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Mirrors > Home > HSE Home > Th. List > norm1exi | Structured version Visualization version Unicode version |
Description: A normalized vector exists in a subspace iff the subspace has a nonzero vector. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
norm1ex.1 |
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Ref | Expression |
---|---|
norm1exi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq1 2685 |
. . 3
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2 | 1 | cbvrexv 3019 |
. 2
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3 | norm1ex.1 |
. . . . . . . . . . 11
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4 | 3 | sheli 26860 |
. . . . . . . . . 10
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5 | normcl 26771 |
. . . . . . . . . 10
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6 | 4, 5 | syl 17 |
. . . . . . . . 9
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7 | 6 | adantr 467 |
. . . . . . . 8
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8 | normne0 26776 |
. . . . . . . . . 10
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9 | 4, 8 | syl 17 |
. . . . . . . . 9
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10 | 9 | biimpar 488 |
. . . . . . . 8
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11 | 7, 10 | rereccld 10431 |
. . . . . . 7
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12 | 11 | recnd 9666 |
. . . . . 6
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13 | simpl 459 |
. . . . . 6
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14 | shmulcl 26864 |
. . . . . . 7
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15 | 3, 14 | mp3an1 1350 |
. . . . . 6
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16 | 12, 13, 15 | syl2anc 666 |
. . . . 5
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17 | norm1 26895 |
. . . . . 6
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18 | 4, 17 | sylan 474 |
. . . . 5
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19 | fveq2 5863 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 19 | eqeq1d 2452 |
. . . . . 6
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21 | 20 | rspcev 3149 |
. . . . 5
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22 | 16, 18, 21 | syl2anc 666 |
. . . 4
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23 | 22 | rexlimiva 2874 |
. . 3
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24 | ax-1ne0 9605 |
. . . . . . . 8
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25 | 24 | neii 2625 |
. . . . . . 7
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26 | eqeq1 2454 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 25, 26 | mtbiri 305 |
. . . . . 6
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28 | 3 | sheli 26860 |
. . . . . . . 8
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29 | norm-i 26775 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | 28, 29 | syl 17 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 30 | necon3bbid 2660 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 27, 31 | syl5ib 223 |
. . . . 5
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33 | 32 | reximia 2852 |
. . . 4
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34 | neeq1 2685 |
. . . . 5
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35 | 34 | cbvrexv 3019 |
. . . 4
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36 | 33, 35 | sylib 200 |
. . 3
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37 | 23, 36 | impbii 191 |
. 2
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38 | 2, 37 | bitri 253 |
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 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 ax-cnex 9592 ax-resscn 9593 ax-1cn 9594 ax-icn 9595 ax-addcl 9596 ax-addrcl 9597 ax-mulcl 9598 ax-mulrcl 9599 ax-mulcom 9600 ax-addass 9601 ax-mulass 9602 ax-distr 9603 ax-i2m1 9604 ax-1ne0 9605 ax-1rid 9606 ax-rnegex 9607 ax-rrecex 9608 ax-cnre 9609 ax-pre-lttri 9610 ax-pre-lttrn 9611 ax-pre-ltadd 9612 ax-pre-mulgt0 9613 ax-pre-sup 9614 ax-hilex 26645 ax-hfvadd 26646 ax-hv0cl 26649 ax-hfvmul 26651 ax-hvmul0 26656 ax-hfi 26725 ax-his1 26728 ax-his3 26730 ax-his4 26731 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 985 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-nel 2624 df-ral 2741 df-rex 2742 df-reu 2743 df-rmo 2744 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-pss 3419 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-tp 3972 df-op 3974 df-uni 4198 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-tr 4497 df-eprel 4744 df-id 4748 df-po 4754 df-so 4755 df-fr 4792 df-we 4794 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-pred 5379 df-ord 5425 df-on 5426 df-lim 5427 df-suc 5428 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-riota 6250 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-om 6690 df-2nd 6791 df-wrecs 7025 df-recs 7087 df-rdg 7125 df-er 7360 df-en 7567 df-dom 7568 df-sdom 7569 df-sup 7953 df-pnf 9674 df-mnf 9675 df-xr 9676 df-ltxr 9677 df-le 9678 df-sub 9859 df-neg 9860 df-div 10267 df-nn 10607 df-2 10665 df-3 10666 df-n0 10867 df-z 10935 df-uz 11157 df-rp 11300 df-seq 12211 df-exp 12270 df-cj 13155 df-re 13156 df-im 13157 df-sqrt 13291 df-abs 13292 df-hnorm 26614 df-sh 26853 |
This theorem is referenced by: norm1hex 26897 pjnmopi 27794 |
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