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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 20560 | . . . . 5 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
2 | eqid 2733 | . . . . . 6 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼) | |
3 | 2 | frlmlmod 20984 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑌) → (𝑅 freeLMod 𝐼) ∈ LMod) |
4 | 1, 3 | sylan 579 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌) → (𝑅 freeLMod 𝐼) ∈ LMod) |
5 | 4 | 3adant3 1130 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → (𝑅 freeLMod 𝐼) ∈ LMod) |
6 | eqid 2733 | . . . . . 6 ⊢ (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼) | |
7 | eqid 2733 | . . . . . 6 ⊢ (LBasis‘(𝑅 freeLMod 𝐼)) = (LBasis‘(𝑅 freeLMod 𝐼)) | |
8 | 2, 6, 7 | frlmlbs 21032 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑌) → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘(𝑅 freeLMod 𝐼))) |
9 | 1, 8 | sylan 579 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌) → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘(𝑅 freeLMod 𝐼))) |
10 | 9 | 3adant3 1130 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘(𝑅 freeLMod 𝐼))) |
11 | simp3 1136 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → 𝐼 ≈ 𝐽) | |
12 | 11 | ensymd 8815 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → 𝐽 ≈ 𝐼) |
13 | 6 | uvcendim 21082 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌) → 𝐼 ≈ ran (𝑅 unitVec 𝐼)) |
14 | 13 | 3adant3 1130 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → 𝐼 ≈ ran (𝑅 unitVec 𝐼)) |
15 | entr 8816 | . . . 4 ⊢ ((𝐽 ≈ 𝐼 ∧ 𝐼 ≈ ran (𝑅 unitVec 𝐼)) → 𝐽 ≈ ran (𝑅 unitVec 𝐼)) | |
16 | 12, 14, 15 | syl2anc 583 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → 𝐽 ≈ ran (𝑅 unitVec 𝐼)) |
17 | eqid 2733 | . . . 4 ⊢ (Scalar‘(𝑅 freeLMod 𝐼)) = (Scalar‘(𝑅 freeLMod 𝐼)) | |
18 | 17, 7 | lbslcic 21076 | . . 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 1369 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → (𝑅 freeLMod 𝐼) ≃𝑚 ((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐽)) |
20 | 2 | frlmsca 20988 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼))) |
21 | 20 | 3adant3 1130 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼))) |
22 | 21 | oveq1d 7310 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → (𝑅 freeLMod 𝐽) = ((Scalar‘(𝑅 freeLMod 𝐼)) freeLMod 𝐽)) |
23 | 19, 22 | breqtrrd 5105 | 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 1085 = wceq 1537 ∈ wcel 2101 class class class wbr 5077 ran crn 5592 ‘cfv 6447 (class class class)co 7295 ≈ cen 8750 Scalarcsca 16993 Ringcrg 19811 LModclmod 20151 ≃𝑚 clmic 20311 LBasisclbs 20364 NzRingcnzr 20556 freeLMod cfrlm 20981 unitVec cuvc 21017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-iin 4930 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-isom 6456 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-of 7553 df-om 7733 df-1st 7851 df-2nd 7852 df-supp 7998 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-map 8637 df-ixp 8706 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-fsupp 9157 df-sup 9229 df-oi 9297 df-card 9725 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-7 12069 df-8 12070 df-9 12071 df-n0 12262 df-z 12348 df-dec 12466 df-uz 12611 df-fz 13268 df-fzo 13411 df-seq 13750 df-hash 14073 df-struct 16876 df-sets 16893 df-slot 16911 df-ndx 16923 df-base 16941 df-ress 16970 df-plusg 17003 df-mulr 17004 df-sca 17006 df-vsca 17007 df-ip 17008 df-tset 17009 df-ple 17010 df-ds 17012 df-hom 17014 df-cco 17015 df-0g 17180 df-gsum 17181 df-prds 17186 df-pws 17188 df-mre 17323 df-mrc 17324 df-acs 17326 df-mgm 18354 df-sgrp 18403 df-mnd 18414 df-mhm 18458 df-submnd 18459 df-grp 18608 df-minusg 18609 df-sbg 18610 df-mulg 18729 df-subg 18780 df-ghm 18860 df-cntz 18951 df-cmn 19416 df-abl 19417 df-mgp 19749 df-ur 19766 df-ring 19813 df-subrg 20050 df-lmod 20153 df-lss 20222 df-lsp 20262 df-lmhm 20312 df-lmim 20313 df-lmic 20314 df-lbs 20365 df-sra 20462 df-rgmod 20463 df-nzr 20557 df-dsmm 20967 df-frlm 20982 df-uvc 21018 df-lindf 21041 df-linds 21042 |
This theorem is referenced by: frlmiscvec 21084 |
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