Proof of Theorem pcadd2
Step | Hyp | Ref
| Expression |
1 | | pcadd2.1 |
. . 3
⊢ (𝜑 → 𝑃 ∈ ℙ) |
2 | | pcadd2.2 |
. . 3
⊢ (𝜑 → 𝐴 ∈ ℚ) |
3 | | pcxcl 16414 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑃 pCnt 𝐴) ∈
ℝ*) |
4 | 1, 2, 3 | syl2anc 587 |
. 2
⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈
ℝ*) |
5 | | pcadd2.3 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ ℚ) |
6 | | qaddcl 12561 |
. . . 4
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ) |
7 | 2, 5, 6 | syl2anc 587 |
. . 3
⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℚ) |
8 | | pcxcl 16414 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧ (𝐴 + 𝐵) ∈ ℚ) → (𝑃 pCnt (𝐴 + 𝐵)) ∈
ℝ*) |
9 | 1, 7, 8 | syl2anc 587 |
. 2
⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ∈
ℝ*) |
10 | | pcxcl 16414 |
. . . . 5
⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑃 pCnt 𝐵) ∈
ℝ*) |
11 | 1, 5, 10 | syl2anc 587 |
. . . 4
⊢ (𝜑 → (𝑃 pCnt 𝐵) ∈
ℝ*) |
12 | | pcadd2.4 |
. . . 4
⊢ (𝜑 → (𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵)) |
13 | 4, 11, 12 | xrltled 12740 |
. . 3
⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) |
14 | 1, 2, 5, 13 | pcadd 16442 |
. 2
⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵))) |
15 | | qnegcl 12562 |
. . . . 5
⊢ (𝐵 ∈ ℚ → -𝐵 ∈
ℚ) |
16 | 5, 15 | syl 17 |
. . . 4
⊢ (𝜑 → -𝐵 ∈ ℚ) |
17 | | xrltnle 10900 |
. . . . . . . . . 10
⊢ (((𝑃 pCnt 𝐴) ∈ ℝ* ∧ (𝑃 pCnt 𝐵) ∈ ℝ*) → ((𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))) |
18 | 4, 11, 17 | syl2anc 587 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))) |
19 | 12, 18 | mpbid 235 |
. . . . . . . 8
⊢ (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)) |
20 | 1 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → 𝑃 ∈ ℙ) |
21 | 16 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → -𝐵 ∈ ℚ) |
22 | 7 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝐴 + 𝐵) ∈ ℚ) |
23 | | pcneg 16427 |
. . . . . . . . . . . . . 14
⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑃 pCnt -𝐵) = (𝑃 pCnt 𝐵)) |
24 | 1, 5, 23 | syl2anc 587 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑃 pCnt -𝐵) = (𝑃 pCnt 𝐵)) |
25 | 24 | breq1d 5063 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) ↔ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))) |
26 | 25 | biimpar 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) |
27 | 20, 21, 22, 26 | pcadd 16442 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵)))) |
28 | 27 | ex 416 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))))) |
29 | | qcn 12559 |
. . . . . . . . . . . . . . 15
⊢ (𝐵 ∈ ℚ → 𝐵 ∈
ℂ) |
30 | 5, 29 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐵 ∈ ℂ) |
31 | 30 | negcld 11176 |
. . . . . . . . . . . . 13
⊢ (𝜑 → -𝐵 ∈ ℂ) |
32 | | qcn 12559 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℚ → 𝐴 ∈
ℂ) |
33 | 2, 32 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐴 ∈ ℂ) |
34 | 31, 33, 30 | add12d 11058 |
. . . . . . . . . . . 12
⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = (𝐴 + (-𝐵 + 𝐵))) |
35 | 31, 30 | addcomd 11034 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (-𝐵 + 𝐵) = (𝐵 + -𝐵)) |
36 | 30 | negidd 11179 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐵 + -𝐵) = 0) |
37 | 35, 36 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (-𝐵 + 𝐵) = 0) |
38 | 37 | oveq2d 7229 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐴 + (-𝐵 + 𝐵)) = (𝐴 + 0)) |
39 | 33 | addid1d 11032 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐴 + 0) = 𝐴) |
40 | 34, 38, 39 | 3eqtrd 2781 |
. . . . . . . . . . 11
⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = 𝐴) |
41 | 40 | oveq2d 7229 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) = (𝑃 pCnt 𝐴)) |
42 | 24, 41 | breq12d 5066 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) ↔ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))) |
43 | 28, 42 | sylibd 242 |
. . . . . . . 8
⊢ (𝜑 → ((𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) → (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))) |
44 | 19, 43 | mtod 201 |
. . . . . . 7
⊢ (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) |
45 | | xrltnle 10900 |
. . . . . . . 8
⊢ (((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℝ* ∧ (𝑃 pCnt 𝐵) ∈ ℝ*) → ((𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))) |
46 | 9, 11, 45 | syl2anc 587 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))) |
47 | 44, 46 | mpbird 260 |
. . . . . 6
⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵)) |
48 | 9, 11, 47 | xrltled 12740 |
. . . . 5
⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐵)) |
49 | 48, 24 | breqtrrd 5081 |
. . . 4
⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt -𝐵)) |
50 | 1, 7, 16, 49 | pcadd 16442 |
. . 3
⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵))) |
51 | 33, 30, 31 | addassd 10855 |
. . . . 5
⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = (𝐴 + (𝐵 + -𝐵))) |
52 | 36 | oveq2d 7229 |
. . . . 5
⊢ (𝜑 → (𝐴 + (𝐵 + -𝐵)) = (𝐴 + 0)) |
53 | 51, 52, 39 | 3eqtrd 2781 |
. . . 4
⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = 𝐴) |
54 | 53 | oveq2d 7229 |
. . 3
⊢ (𝜑 → (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)) = (𝑃 pCnt 𝐴)) |
55 | 50, 54 | breqtrd 5079 |
. 2
⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐴)) |
56 | 4, 9, 14, 55 | xrletrid 12745 |
1
⊢ (𝜑 → (𝑃 pCnt 𝐴) = (𝑃 pCnt (𝐴 + 𝐵))) |