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Theorem seqf1o 14081
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 7135 . . 3 (𝜑 → (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)):(𝑀...𝑁)⟶𝐶)
4 seqf1o.4 . . . . 5 (𝜑𝑁 ∈ (ℤ𝑀))
5 oveq2 7439 . . . . . . . . . . 11 (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀))
6 f1oeq23 6840 . . . . . . . . . . 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 584 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)))
85feq2d 6723 . . . . . . . . . 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 6907 . . . . . . . . . 10 (𝑥 = 𝑀 → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘𝑀))
11 fveq2 6907 . . . . . . . . . 10 (𝑥 = 𝑀 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑀))
1210, 11eqeq12d 2751 . . . . . . . . 9 (𝑥 = 𝑀 → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀)))
139, 12imbi12d 344 . . . . . . . 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 1921 . . . . . . 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 340 . . . . . 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 7439 . . . . . . . . . . 11 (𝑥 = 𝑘 → (𝑀...𝑥) = (𝑀...𝑘))
17 f1oeq23 6840 . . . . . . . . . . 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 584 . . . . . . . . . 10 (𝑥 = 𝑘 → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘)))
1916feq2d 6723 . . . . . . . . . 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 6907 . . . . . . . . . 10 (𝑥 = 𝑘 → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘𝑘))
22 fveq2 6907 . . . . . . . . . 10 (𝑥 = 𝑘 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑘))
2321, 22eqeq12d 2751 . . . . . . . . 9 (𝑥 = 𝑘 → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘)))
2420, 23imbi12d 344 . . . . . . . 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 1921 . . . . . . 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 340 . . . . . 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 7439 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → (𝑀...𝑥) = (𝑀...(𝑘 + 1)))
28 f1oeq23 6840 . . . . . . . . . . 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 584 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1))))
3027feq2d 6723 . . . . . . . . . 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 6907 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)))
33 fveq2 6907 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘(𝑘 + 1)))
3432, 33eqeq12d 2751 . . . . . . . . 9 (𝑥 = (𝑘 + 1) → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘(𝑘 + 1)) = (seq𝑀( + , 𝑔)‘(𝑘 + 1))))
3531, 34imbi12d 344 . . . . . . . 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 1921 . . . . . . 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 340 . . . . . 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 7439 . . . . . . . . . . 11 (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁))
39 f1oeq23 6840 . . . . . . . . . . 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 584 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝑓:(𝑀...𝑥)–1-1-onto→(𝑀...𝑥) ↔ 𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
4138feq2d 6723 . . . . . . . . . 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 6907 . . . . . . . . . 10 (𝑥 = 𝑁 → (seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , (𝑔𝑓))‘𝑁))
44 fveq2 6907 . . . . . . . . . 10 (𝑥 = 𝑁 → (seq𝑀( + , 𝑔)‘𝑥) = (seq𝑀( + , 𝑔)‘𝑁))
4543, 44eqeq12d 2751 . . . . . . . . 9 (𝑥 = 𝑁 → ((seq𝑀( + , (𝑔𝑓))‘𝑥) = (seq𝑀( + , 𝑔)‘𝑥) ↔ (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)))
4642, 45imbi12d 344 . . . . . . . 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 1921 . . . . . . 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 340 . . . . . 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 6849 . . . . . . . . . . . . 13 (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) → 𝑓:(𝑀...𝑀)⟶(𝑀...𝑀))
5049adantr 480 . . . . . . . . . . . 12 ((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → 𝑓:(𝑀...𝑀)⟶(𝑀...𝑀))
51 elfz3 13571 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ (𝑀...𝑀))
52 fvco3 7008 . . . . . . . . . . . 12 ((𝑓:(𝑀...𝑀)⟶(𝑀...𝑀) ∧ 𝑀 ∈ (𝑀...𝑀)) → ((𝑔𝑓)‘𝑀) = (𝑔‘(𝑓𝑀)))
5350, 51, 52syl2anr 597 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → ((𝑔𝑓)‘𝑀) = (𝑔‘(𝑓𝑀)))
54 ffvelcdm 7101 . . . . . . . . . . . . . . 15 ((𝑓:(𝑀...𝑀)⟶(𝑀...𝑀) ∧ 𝑀 ∈ (𝑀...𝑀)) → (𝑓𝑀) ∈ (𝑀...𝑀))
5549, 51, 54syl2anr 597 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)) → (𝑓𝑀) ∈ (𝑀...𝑀))
56 fzsn 13603 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
5756eleq2d 2825 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℤ → ((𝑓𝑀) ∈ (𝑀...𝑀) ↔ (𝑓𝑀) ∈ {𝑀}))
58 elsni 4648 . . . . . . . . . . . . . . . 16 ((𝑓𝑀) ∈ {𝑀} → (𝑓𝑀) = 𝑀)
5957, 58biimtrdi 253 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → ((𝑓𝑀) ∈ (𝑀...𝑀) → (𝑓𝑀) = 𝑀))
6059imp 406 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ (𝑓𝑀) ∈ (𝑀...𝑀)) → (𝑓𝑀) = 𝑀)
6155, 60syldan 591 . . . . . . . . . . . . 13 ((𝑀 ∈ ℤ ∧ 𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀)) → (𝑓𝑀) = 𝑀)
6261adantrr 717 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (𝑓𝑀) = 𝑀)
6362fveq2d 6911 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (𝑔‘(𝑓𝑀)) = (𝑔𝑀))
6453, 63eqtrd 2775 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → ((𝑔𝑓)‘𝑀) = (𝑔𝑀))
65 seq1 14052 . . . . . . . . . . 11 (𝑀 ∈ ℤ → (seq𝑀( + , (𝑔𝑓))‘𝑀) = ((𝑔𝑓)‘𝑀))
6665adantr 480 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = ((𝑔𝑓)‘𝑀))
67 seq1 14052 . . . . . . . . . . 11 (𝑀 ∈ ℤ → (seq𝑀( + , 𝑔)‘𝑀) = (𝑔𝑀))
6867adantr 480 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , 𝑔)‘𝑀) = (𝑔𝑀))
6964, 66, 683eqtr4d 2785 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ (𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶)) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀))
7069ex 412 . . . . . . . 8 (𝑀 ∈ ℤ → ((𝑓:(𝑀...𝑀)–1-1-onto→(𝑀...𝑀) ∧ 𝑔:(𝑀...𝑀)⟶𝐶) → (seq𝑀( + , (𝑔𝑓))‘𝑀) = (seq𝑀( + , 𝑔)‘𝑀)))
7170alrimivv 1926 . . . . . . 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 6837 . . . . . . . . . . . 12 (𝑓 = 𝑡 → (𝑓:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘) ↔ 𝑡:(𝑀...𝑘)–1-1-onto→(𝑀...𝑘)))
74 feq1 6717 . . . . . . . . . . . 12 (𝑔 = 𝑠 → (𝑔:(𝑀...𝑘)⟶𝐶𝑠:(𝑀...𝑘)⟶𝐶))
7573, 74bi2anan9r 639 . . . . . . . . . . 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 5871 . . . . . . . . . . . . . . 15 (𝑔 = 𝑠 → (𝑔𝑓) = (𝑠𝑓))
77 coeq2 5872 . . . . . . . . . . . . . . 15 (𝑓 = 𝑡 → (𝑠𝑓) = (𝑠𝑡))
7876, 77sylan9eq 2795 . . . . . . . . . . . . . 14 ((𝑔 = 𝑠𝑓 = 𝑡) → (𝑔𝑓) = (𝑠𝑡))
7978seqeq3d 14047 . . . . . . . . . . . . 13 ((𝑔 = 𝑠𝑓 = 𝑡) → seq𝑀( + , (𝑔𝑓)) = seq𝑀( + , (𝑠𝑡)))
8079fveq1d 6909 . . . . . . . . . . . 12 ((𝑔 = 𝑠𝑓 = 𝑡) → (seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , (𝑠𝑡))‘𝑘))
81 simpl 482 . . . . . . . . . . . . . 14 ((𝑔 = 𝑠𝑓 = 𝑡) → 𝑔 = 𝑠)
8281seqeq3d 14047 . . . . . . . . . . . . 13 ((𝑔 = 𝑠𝑓 = 𝑡) → seq𝑀( + , 𝑔) = seq𝑀( + , 𝑠))
8382fveq1d 6909 . . . . . . . . . . . 12 ((𝑔 = 𝑠𝑓 = 𝑡) → (seq𝑀( + , 𝑔)‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘))
8480, 83eqeq12d 2751 . . . . . . . . . . 11 ((𝑔 = 𝑠𝑓 = 𝑡) → ((seq𝑀( + , (𝑔𝑓))‘𝑘) = (seq𝑀( + , 𝑔)‘𝑘) ↔ (seq𝑀( + , (𝑠𝑡))‘𝑘) = (seq𝑀( + , 𝑠)‘𝑘)))
8575, 84imbi12d 344 . . . . . . . . . 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 2037 . . . . . . . . 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 775 . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . 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 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)) → 𝑓:(𝑀...(𝑘 + 1))–1-1-onto→(𝑀...(𝑘 + 1)))
98 simprr 773 . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . 14 (𝑓‘(𝑘 + 1)) = (𝑓‘(𝑘 + 1))
101 simplr 769 . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . 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 14080 . . . . . . . . . . . . 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 419 . . . . . . . . . . . 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, 104biimtrrid 243 . . . . . . . . . . 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 1927 . . . . . . . . . 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 1927 . . . . . . . . 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, 107biimtrid 242 . . . . . . . 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 413 . . . . . . 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 12946 . . . . 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 6920 . . . . . . 7 (𝐺𝑥) ∈ V
114 eqid 2735 . . . . . . 7 (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))
115113, 114fnmpti 6712 . . . . . 6 (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) Fn (𝑀...𝑁)
116 fzfi 14010 . . . . . 6 (𝑀...𝑁) ∈ Fin
117 fnfi 9216 . . . . . 6 (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) Fn (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∈ Fin)
118115, 116, 117mp2an 692 . . . . 5 (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∈ Fin
119 f1of 6849 . . . . . . 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 7466 . . . . . 6 (𝜑 → (𝑀...𝑁) ∈ V)
122 fex2 7957 . . . . . 6 ((𝐹:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ V ∧ (𝑀...𝑁) ∈ V) → 𝐹 ∈ V)
123120, 121, 121, 122syl3anc 1370 . . . . 5 (𝜑𝐹 ∈ V)
124 f1oeq1 6837 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
125 feq1 6717 . . . . . . . 8 (𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) → (𝑔:(𝑀...𝑁)⟶𝐶 ↔ (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)):(𝑀...𝑁)⟶𝐶))
126124, 125bi2anan9r 639 . . . . . . 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 5871 . . . . . . . . . . 11 (𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) → (𝑔𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝑓))
128 coeq2 5872 . . . . . . . . . . 11 (𝑓 = 𝐹 → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))
129127, 128sylan9eq 2795 . . . . . . . . . 10 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → (𝑔𝑓) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))
130129seqeq3d 14047 . . . . . . . . 9 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → seq𝑀( + , (𝑔𝑓)) = seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹)))
131130fveq1d 6909 . . . . . . . 8 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → (seq𝑀( + , (𝑔𝑓))‘𝑁) = (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁))
132 simpl 482 . . . . . . . . . 10 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → 𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))
133132seqeq3d 14047 . . . . . . . . 9 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → seq𝑀( + , 𝑔) = seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))))
134133fveq1d 6909 . . . . . . . 8 ((𝑔 = (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∧ 𝑓 = 𝐹) → (seq𝑀( + , 𝑔)‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁))
135131, 134eqeq12d 2751 . . . . . . 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 344 . . . . . 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 587 . . . 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 699 . 2 (𝜑 → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁))
141120ffvelcdmda 7104 . . . . 5 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ (𝑀...𝑁))
142 fveq2 6907 . . . . . 6 (𝑥 = (𝐹𝑘) → (𝐺𝑥) = (𝐺‘(𝐹𝑘)))
143 fvex 6920 . . . . . 6 (𝐺‘(𝐹𝑘)) ∈ V
144142, 114, 143fvmpt 7016 . . . . 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 7008 . . . . 5 ((𝐹:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ 𝑘 ∈ (𝑀...𝑁)) → (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))‘(𝐹𝑘)))
147120, 146sylan 580 . . . 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 2785 . . 3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹)‘𝑘) = (𝐻𝑘))
1504, 149seqfveq 14064 . 2 (𝜑 → (seq𝑀( + , ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)) ∘ 𝐹))‘𝑁) = (seq𝑀( + , 𝐻)‘𝑁))
151 fveq2 6907 . . . . 5 (𝑥 = 𝑘 → (𝐺𝑥) = (𝐺𝑘))
152 fvex 6920 . . . . 5 (𝐺𝑘) ∈ V
153151, 114, 152fvmpt 7016 . . . 4 (𝑘 ∈ (𝑀...𝑁) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))‘𝑘) = (𝐺𝑘))
154153adantl 481 . . 3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → ((𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥))‘𝑘) = (𝐺𝑘))
1554, 154seqfveq 14064 . 2 (𝜑 → (seq𝑀( + , (𝑥 ∈ (𝑀...𝑁) ↦ (𝐺𝑥)))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
156140, 150, 1553eqtr3d 2783 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 1535   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963  ifcif 4531  {csn 4631   class class class wbr 5148  cmpt 5231  ccnv 5688  ccom 5693   Fn wfn 6558  wf 6559  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  Fincfn 8984  1c1 11154   + caddc 11156   < clt 11293  cz 12611  cuz 12876  ...cfz 13544  seqcseq 14039
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040
This theorem is referenced by:  summolem3  15747  prodmolem3  15966  eulerthlem2  16816  gsumval3eu  19937  gsumval3  19940
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