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Theorem pczpre 15759
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 12002 . . 3 (𝑁 ∈ ℤ → 𝑁 ∈ ℚ)
2 eqid 2771 . . . 4 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < )
3 eqid 2771 . . . 4 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )
42, 3pcval 15756 . . 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 661 . 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 754 . . . 4 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℤ)
76zcnd 11690 . . . . . 6 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℂ)
87div1d 10999 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑁 / 1) = 𝑁)
98eqcomd 2777 . . . 4 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 = (𝑁 / 1))
10 prmuz2 15615 . . . . . . . 8 (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))
11 eqid 2771 . . . . . . . 8 1 = 1
12 eqid 2771 . . . . . . . . 9 {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}
13 eqid 2771 . . . . . . . . 9 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < )
1412, 13pcpre1 15754 . . . . . . . 8 ((𝑃 ∈ (ℤ‘2) ∧ 1 = 1) → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ) = 0)
1510, 11, 14sylancl 574 . . . . . . 7 (𝑃 ∈ ℙ → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ) = 0)
1615adantr 466 . . . . . 6 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ) = 0)
1716oveq2d 6812 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < )) = (𝑆 − 0))
18 eqid 2771 . . . . . . . . . 10 {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}
19 pczpre.1 . . . . . . . . . 10 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}, ℝ, < )
2018, 19pcprecl 15751 . . . . . . . . 9 ((𝑃 ∈ (ℤ‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃𝑆) ∥ 𝑁))
2110, 20sylan 569 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃𝑆) ∥ 𝑁))
2221simpld 482 . . . . . . 7 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈ ℕ0)
2322nn0cnd 11560 . . . . . 6 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈ ℂ)
2423subid1d 10587 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 − 0) = 𝑆)
2517, 24eqtr2d 2806 . . . 4 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < )))
26 1nn 11237 . . . . 5 1 ∈ ℕ
27 oveq1 6803 . . . . . . . 8 (𝑥 = 𝑁 → (𝑥 / 𝑦) = (𝑁 / 𝑦))
2827eqeq2d 2781 . . . . . . 7 (𝑥 = 𝑁 → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑁 / 𝑦)))
29 breq2 4791 . . . . . . . . . . . 12 (𝑥 = 𝑁 → ((𝑃𝑛) ∥ 𝑥 ↔ (𝑃𝑛) ∥ 𝑁))
3029rabbidv 3339 . . . . . . . . . . 11 (𝑥 = 𝑁 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁})
3130supeq1d 8512 . . . . . . . . . 10 (𝑥 = 𝑁 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}, ℝ, < ))
3231, 19syl6eqr 2823 . . . . . . . . 9 (𝑥 = 𝑁 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = 𝑆)
3332oveq1d 6811 . . . . . . . 8 (𝑥 = 𝑁 → (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))
3433eqeq2d 2781 . . . . . . 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 616 . . . . . 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 6804 . . . . . . . 8 (𝑦 = 1 → (𝑁 / 𝑦) = (𝑁 / 1))
3736eqeq2d 2781 . . . . . . 7 (𝑦 = 1 → (𝑁 = (𝑁 / 𝑦) ↔ 𝑁 = (𝑁 / 1)))
38 breq2 4791 . . . . . . . . . . 11 (𝑦 = 1 → ((𝑃𝑛) ∥ 𝑦 ↔ (𝑃𝑛) ∥ 1))
3938rabbidv 3339 . . . . . . . . . 10 (𝑦 = 1 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1})
4039supeq1d 8512 . . . . . . . . 9 (𝑦 = 1 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ))
4140oveq2d 6812 . . . . . . . 8 (𝑦 = 1 → (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < )))
4241eqeq2d 2781 . . . . . . 7 (𝑦 = 1 → (𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) ↔ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ))))
4337, 42anbi12d 616 . . . . . 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 3474 . . . . 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 1560 . . . 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 1474 . . 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 10324 . . . . . 6 < Or ℝ
4847supex 8529 . . . . 5 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}, ℝ, < ) ∈ V
4919, 48eqeltri 2846 . . . 4 𝑆 ∈ V
502, 3pceu 15758 . . . . 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 661 . . . 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 2775 . . . . . . 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 614 . . . . . 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 3205 . . . . 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 6019 . . . 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 575 . . 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 222 . 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 2805 1 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  ∃!weu 2618  wne 2943  wrex 3062  {crab 3065  Vcvv 3351   class class class wbr 4787  cio 5991  cfv 6030  (class class class)co 6796  supcsup 8506  cr 10141  0cc0 10142  1c1 10143   < clt 10280  cmin 10472   / cdiv 10890  cn 11226  2c2 11276  0cn0 11499  cz 11584  cuz 11893  cq 11996  cexp 13067  cdvds 15189  cprime 15592   pCnt cpc 15748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8508  df-inf 8509  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-2 11285  df-3 11286  df-n0 11500  df-z 11585  df-uz 11894  df-q 11997  df-rp 12036  df-fl 12801  df-mod 12877  df-seq 13009  df-exp 13068  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-dvds 15190  df-gcd 15425  df-prm 15593  df-pc 15749
This theorem is referenced by:  pczcl  15760  pcmul  15763  pcdiv  15764  pc1  15767  pczdvds  15774  pczndvds  15776  pczndvds2  15778  pcneg  15785
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