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Mirrors > Home > HSE Home > Th. List > hvaddsub4 | Structured version Visualization version Unicode version |
Description: Hilbert vector space addition/subtraction law. (Contributed by NM, 18-May-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvaddsub4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvaddcl 27869 | . . . 4 | |
2 | 1 | adantr 481 | . . 3 |
3 | hvaddcl 27869 | . . . 4 | |
4 | 3 | adantl 482 | . . 3 |
5 | hvaddcl 27869 | . . . . 5 | |
6 | 5 | ancoms 469 | . . . 4 |
7 | 6 | ad2ant2lr 784 | . . 3 |
8 | hvsubcan2 27932 | . . 3 | |
9 | 2, 4, 7, 8 | syl3anc 1326 | . 2 |
10 | simpr 477 | . . . . . . . 8 | |
11 | 10 | anim2i 593 | . . . . . . 7 |
12 | 11 | ancoms 469 | . . . . . 6 |
13 | hvsub4 27894 | . . . . . 6 | |
14 | 12, 13 | syldan 487 | . . . . 5 |
15 | hvsubid 27883 | . . . . . . 7 | |
16 | 15 | ad2antlr 763 | . . . . . 6 |
17 | 16 | oveq2d 6666 | . . . . 5 |
18 | hvsubcl 27874 | . . . . . . 7 | |
19 | ax-hvaddid 27861 | . . . . . . 7 | |
20 | 18, 19 | syl 17 | . . . . . 6 |
21 | 20 | adantlr 751 | . . . . 5 |
22 | 14, 17, 21 | 3eqtrd 2660 | . . . 4 |
23 | 22 | adantrr 753 | . . 3 |
24 | simpl 473 | . . . . . . . 8 | |
25 | 24 | anim1i 592 | . . . . . . 7 |
26 | hvsub4 27894 | . . . . . . 7 | |
27 | 25, 26 | syldan 487 | . . . . . 6 |
28 | hvsubid 27883 | . . . . . . . 8 | |
29 | 28 | ad2antrr 762 | . . . . . . 7 |
30 | 29 | oveq1d 6665 | . . . . . 6 |
31 | hvsubcl 27874 | . . . . . . . 8 | |
32 | hvaddid2 27880 | . . . . . . . 8 | |
33 | 31, 32 | syl 17 | . . . . . . 7 |
34 | 33 | adantll 750 | . . . . . 6 |
35 | 27, 30, 34 | 3eqtrd 2660 | . . . . 5 |
36 | 35 | ancoms 469 | . . . 4 |
37 | 36 | adantll 750 | . . 3 |
38 | 23, 37 | eqeq12d 2637 | . 2 |
39 | 9, 38 | bitr3d 270 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 (class class class)co 6650 chil 27776 cva 27777 c0v 27781 cmv 27782 |
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-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 ax-pre-mulgt0 10013 ax-hfvadd 27857 ax-hvcom 27858 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvmulass 27864 ax-hvdistr1 27865 ax-hvdistr2 27866 ax-hvmul0 27867 |
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-rmo 2920 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-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-hvsub 27828 |
This theorem is referenced by: shuni 28159 cdjreui 29291 |
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