![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > css0 | Structured version Visualization version GIF version |
Description: The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
css0.c | ⊢ 𝐶 = (ClSubSp‘𝑊) |
css0.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
css0 | ⊢ (𝑊 ∈ PreHil → { 0 } ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2772 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2772 | . . 3 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
3 | css0.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
4 | 1, 2, 3 | ocv1 20537 | . 2 ⊢ (𝑊 ∈ PreHil → ((ocv‘𝑊)‘(Base‘𝑊)) = { 0 }) |
5 | ssid 3873 | . . 3 ⊢ (Base‘𝑊) ⊆ (Base‘𝑊) | |
6 | css0.c | . . . 4 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
7 | 1, 6, 2 | ocvcss 20545 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (Base‘𝑊) ⊆ (Base‘𝑊)) → ((ocv‘𝑊)‘(Base‘𝑊)) ∈ 𝐶) |
8 | 5, 7 | mpan2 678 | . 2 ⊢ (𝑊 ∈ PreHil → ((ocv‘𝑊)‘(Base‘𝑊)) ∈ 𝐶) |
9 | 4, 8 | eqeltrrd 2861 | 1 ⊢ (𝑊 ∈ PreHil → { 0 } ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ⊆ wss 3823 {csn 4435 ‘cfv 6185 Basecbs 16337 0gc0g 16567 PreHilcphl 20482 ocvcocv 20518 ClSubSpccss 20519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-tpos 7693 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-plusg 16432 df-mulr 16433 df-sca 16435 df-vsca 16436 df-ip 16437 df-0g 16569 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mhm 17815 df-grp 17906 df-minusg 17907 df-sbg 17908 df-ghm 18139 df-mgp 18975 df-ur 18987 df-ring 19034 df-oppr 19108 df-rnghom 19202 df-staf 19350 df-srng 19351 df-lmod 19370 df-lss 19438 df-lmhm 19528 df-lvec 19609 df-sra 19678 df-rgmod 19679 df-phl 20484 df-ocv 20521 df-css 20522 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |