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Mirrors > Home > MPE Home > Th. List > lmimlbs | Structured version Visualization version GIF version |
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 19561 | . . 3 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇)) | |
2 | eqid 2778 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
3 | eqid 2778 | . . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
4 | 2, 3 | lmimf1o 19560 | . . . 4 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)) |
5 | f1of1 6445 | . . . 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 20682 | . . . 4 ⊢ 𝐽 ⊆ (LIndS‘𝑆) |
9 | 8 | sseli 3856 | . . 3 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ∈ (LIndS‘𝑆)) |
10 | 2, 3 | lindsmm2 20678 | . . 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 1402 | . 2 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ 𝐵) ∈ (LIndS‘𝑇)) |
12 | eqid 2778 | . . . . . 6 ⊢ (LSpan‘𝑆) = (LSpan‘𝑆) | |
13 | 2, 7, 12 | lbssp 19576 | . . . . 5 ⊢ (𝐵 ∈ 𝐽 → ((LSpan‘𝑆)‘𝐵) = (Base‘𝑆)) |
14 | 13 | adantl 474 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → ((LSpan‘𝑆)‘𝐵) = (Base‘𝑆)) |
15 | 14 | imaeq2d 5772 | . . 3 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ ((LSpan‘𝑆)‘𝐵)) = (𝐹 “ (Base‘𝑆))) |
16 | 2, 7 | lbsss 19574 | . . . 4 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ⊆ (Base‘𝑆)) |
17 | eqid 2778 | . . . . 5 ⊢ (LSpan‘𝑇) = (LSpan‘𝑇) | |
18 | 2, 12, 17 | lmhmlsp 19546 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐵 ⊆ (Base‘𝑆)) → (𝐹 “ ((LSpan‘𝑆)‘𝐵)) = ((LSpan‘𝑇)‘(𝐹 “ 𝐵))) |
19 | 1, 16, 18 | syl2an 586 | . . 3 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ ((LSpan‘𝑆)‘𝐵)) = ((LSpan‘𝑇)‘(𝐹 “ 𝐵))) |
20 | 4 | adantr 473 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)) |
21 | f1ofo 6453 | . . . 4 ⊢ (𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇) → 𝐹:(Base‘𝑆)–onto→(Base‘𝑇)) | |
22 | foima 6426 | . . . 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 2822 | . 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 20681 | . 2 ⊢ ((𝐹 “ 𝐵) ∈ 𝐾 ↔ ((𝐹 “ 𝐵) ∈ (LIndS‘𝑇) ∧ ((LSpan‘𝑇)‘(𝐹 “ 𝐵)) = (Base‘𝑇))) |
27 | 11, 24, 26 | sylanbrc 575 | 1 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ 𝐵) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ⊆ wss 3831 “ cima 5411 –1-1→wf1 6187 –onto→wfo 6188 –1-1-onto→wf1o 6189 ‘cfv 6190 (class class class)co 6978 Basecbs 16342 LSpanclspn 19468 LMHom clmhm 19516 LMIso clmim 19517 LBasisclbs 19571 LIndSclinds 20654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-2 11506 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-0g 16574 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-grp 17897 df-minusg 17898 df-sbg 17899 df-subg 18063 df-ghm 18130 df-mgp 18966 df-ur 18978 df-ring 19025 df-lmod 19361 df-lss 19429 df-lsp 19469 df-lmhm 19519 df-lmim 19520 df-lbs 19572 df-lindf 20655 df-linds 20656 |
This theorem is referenced by: lmiclbs 20686 dimkerim 30652 |
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