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Theorem seqf1olem2a 12701
Description: Lemma for seqf1o 12704. (Contributed 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)
seqf1olem2a.1 (𝜑𝐺:𝐴𝐶)
seqf1olem2a.3 (𝜑𝐾𝐴)
seqf1olem2a.4 (𝜑 → (𝑀...𝑁) ⊆ 𝐴)
Assertion
Ref Expression
seqf1olem2a (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺𝐾)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝑥,𝑀,𝑦,𝑧   𝑥, + ,𝑦,𝑧   𝑥,𝑁,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem seqf1olem2a
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seqf1o.4 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 12220 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 6103 . . . . . 6 (𝑚 = 𝑀 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑀))
54oveq2d 6565 . . . . 5 (𝑚 = 𝑀 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑀)))
64oveq1d 6564 . . . . 5 (𝑚 = 𝑀 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺𝐾)))
75, 6eqeq12d 2625 . . . 4 (𝑚 = 𝑀 → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) ↔ ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺𝐾))))
87imbi2d 329 . . 3 (𝑚 = 𝑀 → ((𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾))) ↔ (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺𝐾)))))
9 fveq2 6103 . . . . . 6 (𝑚 = 𝑛 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑛))
109oveq2d 6565 . . . . 5 (𝑚 = 𝑛 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)))
119oveq1d 6564 . . . . 5 (𝑚 = 𝑛 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)))
1210, 11eqeq12d 2625 . . . 4 (𝑚 = 𝑛 → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) ↔ ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾))))
1312imbi2d 329 . . 3 (𝑚 = 𝑛 → ((𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾))) ↔ (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)))))
14 fveq2 6103 . . . . . 6 (𝑚 = (𝑛 + 1) → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘(𝑛 + 1)))
1514oveq2d 6565 . . . . 5 (𝑚 = (𝑛 + 1) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))))
1614oveq1d 6564 . . . . 5 (𝑚 = (𝑛 + 1) → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)))
1715, 16eqeq12d 2625 . . . 4 (𝑚 = (𝑛 + 1) → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) ↔ ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾))))
1817imbi2d 329 . . 3 (𝑚 = (𝑛 + 1) → ((𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾))) ↔ (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)))))
19 fveq2 6103 . . . . . 6 (𝑚 = 𝑁 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑁))
2019oveq2d 6565 . . . . 5 (𝑚 = 𝑁 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑁)))
2119oveq1d 6564 . . . . 5 (𝑚 = 𝑁 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺𝐾)))
2220, 21eqeq12d 2625 . . . 4 (𝑚 = 𝑁 → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) ↔ ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺𝐾))))
2322imbi2d 329 . . 3 (𝑚 = 𝑁 → ((𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾))) ↔ (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺𝐾)))))
24 seqf1o.2 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
25 seqf1olem2a.1 . . . . . 6 (𝜑𝐺:𝐴𝐶)
26 seqf1olem2a.3 . . . . . 6 (𝜑𝐾𝐴)
2725, 26ffvelrnd 6268 . . . . 5 (𝜑 → (𝐺𝐾) ∈ 𝐶)
28 eluzel2 11568 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
29 seq1 12676 . . . . . . 7 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐺)‘𝑀) = (𝐺𝑀))
301, 28, 293syl 18 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) = (𝐺𝑀))
31 seqf1olem2a.4 . . . . . . . 8 (𝜑 → (𝑀...𝑁) ⊆ 𝐴)
32 eluzfz1 12219 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
331, 32syl 17 . . . . . . . 8 (𝜑𝑀 ∈ (𝑀...𝑁))
3431, 33sseldd 3569 . . . . . . 7 (𝜑𝑀𝐴)
3525, 34ffvelrnd 6268 . . . . . 6 (𝜑 → (𝐺𝑀) ∈ 𝐶)
3630, 35eqeltrd 2688 . . . . 5 (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) ∈ 𝐶)
3724, 27, 36caovcomd 6728 . . . 4 (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺𝐾)))
3837a1i 11 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺𝐾))))
39 oveq1 6556 . . . . . 6 (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1))))
40 elfzouz 12343 . . . . . . . . . . 11 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
4140adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
42 seqp1 12678 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
4341, 42syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
4443oveq2d 6565 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((𝐺𝐾) + ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
45 seqf1o.3 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
4645adantlr 747 . . . . . . . . 9 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
47 seqf1o.5 . . . . . . . . . . 11 (𝜑𝐶𝑆)
4847, 27sseldd 3569 . . . . . . . . . 10 (𝜑 → (𝐺𝐾) ∈ 𝑆)
4948adantr 480 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺𝐾) ∈ 𝑆)
5047adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝐶𝑆)
5150adantr 480 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝐶𝑆)
5225adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝐺:𝐴𝐶)
5352adantr 480 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝐺:𝐴𝐶)
54 elfzouz2 12353 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ𝑛))
5554adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑁 ∈ (ℤ𝑛))
56 fzss2 12252 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
5755, 56syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
5831adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑁) ⊆ 𝐴)
5957, 58sstrd 3578 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ 𝐴)
6059sselda 3568 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥𝐴)
6153, 60ffvelrnd 6268 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺𝑥) ∈ 𝐶)
6251, 61sseldd 3569 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺𝑥) ∈ 𝑆)
63 seqf1o.1 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6463adantlr 747 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6541, 62, 64seqcl 12683 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆)
66 fzofzp1 12431 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
6766adantl 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ (𝑀...𝑁))
6858, 67sseldd 3569 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ 𝐴)
6952, 68ffvelrnd 6268 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝐶)
7050, 69sseldd 3569 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
7146, 49, 65, 70caovassd 6731 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))) = ((𝐺𝐾) + ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
7244, 71eqtr4d 2647 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))))
7346, 65, 70, 49caovassd 6731 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺𝐾))))
7443oveq1d 6564 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺𝐾)))
7546, 65, 49, 70caovassd 6731 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺𝐾) + (𝐺‘(𝑛 + 1)))))
7624adantlr 747 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
7727adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺𝐾) ∈ 𝐶)
7876, 69, 77caovcomd 6728 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑛 + 1)) + (𝐺𝐾)) = ((𝐺𝐾) + (𝐺‘(𝑛 + 1))))
7978oveq2d 6565 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺𝐾))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺𝐾) + (𝐺‘(𝑛 + 1)))))
8075, 79eqtr4d 2647 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺𝐾))))
8173, 74, 803eqtr4d 2654 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1))))
8272, 81eqeq12d 2625 . . . . . 6 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)) ↔ (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1)))))
8339, 82syl5ibr 235 . . . . 5 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾))))
8483expcom 450 . . . 4 (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)))))
8584a2d 29 . . 3 (𝑛 ∈ (𝑀..^𝑁) → ((𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾))) → (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)))))
868, 13, 18, 23, 38, 85fzind2 12448 . 2 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺𝐾))))
873, 86mpcom 37 1 (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wss 3540  wf 5800  cfv 5804  (class class class)co 6549  1c1 9816   + caddc 9818  cz 11254  cuz 11563  ...cfz 12197  ..^cfzo 12334  seqcseq 12663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664
This theorem is referenced by:  seqf1olem2  12703
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