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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 3465 | . . 3 β’ (πΎ β π΄ β πΎ β V) | |
3 | fveq2 6846 | . . . . . 6 β’ (π = πΎ β (leβπ) = (leβπΎ)) | |
4 | pltval.l | . . . . . 6 β’ β€ = (leβπΎ) | |
5 | 3, 4 | eqtr4di 2791 | . . . . 5 β’ (π = πΎ β (leβπ) = β€ ) |
6 | 5 | difeq1d 4085 | . . . 4 β’ (π = πΎ β ((leβπ) β I ) = ( β€ β I )) |
7 | df-plt 18227 | . . . 4 β’ lt = (π β V β¦ ((leβπ) β I )) | |
8 | 4 | fvexi 6860 | . . . . 5 β’ β€ β V |
9 | 8 | difexi 5289 | . . . 4 β’ ( β€ β I ) β V |
10 | 6, 7, 9 | fvmpt 6952 | . . 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 2785 | 1 β’ (πΎ β π΄ β < = ( β€ β I )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3447 β cdif 3911 I cid 5534 βcfv 6500 lecple 17148 ltcplt 18205 |
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-pr 5388 |
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-sn 4591 df-pr 4593 df-op 4597 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-iota 6452 df-fun 6502 df-fv 6508 df-plt 18227 |
This theorem is referenced by: pltval 18229 relt 21042 opsrtoslem2 21486 oppglt 31878 xrslt 31923 submarchi 32078 |
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