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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 6278 | . . . . 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 20356 | . . . . . 6 ⊢ ((𝐴 LIndF 𝑇 ∧ 𝑇 ∈ LMod) → 𝐴:dom 𝐴⟶𝐶) |
14 | 12, 6, 13 | syl2anc 696 | . . . . 5 ⊢ (𝜑 → 𝐴:dom 𝐴⟶𝐶) |
15 | frn 6214 | . . . . 5 ⊢ (𝐴:dom 𝐴⟶𝐶 → ran 𝐴 ⊆ 𝐶) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ 𝐶) |
17 | df-f 6053 | . . . 4 ⊢ (𝐴:𝐼⟶𝐶 ↔ (𝐴 Fn 𝐼 ∧ ran 𝐴 ⊆ 𝐶)) | |
18 | 11, 16, 17 | sylanbrc 701 | . . 3 ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 18 | frlmup1 20339 | . 2 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) |
20 | 1, 2, 3, 4, 5, 6, 7, 8, 18 | islindf5 20380 | . . . 4 ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ 𝐸:𝐵–1-1→𝐶)) |
21 | 12, 20 | mpbid 222 | . . 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 20341 | . . . 4 ⊢ (𝜑 → ran 𝐸 = (𝑁‘ran 𝐴)) |
24 | forn 6279 | . . . . . 6 ⊢ (𝐴:𝐼–onto→𝐽 → ran 𝐴 = 𝐽) | |
25 | 9, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → ran 𝐴 = 𝐽) |
26 | 25 | fveq2d 6356 | . . . 4 ⊢ (𝜑 → (𝑁‘ran 𝐴) = (𝑁‘𝐽)) |
27 | indlcim.s | . . . 4 ⊢ (𝜑 → (𝑁‘𝐽) = 𝐶) | |
28 | 23, 26, 27 | 3eqtrd 2798 | . . 3 ⊢ (𝜑 → ran 𝐸 = 𝐶) |
29 | dff1o5 6307 | . . 3 ⊢ (𝐸:𝐵–1-1-onto→𝐶 ↔ (𝐸:𝐵–1-1→𝐶 ∧ ran 𝐸 = 𝐶)) | |
30 | 21, 28, 29 | sylanbrc 701 | . 2 ⊢ (𝜑 → 𝐸:𝐵–1-1-onto→𝐶) |
31 | 2, 3 | islmim 19264 | . 2 ⊢ (𝐸 ∈ (𝐹 LMIso 𝑇) ↔ (𝐸 ∈ (𝐹 LMHom 𝑇) ∧ 𝐸:𝐵–1-1-onto→𝐶)) |
32 | 19, 30, 31 | sylanbrc 701 | 1 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMIso 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 ⊆ wss 3715 class class class wbr 4804 ↦ cmpt 4881 dom cdm 5266 ran crn 5267 Fn wfn 6044 ⟶wf 6045 –1-1→wf1 6046 –onto→wfo 6047 –1-1-onto→wf1o 6048 ‘cfv 6049 (class class class)co 6813 ∘𝑓 cof 7060 Basecbs 16059 Scalarcsca 16146 ·𝑠 cvsca 16147 Σg cgsu 16303 LModclmod 19065 LSpanclspn 19173 LMHom clmhm 19221 LMIso clmim 19222 freeLMod cfrlm 20292 LIndF clindf 20345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-map 8025 df-ixp 8075 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-sup 8513 df-oi 8580 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-fz 12520 df-fzo 12660 df-seq 12996 df-hash 13312 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-hom 16168 df-cco 16169 df-0g 16304 df-gsum 16305 df-prds 16310 df-pws 16312 df-mre 16448 df-mrc 16449 df-acs 16451 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-mhm 17536 df-submnd 17537 df-grp 17626 df-minusg 17627 df-sbg 17628 df-mulg 17742 df-subg 17792 df-ghm 17859 df-cntz 17950 df-cmn 18395 df-abl 18396 df-mgp 18690 df-ur 18702 df-ring 18749 df-subrg 18980 df-lmod 19067 df-lss 19135 df-lsp 19174 df-lmhm 19224 df-lmim 19225 df-lbs 19277 df-sra 19374 df-rgmod 19375 df-nzr 19460 df-dsmm 20278 df-frlm 20293 df-uvc 20324 df-lindf 20347 |
This theorem is referenced by: lbslcic 20382 |
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