Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6592 | . . . . 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 20959 | . . . . . 6 ⊢ ((𝐴 LIndF 𝑇 ∧ 𝑇 ∈ LMod) → 𝐴:dom 𝐴⟶𝐶) |
14 | 12, 6, 13 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝐴:dom 𝐴⟶𝐶) |
15 | 14 | frnd 6521 | . . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ 𝐶) |
16 | df-f 6359 | . . . 4 ⊢ (𝐴:𝐼⟶𝐶 ↔ (𝐴 Fn 𝐼 ∧ ran 𝐴 ⊆ 𝐶)) | |
17 | 11, 15, 16 | sylanbrc 585 | . . 3 ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 17 | frlmup1 20942 | . 2 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 17 | islindf5 20983 | . . . 4 ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ 𝐸:𝐵–1-1→𝐶)) |
20 | 12, 19 | mpbid 234 | . . 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 20944 | . . . 4 ⊢ (𝜑 → ran 𝐸 = (𝑁‘ran 𝐴)) |
23 | forn 6593 | . . . . . 6 ⊢ (𝐴:𝐼–onto→𝐽 → ran 𝐴 = 𝐽) | |
24 | 9, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → ran 𝐴 = 𝐽) |
25 | 24 | fveq2d 6674 | . . . 4 ⊢ (𝜑 → (𝑁‘ran 𝐴) = (𝑁‘𝐽)) |
26 | indlcim.s | . . . 4 ⊢ (𝜑 → (𝑁‘𝐽) = 𝐶) | |
27 | 22, 25, 26 | 3eqtrd 2860 | . . 3 ⊢ (𝜑 → ran 𝐸 = 𝐶) |
28 | dff1o5 6624 | . . 3 ⊢ (𝐸:𝐵–1-1-onto→𝐶 ↔ (𝐸:𝐵–1-1→𝐶 ∧ ran 𝐸 = 𝐶)) | |
29 | 20, 27, 28 | sylanbrc 585 | . 2 ⊢ (𝜑 → 𝐸:𝐵–1-1-onto→𝐶) |
30 | 2, 3 | islmim 19834 | . 2 ⊢ (𝐸 ∈ (𝐹 LMIso 𝑇) ↔ (𝐸 ∈ (𝐹 LMHom 𝑇) ∧ 𝐸:𝐵–1-1-onto→𝐶)) |
31 | 18, 29, 30 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMIso 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 Fn wfn 6350 ⟶wf 6351 –1-1→wf1 6352 –onto→wfo 6353 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 Basecbs 16483 Scalarcsca 16568 ·𝑠 cvsca 16569 Σg cgsu 16714 LModclmod 19634 LSpanclspn 19743 LMHom clmhm 19791 LMIso clmim 19792 freeLMod cfrlm 20890 LIndF clindf 20948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-hom 16589 df-cco 16590 df-0g 16715 df-gsum 16716 df-prds 16721 df-pws 16723 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-subrg 19533 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lmhm 19794 df-lmim 19795 df-lbs 19847 df-sra 19944 df-rgmod 19945 df-nzr 20031 df-dsmm 20876 df-frlm 20891 df-uvc 20927 df-lindf 20950 |
This theorem is referenced by: lbslcic 20985 |
Copyright terms: Public domain | W3C validator |