![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > imsdf | Structured version Visualization version GIF version |
Description: Mapping for the induced metric distance function of a normed complex vector space. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
imsdfn.1 | β’ π = (BaseSetβπ) |
imsdfn.8 | β’ π· = (IndMetβπ) |
Ref | Expression |
---|---|
imsdf | β’ (π β NrmCVec β π·:(π Γ π)βΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imsdfn.1 | . . . 4 β’ π = (BaseSetβπ) | |
2 | eqid 2728 | . . . 4 β’ (normCVβπ) = (normCVβπ) | |
3 | 1, 2 | nvf 30463 | . . 3 β’ (π β NrmCVec β (normCVβπ):πβΆβ) |
4 | eqid 2728 | . . . 4 β’ ( βπ£ βπ) = ( βπ£ βπ) | |
5 | 1, 4 | nvmf 30448 | . . 3 β’ (π β NrmCVec β ( βπ£ βπ):(π Γ π)βΆπ) |
6 | fco 6741 | . . 3 β’ (((normCVβπ):πβΆβ β§ ( βπ£ βπ):(π Γ π)βΆπ) β ((normCVβπ) β ( βπ£ βπ)):(π Γ π)βΆβ) | |
7 | 3, 5, 6 | syl2anc 583 | . 2 β’ (π β NrmCVec β ((normCVβπ) β ( βπ£ βπ)):(π Γ π)βΆβ) |
8 | imsdfn.8 | . . . 4 β’ π· = (IndMetβπ) | |
9 | 4, 2, 8 | imsval 30488 | . . 3 β’ (π β NrmCVec β π· = ((normCVβπ) β ( βπ£ βπ))) |
10 | 9 | feq1d 6701 | . 2 β’ (π β NrmCVec β (π·:(π Γ π)βΆβ β ((normCVβπ) β ( βπ£ βπ)):(π Γ π)βΆβ)) |
11 | 7, 10 | mpbird 257 | 1 β’ (π β NrmCVec β π·:(π Γ π)βΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 Γ cxp 5670 β ccom 5676 βΆwf 6538 βcfv 6542 βcr 11131 NrmCVeccnv 30387 BaseSetcba 30389 βπ£ cnsb 30392 normCVcnmcv 30393 IndMetcims 30394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-ltxr 11277 df-sub 11470 df-neg 11471 df-grpo 30296 df-gid 30297 df-ginv 30298 df-gdiv 30299 df-ablo 30348 df-vc 30362 df-nv 30395 df-va 30398 df-ba 30399 df-sm 30400 df-0v 30401 df-vs 30402 df-nmcv 30403 df-ims 30404 |
This theorem is referenced by: imsmetlem 30493 sspims 30542 |
Copyright terms: Public domain | W3C validator |