Proof of Theorem pcadd2
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | pcadd2.1 | . . 3
⊢ (𝜑 → 𝑃 ∈ ℙ) | 
| 2 |  | pcadd2.2 | . . 3
⊢ (𝜑 → 𝐴 ∈ ℚ) | 
| 3 |  | pcxcl 16899 | . . 3
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑃 pCnt 𝐴) ∈
ℝ*) | 
| 4 | 1, 2, 3 | syl2anc 584 | . 2
⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈
ℝ*) | 
| 5 |  | pcadd2.3 | . . . 4
⊢ (𝜑 → 𝐵 ∈ ℚ) | 
| 6 |  | qaddcl 13007 | . . . 4
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ) | 
| 7 | 2, 5, 6 | syl2anc 584 | . . 3
⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℚ) | 
| 8 |  | pcxcl 16899 | . . 3
⊢ ((𝑃 ∈ ℙ ∧ (𝐴 + 𝐵) ∈ ℚ) → (𝑃 pCnt (𝐴 + 𝐵)) ∈
ℝ*) | 
| 9 | 1, 7, 8 | syl2anc 584 | . 2
⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ∈
ℝ*) | 
| 10 |  | pcxcl 16899 | . . . . 5
⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑃 pCnt 𝐵) ∈
ℝ*) | 
| 11 | 1, 5, 10 | syl2anc 584 | . . . 4
⊢ (𝜑 → (𝑃 pCnt 𝐵) ∈
ℝ*) | 
| 12 |  | pcadd2.4 | . . . 4
⊢ (𝜑 → (𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵)) | 
| 13 | 4, 11, 12 | xrltled 13192 | . . 3
⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) | 
| 14 | 1, 2, 5, 13 | pcadd 16927 | . 2
⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵))) | 
| 15 |  | qnegcl 13008 | . . . . 5
⊢ (𝐵 ∈ ℚ → -𝐵 ∈
ℚ) | 
| 16 | 5, 15 | syl 17 | . . . 4
⊢ (𝜑 → -𝐵 ∈ ℚ) | 
| 17 |  | xrltnle 11328 | . . . . . . . . . 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 584 | . . . . . . . . 9
⊢ (𝜑 → ((𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))) | 
| 19 | 12, 18 | mpbid 232 | . . . . . . . 8
⊢ (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)) | 
| 20 | 1 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → 𝑃 ∈ ℙ) | 
| 21 | 16 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → -𝐵 ∈ ℚ) | 
| 22 | 7 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝐴 + 𝐵) ∈ ℚ) | 
| 23 |  | pcneg 16912 | . . . . . . . . . . . . . 14
⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑃 pCnt -𝐵) = (𝑃 pCnt 𝐵)) | 
| 24 | 1, 5, 23 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ (𝜑 → (𝑃 pCnt -𝐵) = (𝑃 pCnt 𝐵)) | 
| 25 | 24 | breq1d 5153 | . . . . . . . . . . . 12
⊢ (𝜑 → ((𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) ↔ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))) | 
| 26 | 25 | biimpar 477 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) | 
| 27 | 20, 21, 22, 26 | pcadd 16927 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵)))) | 
| 28 | 27 | ex 412 | . . . . . . . . 9
⊢ (𝜑 → ((𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))))) | 
| 29 |  | qcn 13005 | . . . . . . . . . . . . . . 15
⊢ (𝐵 ∈ ℚ → 𝐵 ∈
ℂ) | 
| 30 | 5, 29 | syl 17 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐵 ∈ ℂ) | 
| 31 | 30 | negcld 11607 | . . . . . . . . . . . . 13
⊢ (𝜑 → -𝐵 ∈ ℂ) | 
| 32 |  | qcn 13005 | . . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℚ → 𝐴 ∈
ℂ) | 
| 33 | 2, 32 | syl 17 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝐴 ∈ ℂ) | 
| 34 | 31, 33, 30 | add12d 11488 | . . . . . . . . . . . 12
⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = (𝐴 + (-𝐵 + 𝐵))) | 
| 35 | 31, 30 | addcomd 11463 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (-𝐵 + 𝐵) = (𝐵 + -𝐵)) | 
| 36 | 30 | negidd 11610 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐵 + -𝐵) = 0) | 
| 37 | 35, 36 | eqtrd 2777 | . . . . . . . . . . . . 13
⊢ (𝜑 → (-𝐵 + 𝐵) = 0) | 
| 38 | 37 | oveq2d 7447 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝐴 + (-𝐵 + 𝐵)) = (𝐴 + 0)) | 
| 39 | 33 | addridd 11461 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝐴 + 0) = 𝐴) | 
| 40 | 34, 38, 39 | 3eqtrd 2781 | . . . . . . . . . . 11
⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = 𝐴) | 
| 41 | 40 | oveq2d 7447 | . . . . . . . . . 10
⊢ (𝜑 → (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) = (𝑃 pCnt 𝐴)) | 
| 42 | 24, 41 | breq12d 5156 | . . . . . . . . 9
⊢ (𝜑 → ((𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) ↔ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))) | 
| 43 | 28, 42 | sylibd 239 | . . . . . . . 8
⊢ (𝜑 → ((𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) → (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))) | 
| 44 | 19, 43 | mtod 198 | . . . . . . 7
⊢ (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) | 
| 45 |  | xrltnle 11328 | . . . . . . . 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 584 | . . . . . . 7
⊢ (𝜑 → ((𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))) | 
| 47 | 44, 46 | mpbird 257 | . . . . . 6
⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵)) | 
| 48 | 9, 11, 47 | xrltled 13192 | . . . . 5
⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐵)) | 
| 49 | 48, 24 | breqtrrd 5171 | . . . 4
⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt -𝐵)) | 
| 50 | 1, 7, 16, 49 | pcadd 16927 | . . 3
⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵))) | 
| 51 | 33, 30, 31 | addassd 11283 | . . . . 5
⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = (𝐴 + (𝐵 + -𝐵))) | 
| 52 | 36 | oveq2d 7447 | . . . . 5
⊢ (𝜑 → (𝐴 + (𝐵 + -𝐵)) = (𝐴 + 0)) | 
| 53 | 51, 52, 39 | 3eqtrd 2781 | . . . 4
⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = 𝐴) | 
| 54 | 53 | oveq2d 7447 | . . 3
⊢ (𝜑 → (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)) = (𝑃 pCnt 𝐴)) | 
| 55 | 50, 54 | breqtrd 5169 | . 2
⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐴)) | 
| 56 | 4, 9, 14, 55 | xrletrid 13197 | 1
⊢ (𝜑 → (𝑃 pCnt 𝐴) = (𝑃 pCnt (𝐴 + 𝐵))) |