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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 6030 | . . . . 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 19973 | . . . . . 6 ⊢ ((𝐴 LIndF 𝑇 ∧ 𝑇 ∈ LMod) → 𝐴:dom 𝐴⟶𝐶) |
14 | 12, 6, 13 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → 𝐴:dom 𝐴⟶𝐶) |
15 | frn 5966 | . . . . 5 ⊢ (𝐴:dom 𝐴⟶𝐶 → ran 𝐴 ⊆ 𝐶) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ 𝐶) |
17 | df-f 5808 | . . . 4 ⊢ (𝐴:𝐼⟶𝐶 ↔ (𝐴 Fn 𝐼 ∧ ran 𝐴 ⊆ 𝐶)) | |
18 | 11, 16, 17 | sylanbrc 695 | . . 3 ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 18 | frlmup1 19956 | . 2 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) |
20 | 1, 2, 3, 4, 5, 6, 7, 8, 18 | islindf5 19997 | . . . 4 ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ 𝐸:𝐵–1-1→𝐶)) |
21 | 12, 20 | mpbid 221 | . . 3 ⊢ (𝜑 → 𝐸:𝐵–1-1→𝐶) |
22 | indlcim.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑇) | |
23 | 1, 2, 3, 4, 5, 6, 7, 8, 18, 22 | frlmup3 19958 | . . . 4 ⊢ (𝜑 → ran 𝐸 = (𝑁‘ran 𝐴)) |
24 | forn 6031 | . . . . . 6 ⊢ (𝐴:𝐼–onto→𝐽 → ran 𝐴 = 𝐽) | |
25 | 9, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → ran 𝐴 = 𝐽) |
26 | 25 | fveq2d 6107 | . . . 4 ⊢ (𝜑 → (𝑁‘ran 𝐴) = (𝑁‘𝐽)) |
27 | indlcim.s | . . . 4 ⊢ (𝜑 → (𝑁‘𝐽) = 𝐶) | |
28 | 23, 26, 27 | 3eqtrd 2648 | . . 3 ⊢ (𝜑 → ran 𝐸 = 𝐶) |
29 | dff1o5 6059 | . . 3 ⊢ (𝐸:𝐵–1-1-onto→𝐶 ↔ (𝐸:𝐵–1-1→𝐶 ∧ ran 𝐸 = 𝐶)) | |
30 | 21, 28, 29 | sylanbrc 695 | . 2 ⊢ (𝜑 → 𝐸:𝐵–1-1-onto→𝐶) |
31 | 2, 3 | islmim 18883 | . 2 ⊢ (𝐸 ∈ (𝐹 LMIso 𝑇) ↔ (𝐸 ∈ (𝐹 LMHom 𝑇) ∧ 𝐸:𝐵–1-1-onto→𝐶)) |
32 | 19, 30, 31 | sylanbrc 695 | 1 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMIso 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 Σg cgsu 15924 LModclmod 18686 LSpanclspn 18792 LMHom clmhm 18840 LMIso clmim 18841 freeLMod cfrlm 19909 LIndF clindf 19962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lmhm 18843 df-lmim 18844 df-lbs 18896 df-sra 18993 df-rgmod 18994 df-nzr 19079 df-dsmm 19895 df-frlm 19910 df-uvc 19941 df-lindf 19964 |
This theorem is referenced by: lbslcic 19999 |
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