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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoidifhspval3 | Structured version Visualization version GIF version |
Description: π· is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
hoidifhspval3.d | β’ π· = (π₯ β β β¦ (π β (β βm π) β¦ (π β π β¦ if(π = πΎ, if(π₯ β€ (πβπ), (πβπ), π₯), (πβπ))))) |
hoidifhspval3.y | β’ (π β π β β) |
hoidifhspval3.x | β’ (π β π β π) |
hoidifhspval3.a | β’ (π β π΄:πβΆβ) |
hoidifhspval3.j | β’ (π β π½ β π) |
Ref | Expression |
---|---|
hoidifhspval3 | β’ (π β (((π·βπ)βπ΄)βπ½) = if(π½ = πΎ, if(π β€ (π΄βπ½), (π΄βπ½), π), (π΄βπ½))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoidifhspval3.d | . . 3 β’ π· = (π₯ β β β¦ (π β (β βm π) β¦ (π β π β¦ if(π = πΎ, if(π₯ β€ (πβπ), (πβπ), π₯), (πβπ))))) | |
2 | hoidifhspval3.y | . . 3 β’ (π β π β β) | |
3 | hoidifhspval3.x | . . 3 β’ (π β π β π) | |
4 | hoidifhspval3.a | . . 3 β’ (π β π΄:πβΆβ) | |
5 | 1, 2, 3, 4 | hoidifhspval2 45009 | . 2 β’ (π β ((π·βπ)βπ΄) = (π β π β¦ if(π = πΎ, if(π β€ (π΄βπ), (π΄βπ), π), (π΄βπ)))) |
6 | eqeq1 2735 | . . . 4 β’ (π = π½ β (π = πΎ β π½ = πΎ)) | |
7 | fveq2 6862 | . . . . . 6 β’ (π = π½ β (π΄βπ) = (π΄βπ½)) | |
8 | 7 | breq2d 5137 | . . . . 5 β’ (π = π½ β (π β€ (π΄βπ) β π β€ (π΄βπ½))) |
9 | 8, 7 | ifbieq1d 4530 | . . . 4 β’ (π = π½ β if(π β€ (π΄βπ), (π΄βπ), π) = if(π β€ (π΄βπ½), (π΄βπ½), π)) |
10 | 6, 9, 7 | ifbieq12d 4534 | . . 3 β’ (π = π½ β if(π = πΎ, if(π β€ (π΄βπ), (π΄βπ), π), (π΄βπ)) = if(π½ = πΎ, if(π β€ (π΄βπ½), (π΄βπ½), π), (π΄βπ½))) |
11 | 10 | adantl 482 | . 2 β’ ((π β§ π = π½) β if(π = πΎ, if(π β€ (π΄βπ), (π΄βπ), π), (π΄βπ)) = if(π½ = πΎ, if(π β€ (π΄βπ½), (π΄βπ½), π), (π΄βπ½))) |
12 | hoidifhspval3.j | . 2 β’ (π β π½ β π) | |
13 | fvexd 6877 | . . . 4 β’ (π β (π΄βπ½) β V) | |
14 | 2 | elexd 3479 | . . . 4 β’ (π β π β V) |
15 | 13, 14 | ifcld 4552 | . . 3 β’ (π β if(π β€ (π΄βπ½), (π΄βπ½), π) β V) |
16 | 15, 13 | ifcld 4552 | . 2 β’ (π β if(π½ = πΎ, if(π β€ (π΄βπ½), (π΄βπ½), π), (π΄βπ½)) β V) |
17 | 5, 11, 12, 16 | fvmptd 6975 | 1 β’ (π β (((π·βπ)βπ΄)βπ½) = if(π½ = πΎ, if(π β€ (π΄βπ½), (π΄βπ½), π), (π΄βπ½))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 Vcvv 3459 ifcif 4506 class class class wbr 5125 β¦ cmpt 5208 βΆwf 6512 βcfv 6516 (class class class)co 7377 βm cmap 8787 βcr 11074 β€ cle 11214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-ov 7380 df-oprab 7381 df-mpo 7382 df-map 8789 |
This theorem is referenced by: hoidifhspdmvle 45014 hspmbllem1 45020 hspmbllem2 45021 |
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