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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhvaddcl | Structured version Unicode version |
Description: Closure of the vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) |
Ref | Expression |
---|---|
dvhvaddcl.h |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
dvhvaddcl.t |
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dvhvaddcl.e |
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dvhvaddcl.u |
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dvhvaddcl.d |
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dvhvaddcl.p |
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dvhvaddcl.a |
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Ref | Expression |
---|---|
dvhvaddcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvhvaddcl.h |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | dvhvaddcl.t |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | dvhvaddcl.e |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | dvhvaddcl.u |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | dvhvaddcl.d |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | dvhvaddcl.a |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | dvhvaddcl.p |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 1, 2, 3, 4, 5, 6, 7 | dvhvadd 35095 |
. 2
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9 | simpl 457 |
. . . 4
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10 | xp1st 6719 |
. . . . 5
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11 | 10 | ad2antrl 727 |
. . . 4
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12 | xp1st 6719 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 12 | ad2antll 728 |
. . . 4
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14 | 1, 2 | ltrnco 34721 |
. . . 4
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15 | 9, 11, 13, 14 | syl3anc 1219 |
. . 3
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16 | eqid 2454 |
. . . . . . 7
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17 | 1, 2, 3, 4, 5, 16, 7 | dvhfplusr 35087 |
. . . . . 6
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18 | 17 | adantr 465 |
. . . . 5
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19 | 18 | oveqd 6220 |
. . . 4
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20 | xp2nd 6720 |
. . . . . 6
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21 | 20 | ad2antrl 727 |
. . . . 5
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22 | xp2nd 6720 |
. . . . . 6
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23 | 22 | ad2antll 728 |
. . . . 5
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24 | 1, 2, 3, 16 | tendoplcl 34783 |
. . . . 5
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25 | 9, 21, 23, 24 | syl3anc 1219 |
. . . 4
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26 | 19, 25 | eqeltrd 2542 |
. . 3
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27 | opelxpi 4982 |
. . 3
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28 | 15, 26, 27 | syl2anc 661 |
. 2
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29 | 8, 28 | eqeltrd 2542 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4514 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 ax-cnex 9452 ax-resscn 9453 ax-1cn 9454 ax-icn 9455 ax-addcl 9456 ax-addrcl 9457 ax-mulcl 9458 ax-mulrcl 9459 ax-mulcom 9460 ax-addass 9461 ax-mulass 9462 ax-distr 9463 ax-i2m1 9464 ax-1ne0 9465 ax-1rid 9466 ax-rnegex 9467 ax-rrecex 9468 ax-cnre 9469 ax-pre-lttri 9470 ax-pre-lttrn 9471 ax-pre-ltadd 9472 ax-pre-mulgt0 9473 ax-riotaBAD 32962 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-pss 3455 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-tp 3993 df-op 3995 df-uni 4203 df-int 4240 df-iun 4284 df-iin 4285 df-br 4404 df-opab 4462 df-mpt 4463 df-tr 4497 df-eprel 4743 df-id 4747 df-po 4752 df-so 4753 df-fr 4790 df-we 4792 df-ord 4833 df-on 4834 df-lim 4835 df-suc 4836 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-riota 6164 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-om 6590 df-1st 6690 df-2nd 6691 df-undef 6905 df-recs 6945 df-rdg 6979 df-1o 7033 df-oadd 7037 df-er 7214 df-map 7329 df-en 7424 df-dom 7425 df-sdom 7426 df-fin 7427 df-pnf 9534 df-mnf 9535 df-xr 9536 df-ltxr 9537 df-le 9538 df-sub 9711 df-neg 9712 df-nn 10437 df-2 10494 df-3 10495 df-4 10496 df-5 10497 df-6 10498 df-n0 10694 df-z 10761 df-uz 10976 df-fz 11558 df-struct 14297 df-ndx 14298 df-slot 14299 df-base 14300 df-plusg 14373 df-mulr 14374 df-sca 14376 df-vsca 14377 df-poset 15238 df-plt 15250 df-lub 15266 df-glb 15267 df-join 15268 df-meet 15269 df-p0 15331 df-p1 15332 df-lat 15338 df-clat 15400 df-oposet 33179 df-ol 33181 df-oml 33182 df-covers 33269 df-ats 33270 df-atl 33301 df-cvlat 33325 df-hlat 33354 df-llines 33500 df-lplanes 33501 df-lvols 33502 df-lines 33503 df-psubsp 33505 df-pmap 33506 df-padd 33798 df-lhyp 33990 df-laut 33991 df-ldil 34106 df-ltrn 34107 df-trl 34161 df-tendo 34757 df-edring 34759 df-dvech 35082 |
This theorem is referenced by: dvhvaddass 35100 dvhgrp 35110 dvhlveclem 35111 |
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