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Mirrors > Home > MPE Home > Th. List > Mathboxes > lvecdimfi | Structured version Visualization version GIF version |
Description: Finite version of lvecdim 21045 which does not require the axiom of choice. The axiom of choice is used in acsmapd 18546, which is required in the infinite case. Suggested by Gรฉrard Lang. (Contributed by Thierry Arnoux, 24-May-2023.) |
Ref | Expression |
---|---|
lvecdimfi.j | โข ๐ฝ = (LBasisโ๐) |
lvecdimfi.w | โข (๐ โ ๐ โ LVec) |
lvecdimfi.s | โข (๐ โ ๐ โ ๐ฝ) |
lvecdimfi.t | โข (๐ โ ๐ โ ๐ฝ) |
lvecdimfi.f | โข (๐ โ ๐ โ Fin) |
Ref | Expression |
---|---|
lvecdimfi | โข (๐ โ ๐ โ ๐) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecdimfi.w | . . . . 5 โข (๐ โ ๐ โ LVec) | |
2 | eqid 2728 | . . . . . 6 โข (LSubSpโ๐) = (LSubSpโ๐) | |
3 | eqid 2728 | . . . . . 6 โข (mrClsโ(LSubSpโ๐)) = (mrClsโ(LSubSpโ๐)) | |
4 | eqid 2728 | . . . . . 6 โข (Baseโ๐) = (Baseโ๐) | |
5 | 2, 3, 4 | lssacsex 21032 | . . . . 5 โข (๐ โ LVec โ ((LSubSpโ๐) โ (ACSโ(Baseโ๐)) โง โ๐ฅ โ ๐ซ (Baseโ๐)โ๐ฆ โ (Baseโ๐)โ๐ง โ (((mrClsโ(LSubSpโ๐))โ(๐ฅ โช {๐ฆ})) โ ((mrClsโ(LSubSpโ๐))โ๐ฅ))๐ฆ โ ((mrClsโ(LSubSpโ๐))โ(๐ฅ โช {๐ง})))) |
6 | 1, 5 | syl 17 | . . . 4 โข (๐ โ ((LSubSpโ๐) โ (ACSโ(Baseโ๐)) โง โ๐ฅ โ ๐ซ (Baseโ๐)โ๐ฆ โ (Baseโ๐)โ๐ง โ (((mrClsโ(LSubSpโ๐))โ(๐ฅ โช {๐ฆ})) โ ((mrClsโ(LSubSpโ๐))โ๐ฅ))๐ฆ โ ((mrClsโ(LSubSpโ๐))โ(๐ฅ โช {๐ง})))) |
7 | 6 | simpld 494 | . . 3 โข (๐ โ (LSubSpโ๐) โ (ACSโ(Baseโ๐))) |
8 | 7 | acsmred 17636 | . 2 โข (๐ โ (LSubSpโ๐) โ (Mooreโ(Baseโ๐))) |
9 | eqid 2728 | . 2 โข (mrIndโ(LSubSpโ๐)) = (mrIndโ(LSubSpโ๐)) | |
10 | 6 | simprd 495 | . 2 โข (๐ โ โ๐ฅ โ ๐ซ (Baseโ๐)โ๐ฆ โ (Baseโ๐)โ๐ง โ (((mrClsโ(LSubSpโ๐))โ(๐ฅ โช {๐ฆ})) โ ((mrClsโ(LSubSpโ๐))โ๐ฅ))๐ฆ โ ((mrClsโ(LSubSpโ๐))โ(๐ฅ โช {๐ง}))) |
11 | lvecdimfi.s | . . . 4 โข (๐ โ ๐ โ ๐ฝ) | |
12 | lvecdimfi.j | . . . . . 6 โข ๐ฝ = (LBasisโ๐) | |
13 | 2, 3, 4, 9, 12 | lbsacsbs 21044 | . . . . 5 โข (๐ โ LVec โ (๐ โ ๐ฝ โ (๐ โ (mrIndโ(LSubSpโ๐)) โง ((mrClsโ(LSubSpโ๐))โ๐) = (Baseโ๐)))) |
14 | 13 | biimpa 476 | . . . 4 โข ((๐ โ LVec โง ๐ โ ๐ฝ) โ (๐ โ (mrIndโ(LSubSpโ๐)) โง ((mrClsโ(LSubSpโ๐))โ๐) = (Baseโ๐))) |
15 | 1, 11, 14 | syl2anc 583 | . . 3 โข (๐ โ (๐ โ (mrIndโ(LSubSpโ๐)) โง ((mrClsโ(LSubSpโ๐))โ๐) = (Baseโ๐))) |
16 | 15 | simpld 494 | . 2 โข (๐ โ ๐ โ (mrIndโ(LSubSpโ๐))) |
17 | lvecdimfi.t | . . . 4 โข (๐ โ ๐ โ ๐ฝ) | |
18 | 2, 3, 4, 9, 12 | lbsacsbs 21044 | . . . . 5 โข (๐ โ LVec โ (๐ โ ๐ฝ โ (๐ โ (mrIndโ(LSubSpโ๐)) โง ((mrClsโ(LSubSpโ๐))โ๐) = (Baseโ๐)))) |
19 | 18 | biimpa 476 | . . . 4 โข ((๐ โ LVec โง ๐ โ ๐ฝ) โ (๐ โ (mrIndโ(LSubSpโ๐)) โง ((mrClsโ(LSubSpโ๐))โ๐) = (Baseโ๐))) |
20 | 1, 17, 19 | syl2anc 583 | . . 3 โข (๐ โ (๐ โ (mrIndโ(LSubSpโ๐)) โง ((mrClsโ(LSubSpโ๐))โ๐) = (Baseโ๐))) |
21 | 20 | simpld 494 | . 2 โข (๐ โ ๐ โ (mrIndโ(LSubSpโ๐))) |
22 | lvecdimfi.f | . 2 โข (๐ โ ๐ โ Fin) | |
23 | 15 | simprd 495 | . . 3 โข (๐ โ ((mrClsโ(LSubSpโ๐))โ๐) = (Baseโ๐)) |
24 | 20 | simprd 495 | . . 3 โข (๐ โ ((mrClsโ(LSubSpโ๐))โ๐) = (Baseโ๐)) |
25 | 23, 24 | eqtr4d 2771 | . 2 โข (๐ โ ((mrClsโ(LSubSpโ๐))โ๐) = ((mrClsโ(LSubSpโ๐))โ๐)) |
26 | 8, 3, 9, 10, 16, 21, 22, 25 | mreexfidimd 17630 | 1 โข (๐ โ ๐ โ ๐) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 395 = wceq 1534 โ wcel 2099 โwral 3058 โ cdif 3944 โช cun 3945 ๐ซ cpw 4603 {csn 4629 class class class wbr 5148 โcfv 6548 โ cen 8961 Fincfn 8964 Basecbs 17180 mrClscmrc 17563 mrIndcmri 17564 ACScacs 17565 LSubSpclss 20815 LBasisclbs 20959 LVecclvec 20987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-0g 17423 df-mre 17566 df-mrc 17567 df-mri 17568 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-subg 19078 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-drng 20626 df-lmod 20745 df-lss 20816 df-lsp 20856 df-lbs 20960 df-lvec 20988 |
This theorem is referenced by: dimvalfi 33299 |
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