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Theorem pcqmul 12866

Description: Multiplication property of the prime power function. (Contributed by Mario Carneiro, 9-Sep-2014.)

**Assertion**
Ref	Expression
pcqmul	⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵)))

Proof of Theorem pcqmul Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2l 1047	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℚ)
2		elq 9846	. . 3 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
3	1, 2	sylib 122	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
4		simp3l 1049	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℚ)
5		elq 9846	. . 3 ⊢ (𝐵 ∈ ℚ ↔ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤))
6	4, 5	sylib 122	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤))
7		reeanv 2701	. . 3 ⊢ (∃𝑥 ∈ ℤ ∃𝑧 ∈ ℤ (∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) ↔ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)))
8		reeanv 2701	. . . . 5 ⊢ (∃𝑦 ∈ ℕ ∃𝑤 ∈ ℕ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) ↔ (∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)))
9		simp2r 1048	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → 𝐴 ≠ 0)
10		simp3r 1050	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → 𝐵 ≠ 0)
11	9, 10	jca 306	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → (𝐴 ≠ 0 ∧ 𝐵 ≠ 0))
12	11	ad2antrr 488	. . . . . . 7 ⊢ ((((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (𝐴 ≠ 0 ∧ 𝐵 ≠ 0))
13		simp1 1021	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → 𝑃 ∈ ℙ)
14		simprl 529	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑦 ∈ ℕ)
15	14	nncnd 9147	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑦 ∈ ℂ)
16	14	nnap0d 9179	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑦 # 0)
17	15, 16	div0apd 8957	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (0 / 𝑦) = 0)
18		oveq1 6020	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 / 𝑦) = (0 / 𝑦))
19	18	eqeq1d 2238	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → ((𝑥 / 𝑦) = 0 ↔ (0 / 𝑦) = 0))
20	17, 19	syl5ibrcom 157	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (𝑥 = 0 → (𝑥 / 𝑦) = 0))
21	20	necon3d 2444	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝑥 / 𝑦) ≠ 0 → 𝑥 ≠ 0))
22		simprr 531	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑤 ∈ ℕ)
23	22	nncnd 9147	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑤 ∈ ℂ)
24	22	nnap0d 9179	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑤 # 0)
25	23, 24	div0apd 8957	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (0 / 𝑤) = 0)
26		oveq1 6020	. . . . . . . . . . . . 13 ⊢ (𝑧 = 0 → (𝑧 / 𝑤) = (0 / 𝑤))
27	26	eqeq1d 2238	. . . . . . . . . . . 12 ⊢ (𝑧 = 0 → ((𝑧 / 𝑤) = 0 ↔ (0 / 𝑤) = 0))
28	25, 27	syl5ibrcom 157	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (𝑧 = 0 → (𝑧 / 𝑤) = 0))
29	28	necon3d 2444	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝑧 / 𝑤) ≠ 0 → 𝑧 ≠ 0))
30		simpll 527	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑃 ∈ ℙ)
31		simplrl 535	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑥 ∈ ℤ)
32		simplrr 536	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑧 ∈ ℤ)
33	31, 32	zmulcld 9598	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑥 · 𝑧) ∈ ℤ)
34	31	zcnd 9593	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑥 ∈ ℂ)
35	32	zcnd 9593	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑧 ∈ ℂ)
36		simprrl 539	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑥 ≠ 0)
37		0zd 9481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 0 ∈ ℤ)
38		zapne 9544	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑥 # 0 ↔ 𝑥 ≠ 0))
39	31, 37, 38	syl2anc 411	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑥 # 0 ↔ 𝑥 ≠ 0))
40	36, 39	mpbird 167	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑥 # 0)
41		simprrr 540	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑧 ≠ 0)
42		zapne 9544	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑧 # 0 ↔ 𝑧 ≠ 0))
43	32, 37, 42	syl2anc 411	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑧 # 0 ↔ 𝑧 ≠ 0))
44	41, 43	mpbird 167	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑧 # 0)
45	34, 35, 40, 44	mulap0d 8828	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑥 · 𝑧) # 0)
46		zapne 9544	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 · 𝑧) ∈ ℤ ∧ 0 ∈ ℤ) → ((𝑥 · 𝑧) # 0 ↔ (𝑥 · 𝑧) ≠ 0))
47	33, 37, 46	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → ((𝑥 · 𝑧) # 0 ↔ (𝑥 · 𝑧) ≠ 0))
48	45, 47	mpbid 147	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑥 · 𝑧) ≠ 0)
49	14	adantrr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑦 ∈ ℕ)
50	22	adantrr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑤 ∈ ℕ)
51	49, 50	nnmulcld 9182	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑦 · 𝑤) ∈ ℕ)
52		pcdiv 12865	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ ((𝑥 · 𝑧) ∈ ℤ ∧ (𝑥 · 𝑧) ≠ 0) ∧ (𝑦 · 𝑤) ∈ ℕ) → (𝑃 pCnt ((𝑥 · 𝑧) / (𝑦 · 𝑤))) = ((𝑃 pCnt (𝑥 · 𝑧)) − (𝑃 pCnt (𝑦 · 𝑤))))
53	30, 33, 48, 51, 52	syl121anc 1276	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt ((𝑥 · 𝑧) / (𝑦 · 𝑤))) = ((𝑃 pCnt (𝑥 · 𝑧)) − (𝑃 pCnt (𝑦 · 𝑤))))
54		pcmul 12864	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt (𝑥 · 𝑧)) = ((𝑃 pCnt 𝑥) + (𝑃 pCnt 𝑧)))
55	30, 31, 36, 32, 41, 54	syl122anc 1280	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt (𝑥 · 𝑧)) = ((𝑃 pCnt 𝑥) + (𝑃 pCnt 𝑧)))
56	49	nnzd 9591	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑦 ∈ ℤ)
57	14	nnne0d 9178	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑦 ≠ 0)
58	57	adantrr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑦 ≠ 0)
59	50	nnzd 9591	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑤 ∈ ℤ)
60	22	nnne0d 9178	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑤 ≠ 0)
61	60	adantrr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑤 ≠ 0)
62		pcmul 12864	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ ℙ ∧ (𝑦 ∈ ℤ ∧ 𝑦 ≠ 0) ∧ (𝑤 ∈ ℤ ∧ 𝑤 ≠ 0)) → (𝑃 pCnt (𝑦 · 𝑤)) = ((𝑃 pCnt 𝑦) + (𝑃 pCnt 𝑤)))
63	30, 56, 58, 59, 61, 62	syl122anc 1280	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt (𝑦 · 𝑤)) = ((𝑃 pCnt 𝑦) + (𝑃 pCnt 𝑤)))
64	55, 63	oveq12d 6031	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → ((𝑃 pCnt (𝑥 · 𝑧)) − (𝑃 pCnt (𝑦 · 𝑤))) = (((𝑃 pCnt 𝑥) + (𝑃 pCnt 𝑧)) − ((𝑃 pCnt 𝑦) + (𝑃 pCnt 𝑤))))
65		pczcl 12861	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (𝑃 pCnt 𝑥) ∈ ℕ₀)
66	30, 31, 36, 65	syl12anc 1269	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑥) ∈ ℕ₀)
67	66	nn0cnd 9447	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑥) ∈ ℂ)
68		pczcl 12861	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt 𝑧) ∈ ℕ₀)
69	30, 32, 41, 68	syl12anc 1269	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑧) ∈ ℕ₀)
70	69	nn0cnd 9447	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑧) ∈ ℂ)
71	30, 49	pccld 12863	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑦) ∈ ℕ₀)
72	71	nn0cnd 9447	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑦) ∈ ℂ)
73	30, 50	pccld 12863	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑤) ∈ ℕ₀)
74	73	nn0cnd 9447	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑤) ∈ ℂ)
75	67, 70, 72, 74	addsub4d 8527	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (((𝑃 pCnt 𝑥) + (𝑃 pCnt 𝑧)) − ((𝑃 pCnt 𝑦) + (𝑃 pCnt 𝑤))) = (((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦)) + ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤))))
76	53, 64, 75	3eqtrd 2266	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt ((𝑥 · 𝑧) / (𝑦 · 𝑤))) = (((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦)) + ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤))))
77	15	adantrr 479	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑦 ∈ ℂ)
78	23	adantrr 479	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑤 ∈ ℂ)
79	16	adantrr 479	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑦 # 0)
80	24	adantrr 479	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑤 # 0)
81	34, 77, 35, 78, 79, 80	divmuldivapd 9002	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → ((𝑥 / 𝑦) · (𝑧 / 𝑤)) = ((𝑥 · 𝑧) / (𝑦 · 𝑤)))
82	81	oveq2d 6029	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))) = (𝑃 pCnt ((𝑥 · 𝑧) / (𝑦 · 𝑤))))
83		pcdiv 12865	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (𝑥 / 𝑦)) = ((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦)))
84	30, 31, 36, 49, 83	syl121anc 1276	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt (𝑥 / 𝑦)) = ((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦)))
85		pcdiv 12865	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0) ∧ 𝑤 ∈ ℕ) → (𝑃 pCnt (𝑧 / 𝑤)) = ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤)))
86	30, 32, 41, 50, 85	syl121anc 1276	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt (𝑧 / 𝑤)) = ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤)))
87	84, 86	oveq12d 6031	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → ((𝑃 pCnt (𝑥 / 𝑦)) + (𝑃 pCnt (𝑧 / 𝑤))) = (((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦)) + ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤))))
88	76, 82, 87	3eqtr4d 2272	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))) = ((𝑃 pCnt (𝑥 / 𝑦)) + (𝑃 pCnt (𝑧 / 𝑤))))
89	88	expr 375	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝑥 ≠ 0 ∧ 𝑧 ≠ 0) → (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))) = ((𝑃 pCnt (𝑥 / 𝑦)) + (𝑃 pCnt (𝑧 / 𝑤)))))
90	21, 29, 89	syl2and 295	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (((𝑥 / 𝑦) ≠ 0 ∧ (𝑧 / 𝑤) ≠ 0) → (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))) = ((𝑃 pCnt (𝑥 / 𝑦)) + (𝑃 pCnt (𝑧 / 𝑤)))))
91		neeq1 2413	. . . . . . . . . . 11 ⊢ (𝐴 = (𝑥 / 𝑦) → (𝐴 ≠ 0 ↔ (𝑥 / 𝑦) ≠ 0))
92		neeq1 2413	. . . . . . . . . . 11 ⊢ (𝐵 = (𝑧 / 𝑤) → (𝐵 ≠ 0 ↔ (𝑧 / 𝑤) ≠ 0))
93	91, 92	bi2anan9 608	. . . . . . . . . 10 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) ↔ ((𝑥 / 𝑦) ≠ 0 ∧ (𝑧 / 𝑤) ≠ 0)))
94		oveq12 6022	. . . . . . . . . . . 12 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝐴 · 𝐵) = ((𝑥 / 𝑦) · (𝑧 / 𝑤)))
95	94	oveq2d 6029	. . . . . . . . . . 11 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt (𝐴 · 𝐵)) = (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))))
96		oveq2 6021	. . . . . . . . . . . 12 ⊢ (𝐴 = (𝑥 / 𝑦) → (𝑃 pCnt 𝐴) = (𝑃 pCnt (𝑥 / 𝑦)))
97		oveq2 6021	. . . . . . . . . . . 12 ⊢ (𝐵 = (𝑧 / 𝑤) → (𝑃 pCnt 𝐵) = (𝑃 pCnt (𝑧 / 𝑤)))
98	96, 97	oveqan12d 6032	. . . . . . . . . . 11 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵)) = ((𝑃 pCnt (𝑥 / 𝑦)) + (𝑃 pCnt (𝑧 / 𝑤))))
99	95, 98	eqeq12d 2244	. . . . . . . . . 10 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → ((𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵)) ↔ (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))) = ((𝑃 pCnt (𝑥 / 𝑦)) + (𝑃 pCnt (𝑧 / 𝑤)))))
100	93, 99	imbi12d 234	. . . . . . . . 9 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵))) ↔ (((𝑥 / 𝑦) ≠ 0 ∧ (𝑧 / 𝑤) ≠ 0) → (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))) = ((𝑃 pCnt (𝑥 / 𝑦)) + (𝑃 pCnt (𝑧 / 𝑤))))))
101	90, 100	syl5ibrcom 157	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵)))))
102	13, 101	sylanl1 402	. . . . . . 7 ⊢ ((((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵)))))
103	12, 102	mpid 42	. . . . . 6 ⊢ ((((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵))))
104	103	rexlimdvva 2656	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (∃𝑦 ∈ ℕ ∃𝑤 ∈ ℕ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵))))
105	8, 104	biimtrrid 153	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵))))
106	105	rexlimdvva 2656	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → (∃𝑥 ∈ ℤ ∃𝑧 ∈ ℤ (∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵))))
107	7, 106	biimtrrid 153	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → ((∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵))))
108	3, 6, 107	mp2and 433	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 ∃wrex 2509 class class class wbr 4086 (class class class)co 6013 ℂcc 8020 0cc0 8022 + caddc 8025 · cmul 8027 − cmin 8340 # cap 8751 / cdiv 8842 ℕcn 9133 ℕ₀cn0 9392 ℤcz 9469 ℚcq 9843 ℙcprime 12669 pCnt cpc 12847

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-mulrcl 8121 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-precex 8132 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138 ax-pre-mulgt0 8139 ax-pre-mulext 8140 ax-arch 8141 ax-caucvg 8142

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-frec 6552 df-1o 6577 df-2o 6578 df-er 6697 df-en 6905 df-sup 7174 df-inf 7175 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209 df-le 8210 df-sub 8342 df-neg 8343 df-reap 8745 df-ap 8752 df-div 8843 df-inn 9134 df-2 9192 df-3 9193 df-4 9194 df-n0 9393 df-z 9470 df-uz 9746 df-q 9844 df-rp 9879 df-fz 10234 df-fzo 10368 df-fl 10520 df-mod 10575 df-seqfrec 10700 df-exp 10791 df-cj 11393 df-re 11394 df-im 11395 df-rsqrt 11549 df-abs 11550 df-dvds 12339 df-gcd 12515 df-prm 12670 df-pc 12848

This theorem is referenced by: pcqdiv 12870 pcexp 12872 pcaddlem 12902

W3C validator