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Mirrors > Home > MPE Home > Th. List > indlcim | Structured version Visualization version GIF version |
Description: An independent, spanning family extends to an isomorphism from a free module. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
Ref | Expression |
---|---|
indlcim.f | ⊢ 𝐹 = (𝑅 freeLMod 𝐼) |
indlcim.b | ⊢ 𝐵 = (Base‘𝐹) |
indlcim.c | ⊢ 𝐶 = (Base‘𝑇) |
indlcim.v | ⊢ · = ( ·𝑠 ‘𝑇) |
indlcim.n | ⊢ 𝑁 = (LSpan‘𝑇) |
indlcim.e | ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘f · 𝐴))) |
indlcim.t | ⊢ (𝜑 → 𝑇 ∈ LMod) |
indlcim.i | ⊢ (𝜑 → 𝐼 ∈ 𝑋) |
indlcim.r | ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) |
indlcim.a | ⊢ (𝜑 → 𝐴:𝐼–onto→𝐽) |
indlcim.l | ⊢ (𝜑 → 𝐴 LIndF 𝑇) |
indlcim.s | ⊢ (𝜑 → (𝑁‘𝐽) = 𝐶) |
Ref | Expression |
---|---|
indlcim | ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMIso 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indlcim.f | . . 3 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
2 | indlcim.b | . . 3 ⊢ 𝐵 = (Base‘𝐹) | |
3 | indlcim.c | . . 3 ⊢ 𝐶 = (Base‘𝑇) | |
4 | indlcim.v | . . 3 ⊢ · = ( ·𝑠 ‘𝑇) | |
5 | indlcim.e | . . 3 ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘f · 𝐴))) | |
6 | indlcim.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ LMod) | |
7 | indlcim.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑋) | |
8 | indlcim.r | . . 3 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) | |
9 | indlcim.a | . . . . 5 ⊢ (𝜑 → 𝐴:𝐼–onto→𝐽) | |
10 | fofn 6613 | . . . . 5 ⊢ (𝐴:𝐼–onto→𝐽 → 𝐴 Fn 𝐼) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 Fn 𝐼) |
12 | indlcim.l | . . . . . 6 ⊢ (𝜑 → 𝐴 LIndF 𝑇) | |
13 | 3 | lindff 20731 | . . . . . 6 ⊢ ((𝐴 LIndF 𝑇 ∧ 𝑇 ∈ LMod) → 𝐴:dom 𝐴⟶𝐶) |
14 | 12, 6, 13 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → 𝐴:dom 𝐴⟶𝐶) |
15 | 14 | frnd 6531 | . . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ 𝐶) |
16 | df-f 6362 | . . . 4 ⊢ (𝐴:𝐼⟶𝐶 ↔ (𝐴 Fn 𝐼 ∧ ran 𝐴 ⊆ 𝐶)) | |
17 | 11, 15, 16 | sylanbrc 586 | . . 3 ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 17 | frlmup1 20714 | . 2 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 17 | islindf5 20755 | . . . 4 ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ 𝐸:𝐵–1-1→𝐶)) |
20 | 12, 19 | mpbid 235 | . . 3 ⊢ (𝜑 → 𝐸:𝐵–1-1→𝐶) |
21 | indlcim.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑇) | |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 17, 21 | frlmup3 20716 | . . . 4 ⊢ (𝜑 → ran 𝐸 = (𝑁‘ran 𝐴)) |
23 | forn 6614 | . . . . . 6 ⊢ (𝐴:𝐼–onto→𝐽 → ran 𝐴 = 𝐽) | |
24 | 9, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → ran 𝐴 = 𝐽) |
25 | 24 | fveq2d 6699 | . . . 4 ⊢ (𝜑 → (𝑁‘ran 𝐴) = (𝑁‘𝐽)) |
26 | indlcim.s | . . . 4 ⊢ (𝜑 → (𝑁‘𝐽) = 𝐶) | |
27 | 22, 25, 26 | 3eqtrd 2775 | . . 3 ⊢ (𝜑 → ran 𝐸 = 𝐶) |
28 | dff1o5 6648 | . . 3 ⊢ (𝐸:𝐵–1-1-onto→𝐶 ↔ (𝐸:𝐵–1-1→𝐶 ∧ ran 𝐸 = 𝐶)) | |
29 | 20, 27, 28 | sylanbrc 586 | . 2 ⊢ (𝜑 → 𝐸:𝐵–1-1-onto→𝐶) |
30 | 2, 3 | islmim 20053 | . 2 ⊢ (𝐸 ∈ (𝐹 LMIso 𝑇) ↔ (𝐸 ∈ (𝐹 LMHom 𝑇) ∧ 𝐸:𝐵–1-1-onto→𝐶)) |
31 | 18, 29, 30 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMIso 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 class class class wbr 5039 ↦ cmpt 5120 dom cdm 5536 ran crn 5537 Fn wfn 6353 ⟶wf 6354 –1-1→wf1 6355 –onto→wfo 6356 –1-1-onto→wf1o 6357 ‘cfv 6358 (class class class)co 7191 ∘f cof 7445 Basecbs 16666 Scalarcsca 16752 ·𝑠 cvsca 16753 Σg cgsu 16899 LModclmod 19853 LSpanclspn 19962 LMHom clmhm 20010 LMIso clmim 20011 freeLMod cfrlm 20662 LIndF clindf 20720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-sup 9036 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-hom 16773 df-cco 16774 df-0g 16900 df-gsum 16901 df-prds 16906 df-pws 16908 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-ghm 18574 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-subrg 19752 df-lmod 19855 df-lss 19923 df-lsp 19963 df-lmhm 20013 df-lmim 20014 df-lbs 20066 df-sra 20163 df-rgmod 20164 df-nzr 20250 df-dsmm 20648 df-frlm 20663 df-uvc 20699 df-lindf 20722 |
This theorem is referenced by: lbslcic 20757 |
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