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Theorem seqf1olem2a 13407
 Description: Lemma for seqf1o 13410. (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 12913 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 6646 . . . . . 6 (𝑚 = 𝑀 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑀))
54oveq2d 7152 . . . . 5 (𝑚 = 𝑀 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑀)))
64oveq1d 7151 . . . . 5 (𝑚 = 𝑀 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺𝐾)))
75, 6eqeq12d 2814 . . . 4 (𝑚 = 𝑀 → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) ↔ ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺𝐾))))
87imbi2d 344 . . 3 (𝑚 = 𝑀 → ((𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾))) ↔ (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺𝐾)))))
9 fveq2 6646 . . . . . 6 (𝑚 = 𝑛 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑛))
109oveq2d 7152 . . . . 5 (𝑚 = 𝑛 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)))
119oveq1d 7151 . . . . 5 (𝑚 = 𝑛 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)))
1210, 11eqeq12d 2814 . . . 4 (𝑚 = 𝑛 → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) ↔ ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾))))
1312imbi2d 344 . . 3 (𝑚 = 𝑛 → ((𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾))) ↔ (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)))))
14 fveq2 6646 . . . . . 6 (𝑚 = (𝑛 + 1) → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘(𝑛 + 1)))
1514oveq2d 7152 . . . . 5 (𝑚 = (𝑛 + 1) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))))
1614oveq1d 7151 . . . . 5 (𝑚 = (𝑛 + 1) → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)))
1715, 16eqeq12d 2814 . . . 4 (𝑚 = (𝑛 + 1) → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) ↔ ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾))))
1817imbi2d 344 . . 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 6646 . . . . . 6 (𝑚 = 𝑁 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑁))
2019oveq2d 7152 . . . . 5 (𝑚 = 𝑁 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑁)))
2119oveq1d 7151 . . . . 5 (𝑚 = 𝑁 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺𝐾)))
2220, 21eqeq12d 2814 . . . 4 (𝑚 = 𝑁 → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) ↔ ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺𝐾))))
2322imbi2d 344 . . 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 6830 . . . . 5 (𝜑 → (𝐺𝐾) ∈ 𝐶)
28 eluzel2 12239 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
29 seq1 13380 . . . . . . 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 12912 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
331, 32syl 17 . . . . . . . 8 (𝜑𝑀 ∈ (𝑀...𝑁))
3431, 33sseldd 3916 . . . . . . 7 (𝜑𝑀𝐴)
3525, 34ffvelrnd 6830 . . . . . 6 (𝜑 → (𝐺𝑀) ∈ 𝐶)
3630, 35eqeltrd 2890 . . . . 5 (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) ∈ 𝐶)
3724, 27, 36caovcomd 7326 . . . 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 7143 . . . . . 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 13040 . . . . . . . . . . 11 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
4140adantl 485 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
42 seqp1 13382 . . . . . . . . . 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 7152 . . . . . . . 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 714 . . . . . . . . 9 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
47 seqf1o.5 . . . . . . . . . . 11 (𝜑𝐶𝑆)
4847, 27sseldd 3916 . . . . . . . . . 10 (𝜑 → (𝐺𝐾) ∈ 𝑆)
4948adantr 484 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺𝐾) ∈ 𝑆)
5047adantr 484 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝐶𝑆)
5150adantr 484 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝐶𝑆)
5225adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝐺:𝐴𝐶)
5352adantr 484 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝐺:𝐴𝐶)
54 elfzouz2 13050 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ𝑛))
5554adantl 485 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑁 ∈ (ℤ𝑛))
56 fzss2 12945 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
5755, 56syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
5831adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑁) ⊆ 𝐴)
5957, 58sstrd 3925 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ 𝐴)
6059sselda 3915 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥𝐴)
6153, 60ffvelrnd 6830 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺𝑥) ∈ 𝐶)
6251, 61sseldd 3916 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺𝑥) ∈ 𝑆)
63 seqf1o.1 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6463adantlr 714 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6541, 62, 64seqcl 13389 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆)
66 fzofzp1 13132 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
6766adantl 485 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ (𝑀...𝑁))
6858, 67sseldd 3916 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ 𝐴)
6952, 68ffvelrnd 6830 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝐶)
7050, 69sseldd 3916 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
7146, 49, 65, 70caovassd 7329 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))) = ((𝐺𝐾) + ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
7244, 71eqtr4d 2836 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))))
7346, 65, 70, 49caovassd 7329 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺𝐾))))
7443oveq1d 7151 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺𝐾)))
7546, 65, 49, 70caovassd 7329 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺𝐾) + (𝐺‘(𝑛 + 1)))))
7624adantlr 714 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
7727adantr 484 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺𝐾) ∈ 𝐶)
7876, 69, 77caovcomd 7326 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑛 + 1)) + (𝐺𝐾)) = ((𝐺𝐾) + (𝐺‘(𝑛 + 1))))
7978oveq2d 7152 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺𝐾))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺𝐾) + (𝐺‘(𝑛 + 1)))))
8075, 79eqtr4d 2836 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺𝐾))))
8173, 74, 803eqtr4d 2843 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1))))
8272, 81eqeq12d 2814 . . . . . 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 249 . . . . 5 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾))))
8483expcom 417 . . . 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 13153 . 2 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺𝐾))))
873, 86mpcom 38 1 (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136  1c1 10530   + caddc 10532  ℤcz 11972  ℤ≥cuz 12234  ...cfz 12888  ..^cfzo 13031  seqcseq 13367 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-n0 11889  df-z 11973  df-uz 12235  df-fz 12889  df-fzo 13032  df-seq 13368 This theorem is referenced by:  seqf1olem2  13409
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