Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 19830 | . . 3 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇)) | |
2 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
3 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
4 | 2, 3 | lmimf1o 19829 | . . . 4 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)) |
5 | f1of1 6608 | . . . 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 20971 | . . . 4 ⊢ 𝐽 ⊆ (LIndS‘𝑆) |
9 | 8 | sseli 3962 | . . 3 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ∈ (LIndS‘𝑆)) |
10 | 2, 3 | lindsmm2 20967 | . . 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 1418 | . 2 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ 𝐵) ∈ (LIndS‘𝑇)) |
12 | eqid 2821 | . . . . . 6 ⊢ (LSpan‘𝑆) = (LSpan‘𝑆) | |
13 | 2, 7, 12 | lbssp 19845 | . . . . 5 ⊢ (𝐵 ∈ 𝐽 → ((LSpan‘𝑆)‘𝐵) = (Base‘𝑆)) |
14 | 13 | adantl 484 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → ((LSpan‘𝑆)‘𝐵) = (Base‘𝑆)) |
15 | 14 | imaeq2d 5923 | . . 3 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ ((LSpan‘𝑆)‘𝐵)) = (𝐹 “ (Base‘𝑆))) |
16 | 2, 7 | lbsss 19843 | . . . 4 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ⊆ (Base‘𝑆)) |
17 | eqid 2821 | . . . . 5 ⊢ (LSpan‘𝑇) = (LSpan‘𝑇) | |
18 | 2, 12, 17 | lmhmlsp 19815 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐵 ⊆ (Base‘𝑆)) → (𝐹 “ ((LSpan‘𝑆)‘𝐵)) = ((LSpan‘𝑇)‘(𝐹 “ 𝐵))) |
19 | 1, 16, 18 | syl2an 597 | . . 3 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ ((LSpan‘𝑆)‘𝐵)) = ((LSpan‘𝑇)‘(𝐹 “ 𝐵))) |
20 | 4 | adantr 483 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)) |
21 | f1ofo 6616 | . . . 4 ⊢ (𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇) → 𝐹:(Base‘𝑆)–onto→(Base‘𝑇)) | |
22 | foima 6589 | . . . 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 2864 | . 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 20970 | . 2 ⊢ ((𝐹 “ 𝐵) ∈ 𝐾 ↔ ((𝐹 “ 𝐵) ∈ (LIndS‘𝑇) ∧ ((LSpan‘𝑇)‘(𝐹 “ 𝐵)) = (Base‘𝑇))) |
27 | 11, 24, 26 | sylanbrc 585 | 1 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ 𝐵) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 “ cima 5552 –1-1→wf1 6346 –onto→wfo 6347 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 LSpanclspn 19737 LMHom clmhm 19785 LMIso clmim 19786 LBasisclbs 19840 LIndSclinds 20943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-ghm 18350 df-mgp 19234 df-ur 19246 df-ring 19293 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lmhm 19788 df-lmim 19789 df-lbs 19841 df-lindf 20944 df-linds 20945 |
This theorem is referenced by: lmiclbs 20975 dimkerim 31018 |
Copyright terms: Public domain | W3C validator |