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Theorem pczpre 16187
Description: Connect the prime count pre-function to the actual prime count function, when restricted to the integers. (Contributed by Mario Carneiro, 23-Feb-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
pczpre.1 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}, ℝ, < )
Assertion
Ref Expression
pczpre ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = 𝑆)
Distinct variable groups:   𝑛,𝑁   𝑃,𝑛
Allowed substitution hint:   𝑆(𝑛)

Proof of Theorem pczpre
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zq 12357 . . 3 (𝑁 ∈ ℤ → 𝑁 ∈ ℚ)
2 eqid 2824 . . . 4 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < )
3 eqid 2824 . . . 4 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )
42, 3pcval 16184 . . 3 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))))
51, 4sylanr1 680 . 2 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))))
6 simprl 769 . . . 4 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℤ)
76zcnd 12091 . . . . . 6 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℂ)
87div1d 11411 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑁 / 1) = 𝑁)
98eqcomd 2830 . . . 4 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 = (𝑁 / 1))
10 prmuz2 16043 . . . . . . . 8 (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))
11 eqid 2824 . . . . . . . 8 1 = 1
12 eqid 2824 . . . . . . . . 9 {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}
13 eqid 2824 . . . . . . . . 9 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < )
1412, 13pcpre1 16182 . . . . . . . 8 ((𝑃 ∈ (ℤ‘2) ∧ 1 = 1) → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ) = 0)
1510, 11, 14sylancl 588 . . . . . . 7 (𝑃 ∈ ℙ → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ) = 0)
1615adantr 483 . . . . . 6 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ) = 0)
1716oveq2d 7175 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < )) = (𝑆 − 0))
18 eqid 2824 . . . . . . . . . 10 {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}
19 pczpre.1 . . . . . . . . . 10 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}, ℝ, < )
2018, 19pcprecl 16179 . . . . . . . . 9 ((𝑃 ∈ (ℤ‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃𝑆) ∥ 𝑁))
2110, 20sylan 582 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃𝑆) ∥ 𝑁))
2221simpld 497 . . . . . . 7 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈ ℕ0)
2322nn0cnd 11960 . . . . . 6 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈ ℂ)
2423subid1d 10989 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 − 0) = 𝑆)
2517, 24eqtr2d 2860 . . . 4 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < )))
26 1nn 11652 . . . . 5 1 ∈ ℕ
27 oveq1 7166 . . . . . . . 8 (𝑥 = 𝑁 → (𝑥 / 𝑦) = (𝑁 / 𝑦))
2827eqeq2d 2835 . . . . . . 7 (𝑥 = 𝑁 → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑁 / 𝑦)))
29 breq2 5073 . . . . . . . . . . . 12 (𝑥 = 𝑁 → ((𝑃𝑛) ∥ 𝑥 ↔ (𝑃𝑛) ∥ 𝑁))
3029rabbidv 3483 . . . . . . . . . . 11 (𝑥 = 𝑁 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁})
3130supeq1d 8913 . . . . . . . . . 10 (𝑥 = 𝑁 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}, ℝ, < ))
3231, 19syl6eqr 2877 . . . . . . . . 9 (𝑥 = 𝑁 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = 𝑆)
3332oveq1d 7174 . . . . . . . 8 (𝑥 = 𝑁 → (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))
3433eqeq2d 2835 . . . . . . 7 (𝑥 = 𝑁 → (𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) ↔ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))))
3528, 34anbi12d 632 . . . . . 6 (𝑥 = 𝑁 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (𝑁 = (𝑁 / 𝑦) ∧ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))))
36 oveq2 7167 . . . . . . . 8 (𝑦 = 1 → (𝑁 / 𝑦) = (𝑁 / 1))
3736eqeq2d 2835 . . . . . . 7 (𝑦 = 1 → (𝑁 = (𝑁 / 𝑦) ↔ 𝑁 = (𝑁 / 1)))
38 breq2 5073 . . . . . . . . . . 11 (𝑦 = 1 → ((𝑃𝑛) ∥ 𝑦 ↔ (𝑃𝑛) ∥ 1))
3938rabbidv 3483 . . . . . . . . . 10 (𝑦 = 1 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1})
4039supeq1d 8913 . . . . . . . . 9 (𝑦 = 1 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ))
4140oveq2d 7175 . . . . . . . 8 (𝑦 = 1 → (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < )))
4241eqeq2d 2835 . . . . . . 7 (𝑦 = 1 → (𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) ↔ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ))))
4337, 42anbi12d 632 . . . . . 6 (𝑦 = 1 → ((𝑁 = (𝑁 / 𝑦) ∧ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (𝑁 = (𝑁 / 1) ∧ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < )))))
4435, 43rspc2ev 3638 . . . . 5 ((𝑁 ∈ ℤ ∧ 1 ∈ ℕ ∧ (𝑁 = (𝑁 / 1) ∧ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < )))) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))))
4526, 44mp3an2 1445 . . . 4 ((𝑁 ∈ ℤ ∧ (𝑁 = (𝑁 / 1) ∧ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < )))) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))))
466, 9, 25, 45syl12anc 834 . . 3 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))))
47 ltso 10724 . . . . . 6 < Or ℝ
4847supex 8930 . . . . 5 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}, ℝ, < ) ∈ V
4919, 48eqeltri 2912 . . . 4 𝑆 ∈ V
502, 3pceu 16186 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃!𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))))
511, 50sylanr1 680 . . . 4 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃!𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))))
52 eqeq1 2828 . . . . . . 7 (𝑧 = 𝑆 → (𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) ↔ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))))
5352anbi2d 630 . . . . . 6 (𝑧 = 𝑆 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))))
54532rexbidv 3303 . . . . 5 (𝑧 = 𝑆 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))))
5554iota2 6347 . . . 4 ((𝑆 ∈ V ∧ ∃!𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))) = 𝑆))
5649, 51, 55sylancr 589 . . 3 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))) = 𝑆))
5746, 56mpbid 234 . 2 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))) = 𝑆)
585, 57eqtrd 2859 1 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1536  wcel 2113  ∃!weu 2652  wne 3019  wrex 3142  {crab 3145  Vcvv 3497   class class class wbr 5069  cio 6315  cfv 6358  (class class class)co 7159  supcsup 8907  cr 10539  0cc0 10540  1c1 10541   < clt 10678  cmin 10873   / cdiv 11300  cn 11641  2c2 11695  0cn0 11900  cz 11984  cuz 12246  cq 12351  cexp 13432  cdvds 15610  cprime 16018   pCnt cpc 16176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-pre-sup 10618
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-er 8292  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-sup 8909  df-inf 8910  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-div 11301  df-nn 11642  df-2 11703  df-3 11704  df-n0 11901  df-z 11985  df-uz 12247  df-q 12352  df-rp 12393  df-fl 13165  df-mod 13241  df-seq 13373  df-exp 13433  df-cj 14461  df-re 14462  df-im 14463  df-sqrt 14597  df-abs 14598  df-dvds 15611  df-gcd 15847  df-prm 16019  df-pc 16177
This theorem is referenced by:  pczcl  16188  pcmul  16191  pcdiv  16192  pc1  16195  pczdvds  16202  pczndvds  16204  pczndvds2  16206  pcneg  16213
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