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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mthmval | Structured version Visualization version GIF version | ||
| Description: A theorem is a pre-statement, whose reduct is also the reduct of a provable pre-statement. Unlike the difference between pre-statement and statement, this application of the reduct is not necessarily trivial: there are theorems that are not themselves provable but are provable once enough "dummy variables" are introduced. (Contributed by Mario Carneiro, 18-Jul-2016.) |
| Ref | Expression |
|---|---|
| mthmval.r | ⊢ 𝑅 = (mStRed‘𝑇) |
| mthmval.j | ⊢ 𝐽 = (mPPSt‘𝑇) |
| mthmval.u | ⊢ 𝑈 = (mThm‘𝑇) |
| Ref | Expression |
|---|---|
| mthmval | ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mthmval.u | . 2 ⊢ 𝑈 = (mThm‘𝑇) | |
| 2 | fveq2 6906 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇)) | |
| 3 | mthmval.r | . . . . . . 7 ⊢ 𝑅 = (mStRed‘𝑇) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅) |
| 5 | 4 | cnveqd 5886 | . . . . 5 ⊢ (𝑡 = 𝑇 → ◡(mStRed‘𝑡) = ◡𝑅) |
| 6 | fveq2 6906 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = (mPPSt‘𝑇)) | |
| 7 | mthmval.j | . . . . . . 7 ⊢ 𝐽 = (mPPSt‘𝑇) | |
| 8 | 6, 7 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = 𝐽) |
| 9 | 4, 8 | imaeq12d 6079 | . . . . 5 ⊢ (𝑡 = 𝑇 → ((mStRed‘𝑡) “ (mPPSt‘𝑡)) = (𝑅 “ 𝐽)) |
| 10 | 5, 9 | imaeq12d 6079 | . . . 4 ⊢ (𝑡 = 𝑇 → (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) = (◡𝑅 “ (𝑅 “ 𝐽))) |
| 11 | df-mthm 35504 | . . . 4 ⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡)))) | |
| 12 | fvex 6919 | . . . . . 6 ⊢ (mStRed‘𝑡) ∈ V | |
| 13 | 12 | cnvex 7947 | . . . . 5 ⊢ ◡(mStRed‘𝑡) ∈ V |
| 14 | imaexg 7935 | . . . . 5 ⊢ (◡(mStRed‘𝑡) ∈ V → (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) ∈ V) | |
| 15 | 13, 14 | ax-mp 5 | . . . 4 ⊢ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) ∈ V |
| 16 | 10, 11, 15 | fvmpt3i 7021 | . . 3 ⊢ (𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽))) |
| 17 | 0ima 6096 | . . . . 5 ⊢ (∅ “ (𝑅 “ 𝐽)) = ∅ | |
| 18 | 17 | eqcomi 2746 | . . . 4 ⊢ ∅ = (∅ “ (𝑅 “ 𝐽)) |
| 19 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = ∅) | |
| 20 | fvprc 6898 | . . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅) | |
| 21 | 3, 20 | eqtrid 2789 | . . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅) |
| 22 | 21 | cnveqd 5886 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ◡∅) |
| 23 | cnv0 6160 | . . . . . 6 ⊢ ◡∅ = ∅ | |
| 24 | 22, 23 | eqtrdi 2793 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ∅) |
| 25 | 24 | imaeq1d 6077 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (◡𝑅 “ (𝑅 “ 𝐽)) = (∅ “ (𝑅 “ 𝐽))) |
| 26 | 18, 19, 25 | 3eqtr4a 2803 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽))) |
| 27 | 16, 26 | pm2.61i 182 | . 2 ⊢ (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽)) |
| 28 | 1, 27 | eqtri 2765 | 1 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ◡ccnv 5684 “ cima 5688 ‘cfv 6561 mStRedcmsr 35479 mPPStcmpps 35483 mThmcmthm 35484 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-mthm 35504 |
| This theorem is referenced by: elmthm 35581 mthmsta 35583 mthmblem 35585 |
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