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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 6887 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇)) | |
3 | mthmval.r | . . . . . . 7 ⊢ 𝑅 = (mStRed‘𝑇) | |
4 | 2, 3 | eqtr4di 2791 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅) |
5 | 4 | cnveqd 5872 | . . . . 5 ⊢ (𝑡 = 𝑇 → ◡(mStRed‘𝑡) = ◡𝑅) |
6 | fveq2 6887 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = (mPPSt‘𝑇)) | |
7 | mthmval.j | . . . . . . 7 ⊢ 𝐽 = (mPPSt‘𝑇) | |
8 | 6, 7 | eqtr4di 2791 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = 𝐽) |
9 | 4, 8 | imaeq12d 6057 | . . . . 5 ⊢ (𝑡 = 𝑇 → ((mStRed‘𝑡) “ (mPPSt‘𝑡)) = (𝑅 “ 𝐽)) |
10 | 5, 9 | imaeq12d 6057 | . . . 4 ⊢ (𝑡 = 𝑇 → (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) = (◡𝑅 “ (𝑅 “ 𝐽))) |
11 | df-mthm 34427 | . . . 4 ⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡)))) | |
12 | fvex 6900 | . . . . . 6 ⊢ (mStRed‘𝑡) ∈ V | |
13 | 12 | cnvex 7910 | . . . . 5 ⊢ ◡(mStRed‘𝑡) ∈ V |
14 | imaexg 7900 | . . . . 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 6998 | . . 3 ⊢ (𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽))) |
17 | 0ima 6073 | . . . . 5 ⊢ (∅ “ (𝑅 “ 𝐽)) = ∅ | |
18 | 17 | eqcomi 2742 | . . . 4 ⊢ ∅ = (∅ “ (𝑅 “ 𝐽)) |
19 | fvprc 6879 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = ∅) | |
20 | fvprc 6879 | . . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅) | |
21 | 3, 20 | eqtrid 2785 | . . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅) |
22 | 21 | cnveqd 5872 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ◡∅) |
23 | cnv0 6136 | . . . . . 6 ⊢ ◡∅ = ∅ | |
24 | 22, 23 | eqtrdi 2789 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ∅) |
25 | 24 | imaeq1d 6055 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (◡𝑅 “ (𝑅 “ 𝐽)) = (∅ “ (𝑅 “ 𝐽))) |
26 | 18, 19, 25 | 3eqtr4a 2799 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽))) |
27 | 16, 26 | pm2.61i 182 | . 2 ⊢ (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽)) |
28 | 1, 27 | eqtri 2761 | 1 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∅c0 4320 ◡ccnv 5673 “ cima 5677 ‘cfv 6539 mStRedcmsr 34402 mPPStcmpps 34406 mThmcmthm 34407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fv 6547 df-mthm 34427 |
This theorem is referenced by: elmthm 34504 mthmsta 34506 mthmblem 34508 |
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