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Theorem pcqmul 12568

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 1025	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℚ)
2		elq 9742	. . 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 1027	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℚ)
5		elq 9742	. . 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 2675	. . 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 2675	. . . . 5 ⊢ (∃𝑦 ∈ ℕ ∃𝑤 ∈ ℕ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) ↔ (∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)))
9		simp2r 1026	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → 𝐴 ≠ 0)
10		simp3r 1028	. . . . . . . . 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 999	. . . . . . . 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 9049	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑦 ∈ ℂ)
16	14	nnap0d 9081	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑦 # 0)
17	15, 16	div0apd 8859	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (0 / 𝑦) = 0)
18		oveq1 5950	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 / 𝑦) = (0 / 𝑦))
19	18	eqeq1d 2213	. . . . . . . . . . . 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 2419	. . . . . . . . . 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 9049	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑤 ∈ ℂ)
24	22	nnap0d 9081	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑤 # 0)
25	23, 24	div0apd 8859	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (0 / 𝑤) = 0)
26		oveq1 5950	. . . . . . . . . . . . 13 ⊢ (𝑧 = 0 → (𝑧 / 𝑤) = (0 / 𝑤))
27	26	eqeq1d 2213	. . . . . . . . . . . 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 2419	. . . . . . . . . 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 9500	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑥 · 𝑧) ∈ ℤ)
34	31	zcnd 9495	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑥 ∈ ℂ)
35	32	zcnd 9495	. . . . . . . . . . . . . . . 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 9383	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 0 ∈ ℤ)
38		zapne 9446	. . . . . . . . . . . . . . . . . 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 9446	. . . . . . . . . . . . . . . . . 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 8730	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑥 · 𝑧) # 0)
46		zapne 9446	. . . . . . . . . . . . . . . 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 9084	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑦 · 𝑤) ∈ ℕ)
52		pcdiv 12567	. . . . . . . . . . . . . 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 1254	. . . . . . . . . . . . 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 12566	. . . . . . . . . . . . . . 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 1258	. . . . . . . . . . . . . 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 9493	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑦 ∈ ℤ)
57	14	nnne0d 9080	. . . . . . . . . . . . . . . 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 9493	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑤 ∈ ℤ)
60	22	nnne0d 9080	. . . . . . . . . . . . . . . 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 12566	. . . . . . . . . . . . . . 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 1258	. . . . . . . . . . . . . 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 5961	. . . . . . . . . . . . 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 12563	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (𝑃 pCnt 𝑥) ∈ ℕ₀)
66	30, 31, 36, 65	syl12anc 1247	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑥) ∈ ℕ₀)
67	66	nn0cnd 9349	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑥) ∈ ℂ)
68		pczcl 12563	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt 𝑧) ∈ ℕ₀)
69	30, 32, 41, 68	syl12anc 1247	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑧) ∈ ℕ₀)
70	69	nn0cnd 9349	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑧) ∈ ℂ)
71	30, 49	pccld 12565	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑦) ∈ ℕ₀)
72	71	nn0cnd 9349	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑦) ∈ ℂ)
73	30, 50	pccld 12565	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑤) ∈ ℕ₀)
74	73	nn0cnd 9349	. . . . . . . . . . . . . 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 8429	. . . . . . . . . . . . 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 2241	. . . . . . . . . . . 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 8904	. . . . . . . . . . . . 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 5959	. . . . . . . . . . . 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 12567	. . . . . . . . . . . . . 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 1254	. . . . . . . . . . . . 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 12567	. . . . . . . . . . . . . 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 1254	. . . . . . . . . . . . 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 5961	. . . . . . . . . . . 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 2247	. . . . . . . . . . 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 2388	. . . . . . . . . . 11 ⊢ (𝐴 = (𝑥 / 𝑦) → (𝐴 ≠ 0 ↔ (𝑥 / 𝑦) ≠ 0))
92		neeq1 2388	. . . . . . . . . . 11 ⊢ (𝐵 = (𝑧 / 𝑤) → (𝐵 ≠ 0 ↔ (𝑧 / 𝑤) ≠ 0))
93	91, 92	bi2anan9 606	. . . . . . . . . 10 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) ↔ ((𝑥 / 𝑦) ≠ 0 ∧ (𝑧 / 𝑤) ≠ 0)))
94		oveq12 5952	. . . . . . . . . . . 12 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝐴 · 𝐵) = ((𝑥 / 𝑦) · (𝑧 / 𝑤)))
95	94	oveq2d 5959	. . . . . . . . . . 11 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt (𝐴 · 𝐵)) = (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))))
96		oveq2 5951	. . . . . . . . . . . 12 ⊢ (𝐴 = (𝑥 / 𝑦) → (𝑃 pCnt 𝐴) = (𝑃 pCnt (𝑥 / 𝑦)))
97		oveq2 5951	. . . . . . . . . . . 12 ⊢ (𝐵 = (𝑧 / 𝑤) → (𝑃 pCnt 𝐵) = (𝑃 pCnt (𝑧 / 𝑤)))
98	96, 97	oveqan12d 5962	. . . . . . . . . . 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 2219	. . . . . . . . . 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 2630	. . . . 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 2630	. . 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 980 = wceq 1372 ∈ wcel 2175 ≠ wne 2375 ∃wrex 2484 class class class wbr 4043 (class class class)co 5943 ℂcc 7922 0cc0 7924 + caddc 7927 · cmul 7929 − cmin 8242 # cap 8653 / cdiv 8744 ℕcn 9035 ℕ₀cn0 9294 ℤcz 9371 ℚcq 9739 ℙcprime 12371 pCnt cpc 12549

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 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-iinf 4635 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-mulrcl 8023 ax-addcom 8024 ax-mulcom 8025 ax-addass 8026 ax-mulass 8027 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-1rid 8031 ax-0id 8032 ax-rnegex 8033 ax-precex 8034 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040 ax-pre-mulgt0 8041 ax-pre-mulext 8042 ax-arch 8043 ax-caucvg 8044

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4339 df-po 4342 df-iso 4343 df-iord 4412 df-on 4414 df-ilim 4415 df-suc 4417 df-iom 4638 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-isom 5279 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-1st 6225 df-2nd 6226 df-recs 6390 df-frec 6476 df-1o 6501 df-2o 6502 df-er 6619 df-en 6827 df-sup 7085 df-inf 7086 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-reap 8647 df-ap 8654 df-div 8745 df-inn 9036 df-2 9094 df-3 9095 df-4 9096 df-n0 9295 df-z 9372 df-uz 9648 df-q 9740 df-rp 9775 df-fz 10130 df-fzo 10264 df-fl 10411 df-mod 10466 df-seqfrec 10591 df-exp 10682 df-cj 11095 df-re 11096 df-im 11097 df-rsqrt 11251 df-abs 11252 df-dvds 12041 df-gcd 12217 df-prm 12372 df-pc 12550

This theorem is referenced by: pcqdiv 12572 pcexp 12574 pcaddlem 12604

W3C validator