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Mirrors > Home > ILE Home > Th. List > metres2 | GIF version |
Description: Lemma for metres 13745. (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 13717 | . . 3 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
2 | xmetres2 13741 | . . 3 β’ ((π· β (βMetβπ) β§ π β π) β (π· βΎ (π Γ π )) β (βMetβπ )) | |
3 | 1, 2 | sylan 283 | . 2 β’ ((π· β (Metβπ) β§ π β π) β (π· βΎ (π Γ π )) β (βMetβπ )) |
4 | metf 13713 | . . . 4 β’ (π· β (Metβπ) β π·:(π Γ π)βΆβ) | |
5 | 4 | adantr 276 | . . 3 β’ ((π· β (Metβπ) β§ π β π) β π·:(π Γ π)βΆβ) |
6 | simpr 110 | . . . 4 β’ ((π· β (Metβπ) β§ π β π) β π β π) | |
7 | xpss12 4732 | . . . 4 β’ ((π β π β§ π β π) β (π Γ π ) β (π Γ π)) | |
8 | 6, 7 | sylancom 420 | . . 3 β’ ((π· β (Metβπ) β§ π β π) β (π Γ π ) β (π Γ π)) |
9 | 5, 8 | fssresd 5390 | . 2 β’ ((π· β (Metβπ) β§ π β π) β (π· βΎ (π Γ π )):(π Γ π )βΆβ) |
10 | ismet2 13716 | . 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 417 | 1 β’ ((π· β (Metβπ) β§ π β π) β (π· βΎ (π Γ π )) β (Metβπ )) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 β wcel 2148 β wss 3129 Γ cxp 4623 βΎ cres 4627 βΆwf 5210 βcfv 5214 βcr 7806 βMetcxmet 13300 Metcmet 13301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7898 ax-resscn 7899 ax-1re 7901 ax-addrcl 7904 ax-rnegex 7916 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5176 df-fun 5216 df-fn 5217 df-f 5218 df-fv 5222 df-ov 5874 df-oprab 5875 df-mpo 5876 df-1st 6137 df-2nd 6138 df-map 6646 df-pnf 7989 df-mnf 7990 df-xr 7991 df-xadd 9768 df-xmet 13308 df-met 13309 |
This theorem is referenced by: metres 13745 remet 13902 |
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