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Mirrors > Home > MPE Home > Th. List > ngpds | Structured version Visualization version Unicode version |
Description: Value of the distance function in terms of the norm of a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
ngpds.n |
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ngpds.x |
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ngpds.m |
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ngpds.d |
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Ref | Expression |
---|---|
ngpds |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ngpds.n |
. . . . . 6
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2 | ngpds.m |
. . . . . 6
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3 | ngpds.d |
. . . . . 6
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4 | ngpds.x |
. . . . . 6
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5 | eqid 2451 |
. . . . . 6
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6 | 1, 2, 3, 4, 5 | isngp2 21611 |
. . . . 5
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7 | 6 | simp3bi 1025 |
. . . 4
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8 | 7 | 3ad2ant1 1029 |
. . 3
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9 | 8 | oveqd 6307 |
. 2
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10 | ngpgrp 21613 |
. . . . . 6
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11 | 4, 2 | grpsubf 16733 |
. . . . . 6
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12 | 10, 11 | syl 17 |
. . . . 5
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13 | 12 | 3ad2ant1 1029 |
. . . 4
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14 | opelxpi 4866 |
. . . . 5
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15 | 14 | 3adant1 1026 |
. . . 4
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16 | fvco3 5942 |
. . . 4
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17 | 13, 15, 16 | syl2anc 667 |
. . 3
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18 | df-ov 6293 |
. . 3
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19 | df-ov 6293 |
. . . 4
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20 | 19 | fveq2i 5868 |
. . 3
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21 | 17, 18, 20 | 3eqtr4g 2510 |
. 2
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22 | ovres 6436 |
. . 3
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23 | 22 | 3adant1 1026 |
. 2
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24 | 9, 21, 23 | 3eqtr3rd 2494 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616 ax-pre-sup 9617 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-iun 4280 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-om 6693 df-1st 6793 df-2nd 6794 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-er 7363 df-map 7474 df-en 7570 df-dom 7571 df-sdom 7572 df-sup 7956 df-inf 7957 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 df-div 10270 df-nn 10610 df-2 10668 df-n0 10870 df-z 10938 df-uz 11160 df-q 11265 df-rp 11303 df-xneg 11409 df-xadd 11410 df-xmul 11411 df-0g 15340 df-topgen 15342 df-mgm 16488 df-sgrp 16527 df-mnd 16537 df-grp 16673 df-minusg 16674 df-sbg 16675 df-psmet 18962 df-xmet 18963 df-met 18964 df-bl 18965 df-mopn 18966 df-top 19921 df-bases 19922 df-topon 19923 df-topsp 19924 df-xms 21335 df-ms 21336 df-nm 21597 df-ngp 21598 |
This theorem is referenced by: ngpdsr 21618 ngpds2 21619 ngprcan 21623 ngpinvds 21626 nmmtri 21635 nmrtri 21637 subgngp 21643 nrgdsdi 21668 nrgdsdir 21669 nlmdsdi 21684 nlmdsdir 21685 nrginvrcnlem 21693 nmods 21765 ipcnlem2 22215 minveclem2 22368 minveclem3b 22370 minveclem4 22374 minveclem6 22376 minveclem2OLD 22380 minveclem3bOLD 22382 minveclem4OLD 22386 minveclem6OLD 22388 |
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