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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mpst123 | Structured version Visualization version GIF version |
Description: Decompose a pre-statement into a triple of values. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mpstssv.p | β’ π = (mPreStβπ) |
Ref | Expression |
---|---|
mpst123 | β’ (π β π β π = β¨(1st β(1st βπ)), (2nd β(1st βπ)), (2nd βπ)β©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpstssv.p | . . . 4 β’ π = (mPreStβπ) | |
2 | 1 | mpstssv 34197 | . . 3 β’ π β ((V Γ V) Γ V) |
3 | 2 | sseli 3944 | . 2 β’ (π β π β π β ((V Γ V) Γ V)) |
4 | 1st2nd2 7964 | . . . 4 β’ (π β ((V Γ V) Γ V) β π = β¨(1st βπ), (2nd βπ)β©) | |
5 | xp1st 7957 | . . . . . 6 β’ (π β ((V Γ V) Γ V) β (1st βπ) β (V Γ V)) | |
6 | 1st2nd2 7964 | . . . . . 6 β’ ((1st βπ) β (V Γ V) β (1st βπ) = β¨(1st β(1st βπ)), (2nd β(1st βπ))β©) | |
7 | 5, 6 | syl 17 | . . . . 5 β’ (π β ((V Γ V) Γ V) β (1st βπ) = β¨(1st β(1st βπ)), (2nd β(1st βπ))β©) |
8 | 7 | opeq1d 4840 | . . . 4 β’ (π β ((V Γ V) Γ V) β β¨(1st βπ), (2nd βπ)β© = β¨β¨(1st β(1st βπ)), (2nd β(1st βπ))β©, (2nd βπ)β©) |
9 | 4, 8 | eqtrd 2773 | . . 3 β’ (π β ((V Γ V) Γ V) β π = β¨β¨(1st β(1st βπ)), (2nd β(1st βπ))β©, (2nd βπ)β©) |
10 | df-ot 4599 | . . 3 β’ β¨(1st β(1st βπ)), (2nd β(1st βπ)), (2nd βπ)β© = β¨β¨(1st β(1st βπ)), (2nd β(1st βπ))β©, (2nd βπ)β© | |
11 | 9, 10 | eqtr4di 2791 | . 2 β’ (π β ((V Γ V) Γ V) β π = β¨(1st β(1st βπ)), (2nd β(1st βπ)), (2nd βπ)β©) |
12 | 3, 11 | syl 17 | 1 β’ (π β π β π = β¨(1st β(1st βπ)), (2nd β(1st βπ)), (2nd βπ)β©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3447 β¨cop 4596 β¨cotp 4598 Γ cxp 5635 βcfv 6500 1st c1st 7923 2nd c2nd 7924 mPreStcmpst 34131 |
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 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-ot 4599 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-iota 6452 df-fun 6502 df-fv 6508 df-1st 7925 df-2nd 7926 df-mpst 34151 |
This theorem is referenced by: msrf 34200 msrid 34203 mthmpps 34240 |
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