![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhils0 | Structured version Unicode version |
Description: The scalar ring zero for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) |
Ref | Expression |
---|---|
hlhilsbase.h |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hlhilsbase.l |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hlhilsbase.s |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hlhilsbase.u |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hlhilsbase.r |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hlhilsbase.k |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hlhils0.z |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
hlhils0 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhils0.z |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | eqidd 2455 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | hlhilsbase.h |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | hlhilsbase.l |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | hlhilsbase.s |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | hlhilsbase.u |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | hlhilsbase.r |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | hlhilsbase.k |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | eqid 2454 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 3, 4, 5, 6, 7, 8, 9 | hlhilsbase2 35953 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | eqid 2454 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 3, 4, 5, 6, 7, 8, 11 | hlhilsplus2 35954 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 12 | proplem3 14752 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 2, 10, 13 | grpidpropd 15570 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 1, 14 | syl5eq 2507 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 9453 ax-resscn 9454 ax-1cn 9455 ax-icn 9456 ax-addcl 9457 ax-addrcl 9458 ax-mulcl 9459 ax-mulrcl 9460 ax-mulcom 9461 ax-addass 9462 ax-mulass 9463 ax-distr 9464 ax-i2m1 9465 ax-1ne0 9466 ax-1rid 9467 ax-rnegex 9468 ax-rrecex 9469 ax-cnre 9470 ax-pre-lttri 9471 ax-pre-lttrn 9472 ax-pre-ltadd 9473 ax-pre-mulgt0 9474 |
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-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-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-recs 6945 df-rdg 6979 df-1o 7033 df-oadd 7037 df-er 7214 df-en 7424 df-dom 7425 df-sdom 7426 df-fin 7427 df-pnf 9535 df-mnf 9536 df-xr 9537 df-ltxr 9538 df-le 9539 df-sub 9712 df-neg 9713 df-nn 10438 df-2 10495 df-3 10496 df-4 10497 df-5 10498 df-6 10499 df-7 10500 df-8 10501 df-n0 10695 df-z 10762 df-uz 10977 df-fz 11559 df-struct 14298 df-ndx 14299 df-slot 14300 df-base 14301 df-sets 14302 df-plusg 14374 df-starv 14376 df-sca 14377 df-vsca 14378 df-ip 14379 df-0g 14503 df-dvech 35087 df-hlhil 35944 |
This theorem is referenced by: hlhilocv 35968 hlhilphllem 35970 |
Copyright terms: Public domain | W3C validator |