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| Description: The isomorphic image of a basis is a basis. (Contributed by Stefan O'Rear, 26-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| lmimlbs.j | ⊢ 𝐽 = (LBasis‘𝑆) | 
| lmimlbs.k | ⊢ 𝐾 = (LBasis‘𝑇) | 
| Ref | Expression | 
|---|---|
| lmimlbs | ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ 𝐵) ∈ 𝐾) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lmimlmhm 21064 | . . 3 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇)) | |
| 2 | eqid 2736 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
| 3 | eqid 2736 | . . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
| 4 | 2, 3 | lmimf1o 21063 | . . . 4 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)) | 
| 5 | f1of1 6846 | . . . 4 ⊢ (𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇) → 𝐹:(Base‘𝑆)–1-1→(Base‘𝑇)) | |
| 6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → 𝐹:(Base‘𝑆)–1-1→(Base‘𝑇)) | 
| 7 | lmimlbs.j | . . . . 5 ⊢ 𝐽 = (LBasis‘𝑆) | |
| 8 | 7 | lbslinds 21854 | . . . 4 ⊢ 𝐽 ⊆ (LIndS‘𝑆) | 
| 9 | 8 | sseli 3978 | . . 3 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ∈ (LIndS‘𝑆)) | 
| 10 | 2, 3 | lindsmm2 21850 | . . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:(Base‘𝑆)–1-1→(Base‘𝑇) ∧ 𝐵 ∈ (LIndS‘𝑆)) → (𝐹 “ 𝐵) ∈ (LIndS‘𝑇)) | 
| 11 | 1, 6, 9, 10 | syl2an3an 1423 | . 2 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ 𝐵) ∈ (LIndS‘𝑇)) | 
| 12 | eqid 2736 | . . . . . 6 ⊢ (LSpan‘𝑆) = (LSpan‘𝑆) | |
| 13 | 2, 7, 12 | lbssp 21079 | . . . . 5 ⊢ (𝐵 ∈ 𝐽 → ((LSpan‘𝑆)‘𝐵) = (Base‘𝑆)) | 
| 14 | 13 | adantl 481 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → ((LSpan‘𝑆)‘𝐵) = (Base‘𝑆)) | 
| 15 | 14 | imaeq2d 6077 | . . 3 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ ((LSpan‘𝑆)‘𝐵)) = (𝐹 “ (Base‘𝑆))) | 
| 16 | 2, 7 | lbsss 21077 | . . . 4 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ⊆ (Base‘𝑆)) | 
| 17 | eqid 2736 | . . . . 5 ⊢ (LSpan‘𝑇) = (LSpan‘𝑇) | |
| 18 | 2, 12, 17 | lmhmlsp 21049 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐵 ⊆ (Base‘𝑆)) → (𝐹 “ ((LSpan‘𝑆)‘𝐵)) = ((LSpan‘𝑇)‘(𝐹 “ 𝐵))) | 
| 19 | 1, 16, 18 | syl2an 596 | . . 3 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ ((LSpan‘𝑆)‘𝐵)) = ((LSpan‘𝑇)‘(𝐹 “ 𝐵))) | 
| 20 | 4 | adantr 480 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)) | 
| 21 | f1ofo 6854 | . . . 4 ⊢ (𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇) → 𝐹:(Base‘𝑆)–onto→(Base‘𝑇)) | |
| 22 | foima 6824 | . . . 4 ⊢ (𝐹:(Base‘𝑆)–onto→(Base‘𝑇) → (𝐹 “ (Base‘𝑆)) = (Base‘𝑇)) | |
| 23 | 20, 21, 22 | 3syl 18 | . . 3 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ (Base‘𝑆)) = (Base‘𝑇)) | 
| 24 | 15, 19, 23 | 3eqtr3d 2784 | . 2 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → ((LSpan‘𝑇)‘(𝐹 “ 𝐵)) = (Base‘𝑇)) | 
| 25 | lmimlbs.k | . . 3 ⊢ 𝐾 = (LBasis‘𝑇) | |
| 26 | 3, 25, 17 | islbs4 21853 | . 2 ⊢ ((𝐹 “ 𝐵) ∈ 𝐾 ↔ ((𝐹 “ 𝐵) ∈ (LIndS‘𝑇) ∧ ((LSpan‘𝑇)‘(𝐹 “ 𝐵)) = (Base‘𝑇))) | 
| 27 | 11, 24, 26 | sylanbrc 583 | 1 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ 𝐵) ∈ 𝐾) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 “ cima 5687 –1-1→wf1 6557 –onto→wfo 6558 –1-1-onto→wf1o 6559 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 LSpanclspn 20970 LMHom clmhm 21019 LMIso clmim 21020 LBasisclbs 21074 LIndSclinds 21826 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 df-sbg 18957 df-subg 19142 df-ghm 19232 df-mgp 20139 df-ur 20180 df-ring 20233 df-lmod 20861 df-lss 20931 df-lsp 20971 df-lmhm 21022 df-lmim 21023 df-lbs 21075 df-lindf 21827 df-linds 21828 | 
| This theorem is referenced by: lmiclbs 21858 lmimdim 33655 dimkerim 33679 | 
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