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Mirrors > Home > MPE Home > Th. List > Mathboxes > ellkr2 | Structured version Visualization version Unicode version |
Description: Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.) |
Ref | Expression |
---|---|
lkrfval2.v | |
lkrfval2.d | Scalar |
lkrfval2.o | |
lkrfval2.f | LFnl |
lkrfval2.k | LKer |
ellkr2.w | |
ellkr2.g | |
ellkr2.x |
Ref | Expression |
---|---|
ellkr2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ellkr2.w | . . 3 | |
2 | ellkr2.g | . . 3 | |
3 | lkrfval2.v | . . . 4 | |
4 | lkrfval2.d | . . . 4 Scalar | |
5 | lkrfval2.o | . . . 4 | |
6 | lkrfval2.f | . . . 4 LFnl | |
7 | lkrfval2.k | . . . 4 LKer | |
8 | 3, 4, 5, 6, 7 | ellkr 34376 | . . 3 |
9 | 1, 2, 8 | syl2anc 693 | . 2 |
10 | ellkr2.x | . . 3 | |
11 | 10 | biantrurd 529 | . 2 |
12 | 9, 11 | bitr4d 271 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 cfv 5888 cbs 15857 Scalarcsca 15944 c0g 16100 LFnlclfn 34344 LKerclk 34372 |
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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-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-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-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-lfl 34345 df-lkr 34373 |
This theorem is referenced by: lclkrlem2f 36801 lclkrlem2n 36809 lcfrlem3 36833 lcfrlem25 36856 hdmapellkr 37206 hdmapip0 37207 hdmapinvlem1 37210 |
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