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Theorem seqf1olem2a 13614
Description: Lemma for seqf1o 13617. (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 13120 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 6717 . . . . . 6 (𝑚 = 𝑀 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑀))
54oveq2d 7229 . . . . 5 (𝑚 = 𝑀 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑀)))
64oveq1d 7228 . . . . 5 (𝑚 = 𝑀 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺𝐾)))
75, 6eqeq12d 2753 . . . 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 6717 . . . . . 6 (𝑚 = 𝑛 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑛))
109oveq2d 7229 . . . . 5 (𝑚 = 𝑛 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)))
119oveq1d 7228 . . . . 5 (𝑚 = 𝑛 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)))
1210, 11eqeq12d 2753 . . . 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 6717 . . . . . 6 (𝑚 = (𝑛 + 1) → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘(𝑛 + 1)))
1514oveq2d 7229 . . . . 5 (𝑚 = (𝑛 + 1) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))))
1614oveq1d 7228 . . . . 5 (𝑚 = (𝑛 + 1) → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)))
1715, 16eqeq12d 2753 . . . 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 6717 . . . . . 6 (𝑚 = 𝑁 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑁))
2019oveq2d 7229 . . . . 5 (𝑚 = 𝑁 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑁)))
2119oveq1d 7228 . . . . 5 (𝑚 = 𝑁 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺𝐾)))
2220, 21eqeq12d 2753 . . . 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 6905 . . . . 5 (𝜑 → (𝐺𝐾) ∈ 𝐶)
28 eluzel2 12443 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
29 seq1 13587 . . . . . . 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 13119 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
331, 32syl 17 . . . . . . . 8 (𝜑𝑀 ∈ (𝑀...𝑁))
3431, 33sseldd 3902 . . . . . . 7 (𝜑𝑀𝐴)
3525, 34ffvelrnd 6905 . . . . . 6 (𝜑 → (𝐺𝑀) ∈ 𝐶)
3630, 35eqeltrd 2838 . . . . 5 (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) ∈ 𝐶)
3724, 27, 36caovcomd 7404 . . . 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 7220 . . . . . 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 13247 . . . . . . . . . . 11 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
4140adantl 485 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
42 seqp1 13589 . . . . . . . . . 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 7229 . . . . . . . 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 3902 . . . . . . . . . 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 13257 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ𝑛))
5554adantl 485 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑁 ∈ (ℤ𝑛))
56 fzss2 13152 . . . . . . . . . . . . . . 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 3911 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ 𝐴)
6059sselda 3901 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥𝐴)
6153, 60ffvelrnd 6905 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺𝑥) ∈ 𝐶)
6251, 61sseldd 3902 . . . . . . . . . 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 13596 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆)
66 fzofzp1 13339 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
6766adantl 485 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ (𝑀...𝑁))
6858, 67sseldd 3902 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ 𝐴)
6952, 68ffvelrnd 6905 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝐶)
7050, 69sseldd 3902 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
7146, 49, 65, 70caovassd 7407 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))) = ((𝐺𝐾) + ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
7244, 71eqtr4d 2780 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))))
7346, 65, 70, 49caovassd 7407 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺𝐾))))
7443oveq1d 7228 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺𝐾)))
7546, 65, 49, 70caovassd 7407 . . . . . . . . 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 484 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺𝐾) ∈ 𝐶)
7876, 69, 77caovcomd 7404 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑛 + 1)) + (𝐺𝐾)) = ((𝐺𝐾) + (𝐺‘(𝑛 + 1))))
7978oveq2d 7229 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺𝐾))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺𝐾) + (𝐺‘(𝑛 + 1)))))
8075, 79eqtr4d 2780 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺𝐾))))
8173, 74, 803eqtr4d 2787 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺𝐾)) + (𝐺‘(𝑛 + 1))))
8272, 81eqeq12d 2753 . . . . . 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 13360 . 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 1089   = wceq 1543  wcel 2110  wss 3866  wf 6376  cfv 6380  (class class class)co 7213  1c1 10730   + caddc 10732  cz 12176  cuz 12438  ...cfz 13095  ..^cfzo 13238  seqcseq 13574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-n0 12091  df-z 12177  df-uz 12439  df-fz 13096  df-fzo 13239  df-seq 13575
This theorem is referenced by:  seqf1olem2  13616
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