + , (𝑔𝑓)) = seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹)))
131130fveq1d 6376 . . . . . . . 8 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁))
132 simpl 474 . . . . . . . . . 10 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → 𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))
133132seqeq3d 13015 . . . . . . . . 9 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → seq𝑀( + , 𝑔) = seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))))
134133fveq1d 6376 . . . . . . . 8 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → (seq𝑀( + , 𝑔)‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁))
135131, 134eqeq12d 2779 . . . . . . 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 335 . . . . . 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 3447 . . . . 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 581 . . . 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 690 . 2 (𝜑 → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁))
141120ffvelrnda 6548 . . . . 5 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ (𝑀...𝑁))
142 fveq2 6374 . . . . . 6 (𝑥 = (𝐹𝑘) → (𝐺𝑥) = (𝐺‘(𝐹𝑘)))
143 fvex 6387 . . . . . 6 (𝐺‘(𝐹𝑘)) ∈ V
144142, 114, 143fvmpt 6470 . . . . 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 6463 . . . . 5 ((𝐹:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ 𝑘 ∈ (𝑀...𝑁)) → (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))‘(𝐹𝑘)))
147120, 146sylan 575 . . . 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 2808 . . 3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹)‘𝑘) = (𝐻𝑘))
1504, 149seqfveq 13031 . 2 (𝜑 → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , 𝐻)‘𝑁))
151 fveq2 6374 . . . . 5 (𝑥 = 𝑘 → (𝐺𝑥) = (𝐺𝑘))
152 fvex 6387 . . . . 5 (𝐺𝑘) ∈ V
153151, 114, 152fvmpt 6470 . . . 4 (𝑘 ∈ (𝑀...𝑁) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))‘𝑘) = (𝐺𝑘))
154153adantl 473 . . 3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))‘𝑘) = (𝐺𝑘))
1554, 154seqfveq 13031 . 2 (𝜑 → (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
156140, 150, 1553eqtr3d 2806 1 (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107  wal 1650   = wceq 1652  wcel 2155  Vcvv 3349  wss 3731  ifcif 4242  {csn 4333   class class class wbr 4808  cmpt 4887  ccnv 5275  ccom 5280   Fn wfn 6062  wf 6063  1-1-ontowf1o 6066  cfv 6067  (class class class)co 6841  Fincfn 8159  1c1 10189   + caddc 10191   < clt 10327  cz 11623  cuz 11885  ...cfz 12532  seqcseq 13007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-oadd 7767  df-er 7946  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-nn 11274  df-n0 11538  df-z 11624  df-uz 11886  df-fz 12533  df-fzo 12673  df-seq 13008
This theorem is referenced by:  summolem3  14731  prodmolem3  14947  eulerthlem2  15767  gsumval3eu  18570  gsumval3  18573
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