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Mirrors > Home > MPE Home > Th. List > Mathboxes > qusdimsum | Structured version Visualization version GIF version |
Description: Let 𝑊 be a vector space, and let 𝑋 be a subspace. Then the dimension of 𝑊 is the sum of the dimension of 𝑋 and the dimension of the quotient space of 𝑋. First part of theorem 5.3 in [Lang] p. 141. (Contributed by Thierry Arnoux, 20-May-2023.) |
Ref | Expression |
---|---|
qusdimsum.x | ⊢ 𝑋 = (𝑊 ↾s 𝑈) |
qusdimsum.y | ⊢ 𝑌 = (𝑊 /s (𝑊 ~QG 𝑈)) |
Ref | Expression |
---|---|
qusdimsum | ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (dim‘𝑊) = ((dim‘𝑋) +𝑒 (dim‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusdimsum.y | . . . 4 ⊢ 𝑌 = (𝑊 /s (𝑊 ~QG 𝑈)) | |
2 | eqid 2798 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | lveclmod 19871 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
4 | 3 | adantr 484 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → 𝑊 ∈ LMod) |
5 | simpr 488 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → 𝑈 ∈ (LSubSp‘𝑊)) | |
6 | eqid 2798 | . . . 4 ⊢ (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) = (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) | |
7 | 1, 2, 4, 5, 6 | quslmhm 30975 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) ∈ (𝑊 LMHom 𝑌)) |
8 | eqid 2798 | . . . 4 ⊢ (0g‘𝑌) = (0g‘𝑌) | |
9 | eqid 2798 | . . . 4 ⊢ (𝑊 ↾s (◡(𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) “ {(0g‘𝑌)})) = (𝑊 ↾s (◡(𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) “ {(0g‘𝑌)})) | |
10 | eqid 2798 | . . . 4 ⊢ (𝑌 ↾s ran (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈))) = (𝑌 ↾s ran (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈))) | |
11 | 8, 9, 10 | dimkerim 31111 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) ∈ (𝑊 LMHom 𝑌)) → (dim‘𝑊) = ((dim‘(𝑊 ↾s (◡(𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) “ {(0g‘𝑌)}))) +𝑒 (dim‘(𝑌 ↾s ran (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)))))) |
12 | 7, 11 | syldan 594 | . 2 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (dim‘𝑊) = ((dim‘(𝑊 ↾s (◡(𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) “ {(0g‘𝑌)}))) +𝑒 (dim‘(𝑌 ↾s ran (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)))))) |
13 | eqid 2798 | . . . . . . . . 9 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
14 | 13 | lsssubg 19722 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ (LSubSp‘𝑊)) → 𝑈 ∈ (SubGrp‘𝑊)) |
15 | 3, 14 | sylan 583 | . . . . . . 7 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → 𝑈 ∈ (SubGrp‘𝑊)) |
16 | lmodabl 19674 | . . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
17 | 3, 16 | syl 17 | . . . . . . . . 9 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ Abel) |
18 | 17 | adantr 484 | . . . . . . . 8 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → 𝑊 ∈ Abel) |
19 | ablnsg 18960 | . . . . . . . 8 ⊢ (𝑊 ∈ Abel → (NrmSGrp‘𝑊) = (SubGrp‘𝑊)) | |
20 | 18, 19 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (NrmSGrp‘𝑊) = (SubGrp‘𝑊)) |
21 | 15, 20 | eleqtrrd 2893 | . . . . . 6 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → 𝑈 ∈ (NrmSGrp‘𝑊)) |
22 | 2, 6, 1, 8 | qusker 30969 | . . . . . . 7 ⊢ (𝑈 ∈ (NrmSGrp‘𝑊) → (◡(𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) “ {(0g‘𝑌)}) = 𝑈) |
23 | 22 | oveq2d 7151 | . . . . . 6 ⊢ (𝑈 ∈ (NrmSGrp‘𝑊) → (𝑊 ↾s (◡(𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) “ {(0g‘𝑌)})) = (𝑊 ↾s 𝑈)) |
24 | 21, 23 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (𝑊 ↾s (◡(𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) “ {(0g‘𝑌)})) = (𝑊 ↾s 𝑈)) |
25 | qusdimsum.x | . . . . 5 ⊢ 𝑋 = (𝑊 ↾s 𝑈) | |
26 | 24, 25 | eqtr4di 2851 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (𝑊 ↾s (◡(𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) “ {(0g‘𝑌)})) = 𝑋) |
27 | 26 | fveq2d 6649 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (dim‘(𝑊 ↾s (◡(𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) “ {(0g‘𝑌)}))) = (dim‘𝑋)) |
28 | 1 | ovexi 7169 | . . . . . 6 ⊢ 𝑌 ∈ V |
29 | eqid 2798 | . . . . . . 7 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
30 | 29 | ressid 16551 | . . . . . 6 ⊢ (𝑌 ∈ V → (𝑌 ↾s (Base‘𝑌)) = 𝑌) |
31 | 28, 30 | ax-mp 5 | . . . . 5 ⊢ (𝑌 ↾s (Base‘𝑌)) = 𝑌 |
32 | 1 | a1i 11 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → 𝑌 = (𝑊 /s (𝑊 ~QG 𝑈))) |
33 | 2 | a1i 11 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (Base‘𝑊) = (Base‘𝑊)) |
34 | ovexd 7170 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (𝑊 ~QG 𝑈) ∈ V) | |
35 | simpl 486 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → 𝑊 ∈ LVec) | |
36 | 32, 33, 6, 34, 35 | quslem 16808 | . . . . . . . 8 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)):(Base‘𝑊)–onto→((Base‘𝑊) / (𝑊 ~QG 𝑈))) |
37 | forn 6568 | . . . . . . . 8 ⊢ ((𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)):(Base‘𝑊)–onto→((Base‘𝑊) / (𝑊 ~QG 𝑈)) → ran (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) = ((Base‘𝑊) / (𝑊 ~QG 𝑈))) | |
38 | 36, 37 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → ran (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) = ((Base‘𝑊) / (𝑊 ~QG 𝑈))) |
39 | 32, 33, 34, 35 | qusbas 16810 | . . . . . . 7 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → ((Base‘𝑊) / (𝑊 ~QG 𝑈)) = (Base‘𝑌)) |
40 | 38, 39 | eqtr2d 2834 | . . . . . 6 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (Base‘𝑌) = ran (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈))) |
41 | 40 | oveq2d 7151 | . . . . 5 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (𝑌 ↾s (Base‘𝑌)) = (𝑌 ↾s ran (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)))) |
42 | 31, 41 | syl5reqr 2848 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (𝑌 ↾s ran (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈))) = 𝑌) |
43 | 42 | fveq2d 6649 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (dim‘(𝑌 ↾s ran (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)))) = (dim‘𝑌)) |
44 | 27, 43 | oveq12d 7153 | . 2 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → ((dim‘(𝑊 ↾s (◡(𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈)) “ {(0g‘𝑌)}))) +𝑒 (dim‘(𝑌 ↾s ran (𝑥 ∈ (Base‘𝑊) ↦ [𝑥](𝑊 ~QG 𝑈))))) = ((dim‘𝑋) +𝑒 (dim‘𝑌))) |
45 | 12, 44 | eqtrd 2833 | 1 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (dim‘𝑊) = ((dim‘𝑋) +𝑒 (dim‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 ↦ cmpt 5110 ◡ccnv 5518 ran crn 5520 “ cima 5522 –onto→wfo 6322 ‘cfv 6324 (class class class)co 7135 [cec 8270 / cqs 8271 +𝑒 cxad 12493 Basecbs 16475 ↾s cress 16476 0gc0g 16705 /s cqus 16770 SubGrpcsubg 18265 NrmSGrpcnsg 18266 ~QG cqg 18267 Abelcabl 18899 LModclmod 19627 LSubSpclss 19696 LMHom clmhm 19784 LVecclvec 19867 dimcldim 31087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 ax-ac2 9874 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-rpss 7429 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-ec 8274 df-qs 8278 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-inf 8891 df-oi 8958 df-r1 9177 df-rank 9178 df-dju 9314 df-card 9352 df-acn 9355 df-ac 9527 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-xadd 12496 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ocomp 16578 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-imas 16773 df-qus 16774 df-mre 16849 df-mrc 16850 df-mri 16851 df-acs 16852 df-proset 17530 df-drs 17531 df-poset 17548 df-ipo 17754 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-nsg 18269 df-eqg 18270 df-ghm 18348 df-cntz 18439 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-drng 19497 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lmhm 19787 df-lmim 19788 df-lbs 19840 df-lvec 19868 df-sra 19937 df-rgmod 19938 df-nzr 20024 df-dsmm 20421 df-frlm 20436 df-uvc 20472 df-lindf 20495 df-linds 20496 df-dim 31088 |
This theorem is referenced by: (None) |
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