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Mirrors > Home > MPE Home > Th. List > metres2 | Structured version Visualization version GIF version |
Description: Lemma for metres 24091. (Contributed by FL, 12-Oct-2006.) (Proof shortened by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
metres2 | β’ ((π· β (Metβπ) β§ π β π) β (π· βΎ (π Γ π )) β (Metβπ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metxmet 24060 | . . 3 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
2 | xmetres2 24087 | . . 3 β’ ((π· β (βMetβπ) β§ π β π) β (π· βΎ (π Γ π )) β (βMetβπ )) | |
3 | 1, 2 | sylan 580 | . 2 β’ ((π· β (Metβπ) β§ π β π) β (π· βΎ (π Γ π )) β (βMetβπ )) |
4 | metf 24056 | . . . 4 β’ (π· β (Metβπ) β π·:(π Γ π)βΆβ) | |
5 | 4 | adantr 481 | . . 3 β’ ((π· β (Metβπ) β§ π β π) β π·:(π Γ π)βΆβ) |
6 | simpr 485 | . . . 4 β’ ((π· β (Metβπ) β§ π β π) β π β π) | |
7 | xpss12 5691 | . . . 4 β’ ((π β π β§ π β π) β (π Γ π ) β (π Γ π)) | |
8 | 6, 7 | sylancom 588 | . . 3 β’ ((π· β (Metβπ) β§ π β π) β (π Γ π ) β (π Γ π)) |
9 | 5, 8 | fssresd 6758 | . 2 β’ ((π· β (Metβπ) β§ π β π) β (π· βΎ (π Γ π )):(π Γ π )βΆβ) |
10 | ismet2 24059 | . 2 β’ ((π· βΎ (π Γ π )) β (Metβπ ) β ((π· βΎ (π Γ π )) β (βMetβπ ) β§ (π· βΎ (π Γ π )):(π Γ π )βΆβ)) | |
11 | 3, 9, 10 | sylanbrc 583 | 1 β’ ((π· β (Metβπ) β§ π β π) β (π· βΎ (π Γ π )) β (Metβπ )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β wcel 2106 β wss 3948 Γ cxp 5674 βΎ cres 5678 βΆwf 6539 βcfv 6543 βcr 11111 βMetcxmet 21129 Metcmet 21130 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-mulcl 11174 ax-i2m1 11180 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-xadd 13097 df-xmet 21137 df-met 21138 |
This theorem is referenced by: metres 24091 xpsmet 24108 tmsms 24216 imasf1oms 24219 prdsms 24260 remet 24526 lebnumii 24706 cmetss 25057 sstotbnd2 36945 bndss 36957 equivbnd2 36963 rrnheibor 37008 iccbnd 37011 |
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