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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 6332 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇)) | |
3 | mthmval.r | . . . . . . 7 ⊢ 𝑅 = (mStRed‘𝑇) | |
4 | 2, 3 | syl6eqr 2822 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅) |
5 | 4 | cnveqd 5436 | . . . . 5 ⊢ (𝑡 = 𝑇 → ◡(mStRed‘𝑡) = ◡𝑅) |
6 | fveq2 6332 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = (mPPSt‘𝑇)) | |
7 | mthmval.j | . . . . . . 7 ⊢ 𝐽 = (mPPSt‘𝑇) | |
8 | 6, 7 | syl6eqr 2822 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = 𝐽) |
9 | 4, 8 | imaeq12d 5608 | . . . . 5 ⊢ (𝑡 = 𝑇 → ((mStRed‘𝑡) “ (mPPSt‘𝑡)) = (𝑅 “ 𝐽)) |
10 | 5, 9 | imaeq12d 5608 | . . . 4 ⊢ (𝑡 = 𝑇 → (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) = (◡𝑅 “ (𝑅 “ 𝐽))) |
11 | df-mthm 31728 | . . . 4 ⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡)))) | |
12 | fvex 6342 | . . . . . 6 ⊢ (mStRed‘𝑡) ∈ V | |
13 | 12 | cnvex 7259 | . . . . 5 ⊢ ◡(mStRed‘𝑡) ∈ V |
14 | imaexg 7249 | . . . . 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 6429 | . . 3 ⊢ (𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽))) |
17 | 0ima 5623 | . . . . 5 ⊢ (∅ “ (𝑅 “ 𝐽)) = ∅ | |
18 | 17 | eqcomi 2779 | . . . 4 ⊢ ∅ = (∅ “ (𝑅 “ 𝐽)) |
19 | fvprc 6326 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = ∅) | |
20 | fvprc 6326 | . . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅) | |
21 | 3, 20 | syl5eq 2816 | . . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅) |
22 | 21 | cnveqd 5436 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ◡∅) |
23 | cnv0 5676 | . . . . . 6 ⊢ ◡∅ = ∅ | |
24 | 22, 23 | syl6eq 2820 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ∅) |
25 | 24 | imaeq1d 5606 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (◡𝑅 “ (𝑅 “ 𝐽)) = (∅ “ (𝑅 “ 𝐽))) |
26 | 18, 19, 25 | 3eqtr4a 2830 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽))) |
27 | 16, 26 | pm2.61i 176 | . 2 ⊢ (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽)) |
28 | 1, 27 | eqtri 2792 | 1 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1630 ∈ wcel 2144 Vcvv 3349 ∅c0 4061 ◡ccnv 5248 “ cima 5252 ‘cfv 6031 mStRedcmsr 31703 mPPStcmpps 31707 mThmcmthm 31708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ral 3065 df-rex 3066 df-rab 3069 df-v 3351 df-sbc 3586 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fv 6039 df-mthm 31728 |
This theorem is referenced by: elmthm 31805 mthmsta 31807 mthmblem 31809 |
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