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Mirrors > Home > MPE Home > Th. List > frlmisfrlm | Structured version Visualization version GIF version |
Description: A free module is isomorphic to a free module over the same (nonzero) ring, with the same cardinality. (Contributed by AV, 10-Mar-2019.) |
Ref | Expression |
---|---|
frlmisfrlm | ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → (𝑅 freeLMod 𝐼) ≃𝑚 (𝑅 freeLMod 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nzrring 19584 | . . . . 5 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
2 | eqid 2799 | . . . . . 6 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼) | |
3 | 2 | frlmlmod 20418 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑌) → (𝑅 freeLMod 𝐼) ∈ LMod) |
4 | 1, 3 | sylan 576 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌) → (𝑅 freeLMod 𝐼) ∈ LMod) |
5 | 4 | 3adant3 1163 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → (𝑅 freeLMod 𝐼) ∈ LMod) |
6 | eqid 2799 | . . . . . 6 ⊢ (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼) | |
7 | eqid 2799 | . . . . . 6 ⊢ (LBasis‘(𝑅 freeLMod 𝐼)) = (LBasis‘(𝑅 freeLMod 𝐼)) | |
8 | 2, 6, 7 | frlmlbs 20461 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑌) → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘(𝑅 freeLMod 𝐼))) |
9 | 1, 8 | sylan 576 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌) → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘(𝑅 freeLMod 𝐼))) |
10 | 9 | 3adant3 1163 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘(𝑅 freeLMod 𝐼))) |
11 | simp3 1169 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → 𝐼 ≈ 𝐽) | |
12 | 11 | ensymd 8246 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → 𝐽 ≈ 𝐼) |
13 | 6 | uvcendim 20511 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌) → 𝐼 ≈ ran (𝑅 unitVec 𝐼)) |
14 | 13 | 3adant3 1163 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → 𝐼 ≈ ran (𝑅 unitVec 𝐼)) |
15 | entr 8247 | . . . 4 ⊢ ((𝐽 ≈ 𝐼 ∧ 𝐼 ≈ ran (𝑅 unitVec 𝐼)) → 𝐽 ≈ ran (𝑅 unitVec 𝐼)) | |
16 | 12, 14, 15 | syl2anc 580 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → 𝐽 ≈ ran (𝑅 unitVec 𝐼)) |
17 | eqid 2799 | . . . 4 ⊢ (Scalar‘(𝑅 freeLMod 𝐼)) = (Scalar‘(𝑅 freeLMod 𝐼)) | |
18 | 17, 7 | lbslcic 20505 | . . 3 ⊢ (((𝑅 freeLMod 𝐼) ∈ LMod ∧ ran (𝑅 unitVec 𝐼) ∈ (LBasis‘(𝑅 freeLMod 𝐼)) ∧ 𝐽 ≈ ran (𝑅 unitVec 𝐼)) → (𝑅 freeLMod 𝐼) ≃𝑚 ((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐽)) |
19 | 5, 10, 16, 18 | syl3anc 1491 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → (𝑅 freeLMod 𝐼) ≃𝑚 ((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐽)) |
20 | 2 | frlmsca 20422 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼))) |
21 | 20 | 3adant3 1163 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼))) |
22 | 21 | oveq1d 6893 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → (𝑅 freeLMod 𝐽) = ((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐽)) |
23 | 19, 22 | breqtrrd 4871 | 1 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → (𝑅 freeLMod 𝐼) ≃𝑚 (𝑅 freeLMod 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 class class class wbr 4843 ran crn 5313 ‘cfv 6101 (class class class)co 6878 ≈ cen 8192 Scalarcsca 16270 Ringcrg 18863 LModclmod 19181 ≃𝑚 clmic 19342 LBasisclbs 19395 NzRingcnzr 19580 freeLMod cfrlm 20415 unitVec cuvc 20446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-map 8097 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-sup 8590 df-oi 8657 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-fz 12581 df-fzo 12721 df-seq 13056 df-hash 13371 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-hom 16291 df-cco 16292 df-0g 16417 df-gsum 16418 df-prds 16423 df-pws 16425 df-mre 16561 df-mrc 16562 df-acs 16564 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-mhm 17650 df-submnd 17651 df-grp 17741 df-minusg 17742 df-sbg 17743 df-mulg 17857 df-subg 17904 df-ghm 17971 df-cntz 18062 df-cmn 18510 df-abl 18511 df-mgp 18806 df-ur 18818 df-ring 18865 df-subrg 19096 df-lmod 19183 df-lss 19251 df-lsp 19293 df-lmhm 19343 df-lmim 19344 df-lmic 19345 df-lbs 19396 df-sra 19495 df-rgmod 19496 df-nzr 19581 df-dsmm 20401 df-frlm 20416 df-uvc 20447 df-lindf 20470 df-linds 20471 |
This theorem is referenced by: frlmiscvec 20513 |
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