Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > sspph | Structured version Visualization version Unicode version | ||
| Description: A subspace of an inner product space is an inner product space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| sspph.h |
        |
| Ref | Expression |
|---|---|
| sspph |
   ![/\](wedge.gif "/\"){kind=small} 
 |
are mutually distinct and distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | phnv 27669 |
. . 3
 | |
| 2 | sspph.h |
. . . 4
| |
| 3 | 2 | sspnv 27581 |
. . 3
|
| 4 | 1, 3 | sylan 488 |
. 2
|
| 5 | eqid 2622 |
. . . . . . . . . 10
 | |
| 6 | eqid 2622 |
. . . . . . . . . 10
| |
| 7 | 5, 6, 2 | sspba 27582 |
. . . . . . . . 9
 |
| 8 | 7 | sseld 3602 |
. . . . . . . 8
|
| 9 | 7 | sseld 3602 |
. . . . . . . 8
|
| 10 | 8, 9 | anim12d 586 |
. . . . . . 7
|
| 11 | 1, 10 | sylan 488 |
. . . . . 6
|
| 12 | 11 | imp 445 |
. . . . 5
|
| 13 | eqid 2622 |
. . . . . . . 8
 | |
| 14 | eqid 2622 |
. . . . . . . 8
 | |
| 15 | eqid 2622 |
. . . . . . . 8
 CV CV | |
| 16 | 5, 13, 14, 15 | phpar2 27678 |
. . . . . . 7
CV   CV  CV CV |
| 17 | 16 | 3expb 1266 |
. . . . . 6
CV CV CV CV |
| 18 | 17 | adantlr 751 |
. . . . 5
CV CV CV CV |
| 19 | 12, 18 | syldan 487 |
. . . 4
CV CV CV CV |
| 20 | eqid 2622 |
. . . . . . . . . . . 12
| |
| 21 | 6, 20 | nvgcl 27475 |
. . . . . . . . . . 11
|
| 22 | 21 | 3expb 1266 |
. . . . . . . . . 10
|
| 23 | 3, 22 | sylan 488 |
. . . . . . . . 9
|
| 24 | eqid 2622 |
. . . . . . . . . . 11
CV CV | |
| 25 | 6, 15, 24, 2 | sspnval 27592 |
. . . . . . . . . 10
CV CV |
| 26 | 25 | 3expa 1265 |
. . . . . . . . 9
CV CV |
| 27 | 23, 26 | syldan 487 |
. . . . . . . 8
CV CV |
| 28 | 27 | oveq1d 6665 |
. . . . . . 7
CV CV |
| 29 | 6, 13, 20, 2 | sspgval 27584 |
. . . . . . . . 9
|
| 30 | 29 | fveq2d 6195 |
. . . . . . . 8
CV CV |
| 31 | 30 | oveq1d 6665 |
. . . . . . 7
CV CV |
| 32 | 28, 31 | eqtrd 2656 |
. . . . . 6
CV CV |
| 33 | eqid 2622 |
. . . . . . . . . . . 12
| |
| 34 | 6, 33 | nvmcl 27501 |
. . . . . . . . . . 11
|
| 35 | 34 | 3expb 1266 |
. . . . . . . . . 10
|
| 36 | 3, 35 | sylan 488 |
. . . . . . . . 9
|
| 37 | 6, 15, 24, 2 | sspnval 27592 |
. . . . . . . . . 10
CV CV |
| 38 | 37 | 3expa 1265 |
. . . . . . . . 9
CV CV |
| 39 | 36, 38 | syldan 487 |
. . . . . . . 8
CV CV |
| 40 | 6, 14, 33, 2 | sspmval 27588 |
. . . . . . . . 9
|
| 41 | 40 | fveq2d 6195 |
. . . . . . . 8
CV CV |
| 42 | 39, 41 | eqtrd 2656 |
. . . . . . 7
CV CV |
| 43 | 42 | oveq1d 6665 |
. . . . . 6
CV CV |
| 44 | 32, 43 | oveq12d 6668 |
. . . . 5
CV CV CV CV |
| 45 | 1, 44 | sylanl1 682 |
. . . 4
CV CV CV CV |
| 46 | 6, 15, 24, 2 | sspnval 27592 |
. . . . . . . . . 10
CV CV |
| 47 | 46 | 3expa 1265 |
. . . . . . . . 9
CV CV |
| 48 | 47 | adantrr 753 |
. . . . . . . 8
CV CV |
| 49 | 48 | oveq1d 6665 |
. . . . . . 7
CV CV |
| 50 | 6, 15, 24, 2 | sspnval 27592 |
. . . . . . . . . 10
CV CV |
| 51 | 50 | 3expa 1265 |
. . . . . . . . 9
CV CV |
| 52 | 51 | adantrl 752 |
. . . . . . . 8
CV CV |
| 53 | 52 | oveq1d 6665 |
. . . . . . 7
CV CV |
| 54 | 49, 53 | oveq12d 6668 |
. . . . . 6
CV CV CV CV |
| 55 | 1, 54 | sylanl1 682 |
. . . . 5
CV CV CV CV |
| 56 | 55 | oveq2d 6666 |
. . . 4
CV CV CV CV |
| 57 | 19, 45, 56 | 3eqtr4d 2666 |
. . 3
CV CV CV CV |
| 58 | 57 | ralrimivva 2971 |
. 2
CV CV CV CV |
| 59 | 6, 20, 33, 24 | isph 27677 |
. 2
CV CV CV CV |
| 60 | 4, 58, 59 | sylanbrc 698 |
1
|
| Colors of variables: wff setvar class |
| Syntax hints: wi 4
wa 384
wceq 1483
wcel 1990
wral 2912
cfv 5888
(class class class)co 6650 caddc 9939 cmul 9941 c2 11070 cexp 12860 cnv 27439 cpv 27440 cba 27441 cnsb 27444 CVcnmcv 27445 css 27576 ccphlo 27667 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
| This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ssp 27577 df-ph 27668 |
| This theorem is referenced by: ssphl 27773 hhssph 28131 |
| Copyright terms: Public domain | W3C validator |