![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pltfval | Structured version Visualization version GIF version |
Description: Value of the less-than relation. (Contributed by Mario Carneiro, 8-Feb-2015.) |
Ref | Expression |
---|---|
pltval.l | β’ β€ = (leβπΎ) |
pltval.s | β’ < = (ltβπΎ) |
Ref | Expression |
---|---|
pltfval | β’ (πΎ β π΄ β < = ( β€ β I )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pltval.s | . 2 β’ < = (ltβπΎ) | |
2 | elex 3491 | . . 3 β’ (πΎ β π΄ β πΎ β V) | |
3 | fveq2 6890 | . . . . . 6 β’ (π = πΎ β (leβπ) = (leβπΎ)) | |
4 | pltval.l | . . . . . 6 β’ β€ = (leβπΎ) | |
5 | 3, 4 | eqtr4di 2788 | . . . . 5 β’ (π = πΎ β (leβπ) = β€ ) |
6 | 5 | difeq1d 4120 | . . . 4 β’ (π = πΎ β ((leβπ) β I ) = ( β€ β I )) |
7 | df-plt 18287 | . . . 4 β’ lt = (π β V β¦ ((leβπ) β I )) | |
8 | 4 | fvexi 6904 | . . . . 5 β’ β€ β V |
9 | 8 | difexi 5327 | . . . 4 β’ ( β€ β I ) β V |
10 | 6, 7, 9 | fvmpt 6997 | . . 3 β’ (πΎ β V β (ltβπΎ) = ( β€ β I )) |
11 | 2, 10 | syl 17 | . 2 β’ (πΎ β π΄ β (ltβπΎ) = ( β€ β I )) |
12 | 1, 11 | eqtrid 2782 | 1 β’ (πΎ β π΄ β < = ( β€ β I )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 Vcvv 3472 β cdif 3944 I cid 5572 βcfv 6542 lecple 17208 ltcplt 18265 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-plt 18287 |
This theorem is referenced by: pltval 18289 relt 21387 opsrtoslem2 21836 oppglt 32399 xrslt 32444 submarchi 32602 |
Copyright terms: Public domain | W3C validator |