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Mirrors > Home > ILE Home > Th. List > bdmetval | GIF version |
Description: Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.) |
Ref | Expression |
---|---|
stdbdmet.1 | β’ π· = (π₯ β π, π¦ β π β¦ inf({(π₯πΆπ¦), π }, β*, < )) |
Ref | Expression |
---|---|
bdmetval | β’ (((πΆ:(π Γ π)βΆβ* β§ π β β*) β§ (π΄ β π β§ π΅ β π)) β (π΄π·π΅) = inf({(π΄πΆπ΅), π }, β*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 529 | . 2 β’ (((πΆ:(π Γ π)βΆβ* β§ π β β*) β§ (π΄ β π β§ π΅ β π)) β π΄ β π) | |
2 | simprr 531 | . 2 β’ (((πΆ:(π Γ π)βΆβ* β§ π β β*) β§ (π΄ β π β§ π΅ β π)) β π΅ β π) | |
3 | simpll 527 | . . . 4 β’ (((πΆ:(π Γ π)βΆβ* β§ π β β*) β§ (π΄ β π β§ π΅ β π)) β πΆ:(π Γ π)βΆβ*) | |
4 | 3, 1, 2 | fovcdmd 6021 | . . 3 β’ (((πΆ:(π Γ π)βΆβ* β§ π β β*) β§ (π΄ β π β§ π΅ β π)) β (π΄πΆπ΅) β β*) |
5 | simplr 528 | . . 3 β’ (((πΆ:(π Γ π)βΆβ* β§ π β β*) β§ (π΄ β π β§ π΅ β π)) β π β β*) | |
6 | xrmincl 11276 | . . 3 β’ (((π΄πΆπ΅) β β* β§ π β β*) β inf({(π΄πΆπ΅), π }, β*, < ) β β*) | |
7 | 4, 5, 6 | syl2anc 411 | . 2 β’ (((πΆ:(π Γ π)βΆβ* β§ π β β*) β§ (π΄ β π β§ π΅ β π)) β inf({(π΄πΆπ΅), π }, β*, < ) β β*) |
8 | oveq12 5886 | . . . . 5 β’ ((π₯ = π΄ β§ π¦ = π΅) β (π₯πΆπ¦) = (π΄πΆπ΅)) | |
9 | 8 | preq1d 3677 | . . . 4 β’ ((π₯ = π΄ β§ π¦ = π΅) β {(π₯πΆπ¦), π } = {(π΄πΆπ΅), π }) |
10 | 9 | infeq1d 7013 | . . 3 β’ ((π₯ = π΄ β§ π¦ = π΅) β inf({(π₯πΆπ¦), π }, β*, < ) = inf({(π΄πΆπ΅), π }, β*, < )) |
11 | stdbdmet.1 | . . 3 β’ π· = (π₯ β π, π¦ β π β¦ inf({(π₯πΆπ¦), π }, β*, < )) | |
12 | 10, 11 | ovmpoga 6006 | . 2 β’ ((π΄ β π β§ π΅ β π β§ inf({(π΄πΆπ΅), π }, β*, < ) β β*) β (π΄π·π΅) = inf({(π΄πΆπ΅), π }, β*, < )) |
13 | 1, 2, 7, 12 | syl3anc 1238 | 1 β’ (((πΆ:(π Γ π)βΆβ* β§ π β β*) β§ (π΄ β π β§ π΅ β π)) β (π΄π·π΅) = inf({(π΄πΆπ΅), π }, β*, < )) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 {cpr 3595 Γ cxp 4626 βΆwf 5214 (class class class)co 5877 β cmpo 5879 infcinf 6984 β*cxr 7993 < clt 7994 |
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-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933 |
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-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-frec 6394 df-sup 6985 df-inf 6986 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-rp 9656 df-xneg 9774 df-seqfrec 10448 df-exp 10522 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 |
This theorem is referenced by: bdbl 14088 |
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