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Mirrors > Home > MPE Home > Th. List > lbslcic | Structured version Visualization version GIF version |
Description: A module with a basis is isomorphic to a free module with the same cardinality. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
Ref | Expression |
---|---|
lbslcic.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lbslcic.j | ⊢ 𝐽 = (LBasis‘𝑊) |
Ref | Expression |
---|---|
lbslcic | ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝑊 ≃𝑚 (𝐹 freeLMod 𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1134 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝐼 ≈ 𝐵) | |
2 | bren 8520 | . . 3 ⊢ (𝐼 ≈ 𝐵 ↔ ∃𝑒 𝑒:𝐼–1-1-onto→𝐵) | |
3 | 1, 2 | sylib 220 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → ∃𝑒 𝑒:𝐼–1-1-onto→𝐵) |
4 | eqid 2823 | . . . 4 ⊢ (𝐹 freeLMod 𝐼) = (𝐹 freeLMod 𝐼) | |
5 | eqid 2823 | . . . 4 ⊢ (Base‘(𝐹 freeLMod 𝐼)) = (Base‘(𝐹 freeLMod 𝐼)) | |
6 | eqid 2823 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
7 | eqid 2823 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
8 | eqid 2823 | . . . 4 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
9 | eqid 2823 | . . . 4 ⊢ (𝑥 ∈ (Base‘(𝐹 freeLMod 𝐼)) ↦ (𝑊 Σg (𝑥 ∘f ( ·𝑠 ‘𝑊)𝑒))) = (𝑥 ∈ (Base‘(𝐹 freeLMod 𝐼)) ↦ (𝑊 Σg (𝑥 ∘f ( ·𝑠 ‘𝑊)𝑒))) | |
10 | simpl1 1187 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑊 ∈ LMod) | |
11 | relen 8516 | . . . . . . 7 ⊢ Rel ≈ | |
12 | 11 | brrelex1i 5610 | . . . . . 6 ⊢ (𝐼 ≈ 𝐵 → 𝐼 ∈ V) |
13 | 12 | 3ad2ant3 1131 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝐼 ∈ V) |
14 | 13 | adantr 483 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝐼 ∈ V) |
15 | lbslcic.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
16 | 15 | a1i 11 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝐹 = (Scalar‘𝑊)) |
17 | f1ofo 6624 | . . . . 5 ⊢ (𝑒:𝐼–1-1-onto→𝐵 → 𝑒:𝐼–onto→𝐵) | |
18 | 17 | adantl 484 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑒:𝐼–onto→𝐵) |
19 | lbslcic.j | . . . . . . . . 9 ⊢ 𝐽 = (LBasis‘𝑊) | |
20 | 19 | lbslinds 20979 | . . . . . . . 8 ⊢ 𝐽 ⊆ (LIndS‘𝑊) |
21 | 20 | sseli 3965 | . . . . . . 7 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ∈ (LIndS‘𝑊)) |
22 | 21 | 3ad2ant2 1130 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝐵 ∈ (LIndS‘𝑊)) |
23 | 22 | adantr 483 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝐵 ∈ (LIndS‘𝑊)) |
24 | f1of1 6616 | . . . . . 6 ⊢ (𝑒:𝐼–1-1-onto→𝐵 → 𝑒:𝐼–1-1→𝐵) | |
25 | 24 | adantl 484 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑒:𝐼–1-1→𝐵) |
26 | f1linds 20971 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ (LIndS‘𝑊) ∧ 𝑒:𝐼–1-1→𝐵) → 𝑒 LIndF 𝑊) | |
27 | 10, 23, 25, 26 | syl3anc 1367 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑒 LIndF 𝑊) |
28 | 6, 19, 8 | lbssp 19853 | . . . . . 6 ⊢ (𝐵 ∈ 𝐽 → ((LSpan‘𝑊)‘𝐵) = (Base‘𝑊)) |
29 | 28 | 3ad2ant2 1130 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → ((LSpan‘𝑊)‘𝐵) = (Base‘𝑊)) |
30 | 29 | adantr 483 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → ((LSpan‘𝑊)‘𝐵) = (Base‘𝑊)) |
31 | 4, 5, 6, 7, 8, 9, 10, 14, 16, 18, 27, 30 | indlcim 20986 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → (𝑥 ∈ (Base‘(𝐹 freeLMod 𝐼)) ↦ (𝑊 Σg (𝑥 ∘f ( ·𝑠 ‘𝑊)𝑒))) ∈ ((𝐹 freeLMod 𝐼) LMIso 𝑊)) |
32 | lmimcnv 19841 | . . 3 ⊢ ((𝑥 ∈ (Base‘(𝐹 freeLMod 𝐼)) ↦ (𝑊 Σg (𝑥 ∘f ( ·𝑠 ‘𝑊)𝑒))) ∈ ((𝐹 freeLMod 𝐼) LMIso 𝑊) → ◡(𝑥 ∈ (Base‘(𝐹 freeLMod 𝐼)) ↦ (𝑊 Σg (𝑥 ∘f ( ·𝑠 ‘𝑊)𝑒))) ∈ (𝑊 LMIso (𝐹 freeLMod 𝐼))) | |
33 | brlmici 19843 | . . 3 ⊢ (◡(𝑥 ∈ (Base‘(𝐹 freeLMod 𝐼)) ↦ (𝑊 Σg (𝑥 ∘f ( ·𝑠 ‘𝑊)𝑒))) ∈ (𝑊 LMIso (𝐹 freeLMod 𝐼)) → 𝑊 ≃𝑚 (𝐹 freeLMod 𝐼)) | |
34 | 31, 32, 33 | 3syl 18 | . 2 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑊 ≃𝑚 (𝐹 freeLMod 𝐼)) |
35 | 3, 34 | exlimddv 1936 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝑊 ≃𝑚 (𝐹 freeLMod 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3496 class class class wbr 5068 ↦ cmpt 5148 ◡ccnv 5556 –1-1→wf1 6354 –onto→wfo 6355 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 ≈ cen 8508 Basecbs 16485 Scalarcsca 16570 ·𝑠 cvsca 16571 Σg cgsu 16716 LModclmod 19636 LSpanclspn 19745 LMIso clmim 19794 ≃𝑚 clmic 19795 LBasisclbs 19848 freeLMod cfrlm 20892 LIndF clindf 20950 LIndSclinds 20951 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-hom 16591 df-cco 16592 df-0g 16717 df-gsum 16718 df-prds 16723 df-pws 16725 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lmhm 19796 df-lmim 19797 df-lmic 19798 df-lbs 19849 df-sra 19946 df-rgmod 19947 df-nzr 20033 df-dsmm 20878 df-frlm 20893 df-uvc 20929 df-lindf 20952 df-linds 20953 |
This theorem is referenced by: lmisfree 20988 frlmisfrlm 20994 |
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