Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lvecdimfi | Structured version Visualization version GIF version |
Description: Finite version of lvecdim 19929 which does not require the axiom of choice. The axiom of choice is used in acsmapd 17788, which is required in the infinite case. Suggested by Gérard Lang. (Contributed by Thierry Arnoux, 24-May-2023.) |
Ref | Expression |
---|---|
lvecdimfi.j | ⊢ 𝐽 = (LBasis‘𝑊) |
lvecdimfi.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lvecdimfi.s | ⊢ (𝜑 → 𝑆 ∈ 𝐽) |
lvecdimfi.t | ⊢ (𝜑 → 𝑇 ∈ 𝐽) |
lvecdimfi.f | ⊢ (𝜑 → 𝑆 ∈ Fin) |
Ref | Expression |
---|---|
lvecdimfi | ⊢ (𝜑 → 𝑆 ≈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecdimfi.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
2 | eqid 2821 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
3 | eqid 2821 | . . . . . 6 ⊢ (mrCls‘(LSubSp‘𝑊)) = (mrCls‘(LSubSp‘𝑊)) | |
4 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
5 | 2, 3, 4 | lssacsex 19916 | . . . . 5 ⊢ (𝑊 ∈ LVec → ((LSubSp‘𝑊) ∈ (ACS‘(Base‘𝑊)) ∧ ∀𝑥 ∈ 𝒫 (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)∀𝑧 ∈ (((mrCls‘(LSubSp‘𝑊))‘(𝑥 ∪ {𝑦})) ∖ ((mrCls‘(LSubSp‘𝑊))‘𝑥))𝑦 ∈ ((mrCls‘(LSubSp‘𝑊))‘(𝑥 ∪ {𝑧})))) |
6 | 1, 5 | syl 17 | . . . 4 ⊢ (𝜑 → ((LSubSp‘𝑊) ∈ (ACS‘(Base‘𝑊)) ∧ ∀𝑥 ∈ 𝒫 (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)∀𝑧 ∈ (((mrCls‘(LSubSp‘𝑊))‘(𝑥 ∪ {𝑦})) ∖ ((mrCls‘(LSubSp‘𝑊))‘𝑥))𝑦 ∈ ((mrCls‘(LSubSp‘𝑊))‘(𝑥 ∪ {𝑧})))) |
7 | 6 | simpld 497 | . . 3 ⊢ (𝜑 → (LSubSp‘𝑊) ∈ (ACS‘(Base‘𝑊))) |
8 | 7 | acsmred 16927 | . 2 ⊢ (𝜑 → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊))) |
9 | eqid 2821 | . 2 ⊢ (mrInd‘(LSubSp‘𝑊)) = (mrInd‘(LSubSp‘𝑊)) | |
10 | 6 | simprd 498 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)∀𝑧 ∈ (((mrCls‘(LSubSp‘𝑊))‘(𝑥 ∪ {𝑦})) ∖ ((mrCls‘(LSubSp‘𝑊))‘𝑥))𝑦 ∈ ((mrCls‘(LSubSp‘𝑊))‘(𝑥 ∪ {𝑧}))) |
11 | lvecdimfi.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐽) | |
12 | lvecdimfi.j | . . . . . 6 ⊢ 𝐽 = (LBasis‘𝑊) | |
13 | 2, 3, 4, 9, 12 | lbsacsbs 19928 | . . . . 5 ⊢ (𝑊 ∈ LVec → (𝑆 ∈ 𝐽 ↔ (𝑆 ∈ (mrInd‘(LSubSp‘𝑊)) ∧ ((mrCls‘(LSubSp‘𝑊))‘𝑆) = (Base‘𝑊)))) |
14 | 13 | biimpa 479 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (𝑆 ∈ (mrInd‘(LSubSp‘𝑊)) ∧ ((mrCls‘(LSubSp‘𝑊))‘𝑆) = (Base‘𝑊))) |
15 | 1, 11, 14 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑆 ∈ (mrInd‘(LSubSp‘𝑊)) ∧ ((mrCls‘(LSubSp‘𝑊))‘𝑆) = (Base‘𝑊))) |
16 | 15 | simpld 497 | . 2 ⊢ (𝜑 → 𝑆 ∈ (mrInd‘(LSubSp‘𝑊))) |
17 | lvecdimfi.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐽) | |
18 | 2, 3, 4, 9, 12 | lbsacsbs 19928 | . . . . 5 ⊢ (𝑊 ∈ LVec → (𝑇 ∈ 𝐽 ↔ (𝑇 ∈ (mrInd‘(LSubSp‘𝑊)) ∧ ((mrCls‘(LSubSp‘𝑊))‘𝑇) = (Base‘𝑊)))) |
19 | 18 | biimpa 479 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝑇 ∈ 𝐽) → (𝑇 ∈ (mrInd‘(LSubSp‘𝑊)) ∧ ((mrCls‘(LSubSp‘𝑊))‘𝑇) = (Base‘𝑊))) |
20 | 1, 17, 19 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑇 ∈ (mrInd‘(LSubSp‘𝑊)) ∧ ((mrCls‘(LSubSp‘𝑊))‘𝑇) = (Base‘𝑊))) |
21 | 20 | simpld 497 | . 2 ⊢ (𝜑 → 𝑇 ∈ (mrInd‘(LSubSp‘𝑊))) |
22 | lvecdimfi.f | . 2 ⊢ (𝜑 → 𝑆 ∈ Fin) | |
23 | 15 | simprd 498 | . . 3 ⊢ (𝜑 → ((mrCls‘(LSubSp‘𝑊))‘𝑆) = (Base‘𝑊)) |
24 | 20 | simprd 498 | . . 3 ⊢ (𝜑 → ((mrCls‘(LSubSp‘𝑊))‘𝑇) = (Base‘𝑊)) |
25 | 23, 24 | eqtr4d 2859 | . 2 ⊢ (𝜑 → ((mrCls‘(LSubSp‘𝑊))‘𝑆) = ((mrCls‘(LSubSp‘𝑊))‘𝑇)) |
26 | 8, 3, 9, 10, 16, 21, 22, 25 | mreexfidimd 16921 | 1 ⊢ (𝜑 → 𝑆 ≈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∖ cdif 3933 ∪ cun 3934 𝒫 cpw 4539 {csn 4567 class class class wbr 5066 ‘cfv 6355 ≈ cen 8506 Fincfn 8509 Basecbs 16483 mrClscmrc 16854 mrIndcmri 16855 ACScacs 16856 LSubSpclss 19703 LBasisclbs 19846 LVecclvec 19874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-mre 16857 df-mrc 16858 df-mri 16859 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lbs 19847 df-lvec 19875 |
This theorem is referenced by: dimvalfi 31002 |
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