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Theorem pczpre 15845
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 12000 . . 3 (𝑁 ∈ ℤ → 𝑁 ∈ ℚ)
2 eqid 2765 . . . 4 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < )
3 eqid 2765 . . . 4 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )
42, 3pcval 15842 . . 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 672 . 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 787 . . . 4 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℤ)
76zcnd 11735 . . . . . 6 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℂ)
87div1d 11051 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑁 / 1) = 𝑁)
98eqcomd 2771 . . . 4 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 = (𝑁 / 1))
10 prmuz2 15702 . . . . . . . 8 (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))
11 eqid 2765 . . . . . . . 8 1 = 1
12 eqid 2765 . . . . . . . . 9 {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}
13 eqid 2765 . . . . . . . . 9 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < )
1412, 13pcpre1 15840 . . . . . . . 8 ((𝑃 ∈ (ℤ‘2) ∧ 1 = 1) → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ) = 0)
1510, 11, 14sylancl 580 . . . . . . 7 (𝑃 ∈ ℙ → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ) = 0)
1615adantr 472 . . . . . 6 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ) = 0)
1716oveq2d 6862 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < )) = (𝑆 − 0))
18 eqid 2765 . . . . . . . . . 10 {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}
19 pczpre.1 . . . . . . . . . 10 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}, ℝ, < )
2018, 19pcprecl 15837 . . . . . . . . 9 ((𝑃 ∈ (ℤ‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃𝑆) ∥ 𝑁))
2110, 20sylan 575 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃𝑆) ∥ 𝑁))
2221simpld 488 . . . . . . 7 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈ ℕ0)
2322nn0cnd 11604 . . . . . 6 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈ ℂ)
2423subid1d 10639 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 − 0) = 𝑆)
2517, 24eqtr2d 2800 . . . 4 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < )))
26 1nn 11291 . . . . 5 1 ∈ ℕ
27 oveq1 6853 . . . . . . . 8 (𝑥 = 𝑁 → (𝑥 / 𝑦) = (𝑁 / 𝑦))
2827eqeq2d 2775 . . . . . . 7 (𝑥 = 𝑁 → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑁 / 𝑦)))
29 breq2 4815 . . . . . . . . . . . 12 (𝑥 = 𝑁 → ((𝑃𝑛) ∥ 𝑥 ↔ (𝑃𝑛) ∥ 𝑁))
3029rabbidv 3338 . . . . . . . . . . 11 (𝑥 = 𝑁 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁})
3130supeq1d 8563 . . . . . . . . . 10 (𝑥 = 𝑁 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}, ℝ, < ))
3231, 19syl6eqr 2817 . . . . . . . . 9 (𝑥 = 𝑁 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = 𝑆)
3332oveq1d 6861 . . . . . . . 8 (𝑥 = 𝑁 → (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))
3433eqeq2d 2775 . . . . . . 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 624 . . . . . 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 6854 . . . . . . . 8 (𝑦 = 1 → (𝑁 / 𝑦) = (𝑁 / 1))
3736eqeq2d 2775 . . . . . . 7 (𝑦 = 1 → (𝑁 = (𝑁 / 𝑦) ↔ 𝑁 = (𝑁 / 1)))
38 breq2 4815 . . . . . . . . . . 11 (𝑦 = 1 → ((𝑃𝑛) ∥ 𝑦 ↔ (𝑃𝑛) ∥ 1))
3938rabbidv 3338 . . . . . . . . . 10 (𝑦 = 1 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1})
4039supeq1d 8563 . . . . . . . . 9 (𝑦 = 1 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ))
4140oveq2d 6862 . . . . . . . 8 (𝑦 = 1 → (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < )))
4241eqeq2d 2775 . . . . . . 7 (𝑦 = 1 → (𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) ↔ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 1}, ℝ, < ))))
4337, 42anbi12d 624 . . . . . 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 3477 . . . . 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 1573 . . . 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 865 . . 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 10376 . . . . . 6 < Or ℝ
4847supex 8580 . . . . 5 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}, ℝ, < ) ∈ V
4919, 48eqeltri 2840 . . . 4 𝑆 ∈ V
502, 3pceu 15844 . . . . 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 672 . . . 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 2769 . . . . . . 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 622 . . . . . 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 3204 . . . . 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 6059 . . . 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 581 . . 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 223 . 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 2799 1 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  ∃!weu 2581  wne 2937  wrex 3056  {crab 3059  Vcvv 3350   class class class wbr 4811  cio 6031  cfv 6070  (class class class)co 6846  supcsup 8557  cr 10192  0cc0 10193  1c1 10194   < clt 10332  cmin 10524   / cdiv 10942  cn 11278  2c2 11331  0cn0 11542  cz 11628  cuz 11891  cq 11994  cexp 13072  cdvds 15279  cprime 15679   pCnt cpc 15834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-er 7951  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-sup 8559  df-inf 8560  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-3 11340  df-n0 11543  df-z 11629  df-uz 11892  df-q 11995  df-rp 12034  df-fl 12806  df-mod 12882  df-seq 13014  df-exp 13073  df-cj 14138  df-re 14139  df-im 14140  df-sqrt 14274  df-abs 14275  df-dvds 15280  df-gcd 15512  df-prm 15680  df-pc 15835
This theorem is referenced by:  pczcl  15846  pcmul  15849  pcdiv  15850  pc1  15853  pczdvds  15860  pczndvds  15862  pczndvds2  15864  pcneg  15871
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