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Theorem seqf1olem2a 14078
Description: Lemma for seqf1o 14081. (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 13569 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 6907 . . . . . 6 (𝑚 = 𝑀 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑀))
54oveq2d 7447 . . . . 5 (𝑚 = 𝑀 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑀)))
64oveq1d 7446 . . . . 5 (𝑚 = 𝑀 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺𝐾)))
75, 6eqeq12d 2751 . . . 4 (𝑚 = 𝑀 → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) ↔ ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺𝐾))))
87imbi2d 340 . . 3 (𝑚 = 𝑀 → ((𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾))) ↔ (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺𝐾)))))
9 fveq2 6907 . . . . . 6 (𝑚 = 𝑛 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑛))
109oveq2d 7447 . . . . 5 (𝑚 = 𝑛 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)))
119oveq1d 7446 . . . . 5 (𝑚 = 𝑛 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)))
1210, 11eqeq12d 2751 . . . 4 (𝑚 = 𝑛 → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) ↔ ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾))))
1312imbi2d 340 . . 3 (𝑚 = 𝑛 → ((𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾))) ↔ (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)))))
14 fveq2 6907 . . . . . 6 (𝑚 = (𝑛 + 1) → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘(𝑛 + 1)))
1514oveq2d 7447 . . . . 5 (𝑚 = (𝑛 + 1) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))))
1614oveq1d 7446 . . . . 5 (𝑚 = (𝑛 + 1) → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)))
1715, 16eqeq12d 2751 . . . 4 (𝑚 = (𝑛 + 1) → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) ↔ ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾))))
1817imbi2d 340 . . 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 6907 . . . . . 6 (𝑚 = 𝑁 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑁))
2019oveq2d 7447 . . . . 5 (𝑚 = 𝑁 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑁)))
2119oveq1d 7446 . . . . 5 (𝑚 = 𝑁 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺𝐾)))
2220, 21eqeq12d 2751 . . . 4 (𝑚 = 𝑁 → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) ↔ ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺𝐾))))
2322imbi2d 340 . . 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, 26ffvelcdmd 7105 . . . . 5 (𝜑 → (𝐺𝐾) ∈ 𝐶)
28 eluzel2 12881 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
29 seq1 14052 . . . . . . 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 13568 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
331, 32syl 17 . . . . . . . 8 (𝜑𝑀 ∈ (𝑀...𝑁))
3431, 33sseldd 3996 . . . . . . 7 (𝜑𝑀𝐴)
3525, 34ffvelcdmd 7105 . . . . . 6 (𝜑 → (𝐺𝑀) ∈ 𝐶)
3630, 35eqeltrd 2839 . . . . 5 (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) ∈ 𝐶)
3724, 27, 36caovcomd 7629 . . . 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 7438 . . . . . 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 13700 . . . . . . . . . . 11 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
4140adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
42 seqp1 14054 . . . . . . . . . 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 7447 . . . . . . . 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 715 . . . . . . . . 9 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
47 seqf1o.5 . . . . . . . . . . 11 (𝜑𝐶𝑆)
4847, 27sseldd 3996 . . . . . . . . . 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 13711 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ𝑛))
5554adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑁 ∈ (ℤ𝑛))
56 fzss2 13601 . . . . . . . . . . . . . . 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 4006 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ 𝐴)
6059sselda 3995 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥𝐴)
6153, 60ffvelcdmd 7105 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺𝑥) ∈ 𝐶)
6251, 61sseldd 3996 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺𝑥) ∈ 𝑆)
63 seqf1o.1 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6463adantlr 715 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6541, 62, 64seqcl 14060 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆)
66 fzofzp1 13800 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
6766adantl 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ (𝑀...𝑁))
6858, 67sseldd 3996 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ 𝐴)
6952, 68ffvelcdmd 7105 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝐶)
7050, 69sseldd 3996 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
7146, 49, 65, 70caovassd 7632 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))) = ((𝐺𝐾) + ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
7244, 71eqtr4d 2778 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))))
7346, 65, 70, 49caovassd 7632 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺𝐾))))
7443oveq1d 7446 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺𝐾)))
7546, 65, 49, 70caovassd 7632 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺𝐾) + (𝐺‘(𝑛 + 1)))))
7624adantlr 715 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
7727adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺𝐾) ∈ 𝐶)
7876, 69, 77caovcomd 7629 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑛 + 1)) + (𝐺𝐾)) = ((𝐺𝐾) + (𝐺‘(𝑛 + 1))))
7978oveq2d 7447 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺𝐾))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺𝐾) + (𝐺‘(𝑛 + 1)))))
8075, 79eqtr4d 2778 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺𝐾))))
8173, 74, 803eqtr4d 2785 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1))))
8272, 81eqeq12d 2751 . . . . . 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, 82imbitrrid 246 . . . . 5 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾))))
8483expcom 413 . . . 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 13821 . 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 395  w3a 1086   = wceq 1537  wcel 2106  wss 3963  wf 6559  cfv 6563  (class class class)co 7431  1c1 11154   + caddc 11156  cz 12611  cuz 12876  ...cfz 13544  ..^cfzo 13691  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-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  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:  seqf1olem2  14080
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