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Mirrors > Home > MPE Home > Th. List > Mathboxes > lvecdimfi | Structured version Visualization version GIF version |
Description: Finite version of lvecdim 21004 which does not require the axiom of choice. The axiom of choice is used in acsmapd 18515, 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 2724 | . . . . . 6 โข (LSubSpโ๐) = (LSubSpโ๐) | |
3 | eqid 2724 | . . . . . 6 โข (mrClsโ(LSubSpโ๐)) = (mrClsโ(LSubSpโ๐)) | |
4 | eqid 2724 | . . . . . 6 โข (Baseโ๐) = (Baseโ๐) | |
5 | 2, 3, 4 | lssacsex 20991 | . . . . 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 17605 | . 2 โข (๐ โ (LSubSpโ๐) โ (Mooreโ(Baseโ๐))) |
9 | eqid 2724 | . 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 21003 | . . . . 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 21003 | . . . . 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 2767 | . 2 โข (๐ โ ((mrClsโ(LSubSpโ๐))โ๐) = ((mrClsโ(LSubSpโ๐))โ๐)) |
26 | 8, 3, 9, 10, 16, 21, 22, 25 | mreexfidimd 17599 | 1 โข (๐ โ ๐ โ ๐) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 395 = wceq 1533 โ wcel 2098 โwral 3053 โ cdif 3938 โช cun 3939 ๐ซ cpw 4595 {csn 4621 class class class wbr 5139 โcfv 6534 โ cen 8933 Fincfn 8936 Basecbs 17149 mrClscmrc 17532 mrIndcmri 17533 ACScacs 17534 LSubSpclss 20774 LBasisclbs 20918 LVecclvec 20946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-0g 17392 df-mre 17535 df-mrc 17536 df-mri 17537 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lbs 20919 df-lvec 20947 |
This theorem is referenced by: dimvalfi 33194 |
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