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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 6673 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇)) | |
3 | mthmval.r | . . . . . . 7 ⊢ 𝑅 = (mStRed‘𝑇) | |
4 | 2, 3 | syl6eqr 2877 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅) |
5 | 4 | cnveqd 5749 | . . . . 5 ⊢ (𝑡 = 𝑇 → ◡(mStRed‘𝑡) = ◡𝑅) |
6 | fveq2 6673 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = (mPPSt‘𝑇)) | |
7 | mthmval.j | . . . . . . 7 ⊢ 𝐽 = (mPPSt‘𝑇) | |
8 | 6, 7 | syl6eqr 2877 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = 𝐽) |
9 | 4, 8 | imaeq12d 5933 | . . . . 5 ⊢ (𝑡 = 𝑇 → ((mStRed‘𝑡) “ (mPPSt‘𝑡)) = (𝑅 “ 𝐽)) |
10 | 5, 9 | imaeq12d 5933 | . . . 4 ⊢ (𝑡 = 𝑇 → (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) = (◡𝑅 “ (𝑅 “ 𝐽))) |
11 | df-mthm 32750 | . . . 4 ⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡)))) | |
12 | fvex 6686 | . . . . . 6 ⊢ (mStRed‘𝑡) ∈ V | |
13 | 12 | cnvex 7633 | . . . . 5 ⊢ ◡(mStRed‘𝑡) ∈ V |
14 | imaexg 7623 | . . . . 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 6776 | . . 3 ⊢ (𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽))) |
17 | 0ima 5949 | . . . . 5 ⊢ (∅ “ (𝑅 “ 𝐽)) = ∅ | |
18 | 17 | eqcomi 2833 | . . . 4 ⊢ ∅ = (∅ “ (𝑅 “ 𝐽)) |
19 | fvprc 6666 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = ∅) | |
20 | fvprc 6666 | . . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅) | |
21 | 3, 20 | syl5eq 2871 | . . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅) |
22 | 21 | cnveqd 5749 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ◡∅) |
23 | cnv0 6002 | . . . . . 6 ⊢ ◡∅ = ∅ | |
24 | 22, 23 | syl6eq 2875 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ∅) |
25 | 24 | imaeq1d 5931 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (◡𝑅 “ (𝑅 “ 𝐽)) = (∅ “ (𝑅 “ 𝐽))) |
26 | 18, 19, 25 | 3eqtr4a 2885 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽))) |
27 | 16, 26 | pm2.61i 184 | . 2 ⊢ (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽)) |
28 | 1, 27 | eqtri 2847 | 1 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∅c0 4294 ◡ccnv 5557 “ cima 5561 ‘cfv 6358 mStRedcmsr 32725 mPPStcmpps 32729 mThmcmthm 32730 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fv 6366 df-mthm 32750 |
This theorem is referenced by: elmthm 32827 mthmsta 32829 mthmblem 32831 |
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