| Step | Hyp | Ref
| Expression |
| 1 | | wemaplem3.px |
. . 3
⊢ (𝜑 → 𝑃𝑇𝑋) |
| 2 | | wemaplem2.p |
. . . 4
⊢ (𝜑 → 𝑃 ∈ (𝐵 ↑m 𝐴)) |
| 3 | | wemaplem2.x |
. . . 4
⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m 𝐴)) |
| 4 | | wemapso.t |
. . . . 5
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} |
| 5 | 4 | wemaplem1 9586 |
. . . 4
⊢ ((𝑃 ∈ (𝐵 ↑m 𝐴) ∧ 𝑋 ∈ (𝐵 ↑m 𝐴)) → (𝑃𝑇𝑋 ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) |
| 6 | 2, 3, 5 | syl2anc 584 |
. . 3
⊢ (𝜑 → (𝑃𝑇𝑋 ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) |
| 7 | 1, 6 | mpbid 232 |
. 2
⊢ (𝜑 → ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))) |
| 8 | | wemaplem3.xq |
. . 3
⊢ (𝜑 → 𝑋𝑇𝑄) |
| 9 | | wemaplem2.q |
. . . 4
⊢ (𝜑 → 𝑄 ∈ (𝐵 ↑m 𝐴)) |
| 10 | 4 | wemaplem1 9586 |
. . . 4
⊢ ((𝑋 ∈ (𝐵 ↑m 𝐴) ∧ 𝑄 ∈ (𝐵 ↑m 𝐴)) → (𝑋𝑇𝑄 ↔ ∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) |
| 11 | 3, 9, 10 | syl2anc 584 |
. . 3
⊢ (𝜑 → (𝑋𝑇𝑄 ↔ ∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) |
| 12 | 8, 11 | mpbid 232 |
. 2
⊢ (𝜑 → ∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)))) |
| 13 | 2 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑃 ∈ (𝐵 ↑m 𝐴)) |
| 14 | 3 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑋 ∈ (𝐵 ↑m 𝐴)) |
| 15 | 9 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑄 ∈ (𝐵 ↑m 𝐴)) |
| 16 | | wemaplem2.r |
. . . . . 6
⊢ (𝜑 → 𝑅 Or 𝐴) |
| 17 | 16 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑅 Or 𝐴) |
| 18 | | wemaplem2.s |
. . . . . 6
⊢ (𝜑 → 𝑆 Po 𝐵) |
| 19 | 18 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑆 Po 𝐵) |
| 20 | | simplrl 777 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑎 ∈ 𝐴) |
| 21 | | simp2rl 1243 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → (𝑃‘𝑎)𝑆(𝑋‘𝑎)) |
| 22 | 21 | 3expa 1119 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → (𝑃‘𝑎)𝑆(𝑋‘𝑎)) |
| 23 | | simprr 773 |
. . . . . 6
⊢ ((𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))) → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))) |
| 24 | 23 | ad2antlr 727 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))) |
| 25 | | simprl 771 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑏 ∈ 𝐴) |
| 26 | | simprrl 781 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → (𝑋‘𝑏)𝑆(𝑄‘𝑏)) |
| 27 | | simprrr 782 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) |
| 28 | 4, 13, 14, 15, 17, 19, 20, 22, 24, 25, 26, 27 | wemaplem2 9587 |
. . . 4
⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑃𝑇𝑄) |
| 29 | 28 | rexlimdvaa 3156 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) → (∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) → 𝑃𝑇𝑄)) |
| 30 | 29 | rexlimdvaa 3156 |
. 2
⊢ (𝜑 → (∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))) → (∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) → 𝑃𝑇𝑄))) |
| 31 | 7, 12, 30 | mp2d 49 |
1
⊢ (𝜑 → 𝑃𝑇𝑄) |