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Theorem seqf1olem2a 13946
Description: Lemma for seqf1o 13949. (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 13449 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 6842 . . . . . 6 (𝑚 = 𝑀 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑀))
54oveq2d 7373 . . . . 5 (𝑚 = 𝑀 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑀)))
64oveq1d 7372 . . . . 5 (𝑚 = 𝑀 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺𝐾)))
75, 6eqeq12d 2752 . . . 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 6842 . . . . . 6 (𝑚 = 𝑛 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑛))
109oveq2d 7373 . . . . 5 (𝑚 = 𝑛 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)))
119oveq1d 7372 . . . . 5 (𝑚 = 𝑛 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)))
1210, 11eqeq12d 2752 . . . 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 6842 . . . . . 6 (𝑚 = (𝑛 + 1) → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘(𝑛 + 1)))
1514oveq2d 7373 . . . . 5 (𝑚 = (𝑛 + 1) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))))
1614oveq1d 7372 . . . . 5 (𝑚 = (𝑛 + 1) → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)))
1715, 16eqeq12d 2752 . . . 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 6842 . . . . . 6 (𝑚 = 𝑁 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑁))
2019oveq2d 7373 . . . . 5 (𝑚 = 𝑁 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑁)))
2119oveq1d 7372 . . . . 5 (𝑚 = 𝑁 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺𝐾)))
2220, 21eqeq12d 2752 . . . 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 7036 . . . . 5 (𝜑 → (𝐺𝐾) ∈ 𝐶)
28 eluzel2 12768 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
29 seq1 13919 . . . . . . 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 13448 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
331, 32syl 17 . . . . . . . 8 (𝜑𝑀 ∈ (𝑀...𝑁))
3431, 33sseldd 3945 . . . . . . 7 (𝜑𝑀𝐴)
3525, 34ffvelcdmd 7036 . . . . . 6 (𝜑 → (𝐺𝑀) ∈ 𝐶)
3630, 35eqeltrd 2838 . . . . 5 (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) ∈ 𝐶)
3724, 27, 36caovcomd 7550 . . . 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 7364 . . . . . 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 13576 . . . . . . . . . . 11 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
4140adantl 482 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
42 seqp1 13921 . . . . . . . . . 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 7373 . . . . . . . 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 713 . . . . . . . . 9 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
47 seqf1o.5 . . . . . . . . . . 11 (𝜑𝐶𝑆)
4847, 27sseldd 3945 . . . . . . . . . 10 (𝜑 → (𝐺𝐾) ∈ 𝑆)
4948adantr 481 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺𝐾) ∈ 𝑆)
5047adantr 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝐶𝑆)
5150adantr 481 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝐶𝑆)
5225adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝐺:𝐴𝐶)
5352adantr 481 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝐺:𝐴𝐶)
54 elfzouz2 13587 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ𝑛))
5554adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑁 ∈ (ℤ𝑛))
56 fzss2 13481 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
5755, 56syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
5831adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑁) ⊆ 𝐴)
5957, 58sstrd 3954 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ 𝐴)
6059sselda 3944 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥𝐴)
6153, 60ffvelcdmd 7036 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺𝑥) ∈ 𝐶)
6251, 61sseldd 3945 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺𝑥) ∈ 𝑆)
63 seqf1o.1 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6463adantlr 713 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6541, 62, 64seqcl 13928 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆)
66 fzofzp1 13669 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
6766adantl 482 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ (𝑀...𝑁))
6858, 67sseldd 3945 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ 𝐴)
6952, 68ffvelcdmd 7036 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝐶)
7050, 69sseldd 3945 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
7146, 49, 65, 70caovassd 7553 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))) = ((𝐺𝐾) + ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
7244, 71eqtr4d 2779 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))))
7346, 65, 70, 49caovassd 7553 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺𝐾))))
7443oveq1d 7372 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺𝐾)))
7546, 65, 49, 70caovassd 7553 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺𝐾) + (𝐺‘(𝑛 + 1)))))
7624adantlr 713 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
7727adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺𝐾) ∈ 𝐶)
7876, 69, 77caovcomd 7550 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑛 + 1)) + (𝐺𝐾)) = ((𝐺𝐾) + (𝐺‘(𝑛 + 1))))
7978oveq2d 7373 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺𝐾))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺𝐾) + (𝐺‘(𝑛 + 1)))))
8075, 79eqtr4d 2779 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺𝐾))))
8173, 74, 803eqtr4d 2786 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1))))
8272, 81eqeq12d 2752 . . . . . 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 245 . . . . 5 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾))))
8483expcom 414 . . . 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 13690 . 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 396  w3a 1087   = wceq 1541  wcel 2106  wss 3910  wf 6492  cfv 6496  (class class class)co 7357  1c1 11052   + caddc 11054  cz 12499  cuz 12763  ...cfz 13424  ..^cfzo 13567  seqcseq 13906
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907
This theorem is referenced by:  seqf1olem2  13948
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