lbslcic - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lbslcic		Structured version Visualization version GIF version

Theorem lbslcic 20403

Description: A module with a basis is isomorphic to a free module with the same cardinality. (Contributed by Stefan O'Rear, 26-Feb-2015.)

**Hypotheses**
Ref	Expression
lbslcic.f	⊢ 𝐹 = (Scalar‘𝑊)
lbslcic.j	⊢ 𝐽 = (LBasis‘𝑊)

**Assertion**
Ref	Expression
lbslcic	⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝑊 ≃_𝑚 (𝐹 freeLMod 𝐼))

Proof of Theorem lbslcic Dummy variables 𝑒 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1133	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝐼 ≈ 𝐵)
2		bren 8133	. . 3 ⊢ (𝐼 ≈ 𝐵 ↔ ∃𝑒 𝑒:𝐼–1-1-onto→𝐵)
3	1, 2	sylib 208	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → ∃𝑒 𝑒:𝐼–1-1-onto→𝐵)
4		eqid 2761	. . . 4 ⊢ (𝐹 freeLMod 𝐼) = (𝐹 freeLMod 𝐼)
5		eqid 2761	. . . 4 ⊢ (Base‘(𝐹 freeLMod 𝐼)) = (Base‘(𝐹 freeLMod 𝐼))
6		eqid 2761	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
7		eqid 2761	. . . 4 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
8		eqid 2761	. . . 4 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
9		eqid 2761	. . . 4 ⊢ (𝑥 ∈ (Base‘(𝐹 freeLMod 𝐼)) ↦ (𝑊 Σ_g (𝑥 ∘_𝑓 ( ·_𝑠 ‘𝑊)𝑒))) = (𝑥 ∈ (Base‘(𝐹 freeLMod 𝐼)) ↦ (𝑊 Σ_g (𝑥 ∘_𝑓 ( ·_𝑠 ‘𝑊)𝑒)))
10		simpl1 1228	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑊 ∈ LMod)
11		relen 8129	. . . . . . 7 ⊢ Rel ≈
12	11	brrelexi 5316	. . . . . 6 ⊢ (𝐼 ≈ 𝐵 → 𝐼 ∈ V)
13	12	3ad2ant3 1130	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝐼 ∈ V)
14	13	adantr 472	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝐼 ∈ V)
15		lbslcic.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
16	15	a1i 11	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝐹 = (Scalar‘𝑊))
17		f1ofo 6307	. . . . 5 ⊢ (𝑒:𝐼–1-1-onto→𝐵 → 𝑒:𝐼–onto→𝐵)
18	17	adantl 473	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑒:𝐼–onto→𝐵)
19		lbslcic.j	. . . . . . . . 9 ⊢ 𝐽 = (LBasis‘𝑊)
20	19	lbslinds 20395	. . . . . . . 8 ⊢ 𝐽 ⊆ (LIndS‘𝑊)
21	20	sseli 3741	. . . . . . 7 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ∈ (LIndS‘𝑊))
22	21	3ad2ant2 1129	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝐵 ∈ (LIndS‘𝑊))
23	22	adantr 472	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝐵 ∈ (LIndS‘𝑊))
24		f1of1 6299	. . . . . 6 ⊢ (𝑒:𝐼–1-1-onto→𝐵 → 𝑒:𝐼–1-1→𝐵)
25	24	adantl 473	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑒:𝐼–1-1→𝐵)
26		f1linds 20387	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ (LIndS‘𝑊) ∧ 𝑒:𝐼–1-1→𝐵) → 𝑒 LIndF 𝑊)
27	10, 23, 25, 26	syl3anc 1477	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑒 LIndF 𝑊)
28	6, 19, 8	lbssp 19302	. . . . . 6 ⊢ (𝐵 ∈ 𝐽 → ((LSpan‘𝑊)‘𝐵) = (Base‘𝑊))
29	28	3ad2ant2 1129	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → ((LSpan‘𝑊)‘𝐵) = (Base‘𝑊))
30	29	adantr 472	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → ((LSpan‘𝑊)‘𝐵) = (Base‘𝑊))
31	4, 5, 6, 7, 8, 9, 10, 14, 16, 18, 27, 30	indlcim 20402	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → (𝑥 ∈ (Base‘(𝐹 freeLMod 𝐼)) ↦ (𝑊 Σ_g (𝑥 ∘_𝑓 ( ·_𝑠 ‘𝑊)𝑒))) ∈ ((𝐹 freeLMod 𝐼) LMIso 𝑊))
32		lmimcnv 19290	. . 3 ⊢ ((𝑥 ∈ (Base‘(𝐹 freeLMod 𝐼)) ↦ (𝑊 Σ_g (𝑥 ∘_𝑓 ( ·_𝑠 ‘𝑊)𝑒))) ∈ ((𝐹 freeLMod 𝐼) LMIso 𝑊) → ^◡(𝑥 ∈ (Base‘(𝐹 freeLMod 𝐼)) ↦ (𝑊 Σ_g (𝑥 ∘_𝑓 ( ·_𝑠 ‘𝑊)𝑒))) ∈ (𝑊 LMIso (𝐹 freeLMod 𝐼)))
33		brlmici 19292	. . 3 ⊢ (^◡(𝑥 ∈ (Base‘(𝐹 freeLMod 𝐼)) ↦ (𝑊 Σ_g (𝑥 ∘_𝑓 ( ·_𝑠 ‘𝑊)𝑒))) ∈ (𝑊 LMIso (𝐹 freeLMod 𝐼)) → 𝑊 ≃_𝑚 (𝐹 freeLMod 𝐼))
34	31, 32, 33	3syl 18	. 2 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑊 ≃_𝑚 (𝐹 freeLMod 𝐼))
35	3, 34	exlimddv 2013	1 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝑊 ≃_𝑚 (𝐹 freeLMod 𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∃wex 1853 ∈ wcel 2140 Vcvv 3341 class class class wbr 4805 ↦ cmpt 4882 ^◡ccnv 5266 –1-1→wf1 6047 –onto→wfo 6048 –1-1-onto→wf1o 6049 ‘cfv 6050 (class class class)co 6815 ∘_𝑓 cof 7062 ≈ cen 8121 Basecbs 16080 Scalarcsca 16167 ·_𝑠 cvsca 16168 Σ_g cgsu 16324 LModclmod 19086 LSpanclspn 19194 LMIso clmim 19243 ≃_𝑚 clmic 19244 LBasisclbs 19297 freeLMod cfrlm 20313 LIndF clindf 20366 LIndSclinds 20367

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 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-rep 4924 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 ax-inf2 8714 ax-cnex 10205 ax-resscn 10206 ax-1cn 10207 ax-icn 10208 ax-addcl 10209 ax-addrcl 10210 ax-mulcl 10211 ax-mulrcl 10212 ax-mulcom 10213 ax-addass 10214 ax-mulass 10215 ax-distr 10216 ax-i2m1 10217 ax-1ne0 10218 ax-1rid 10219 ax-rnegex 10220 ax-rrecex 10221 ax-cnre 10222 ax-pre-lttri 10223 ax-pre-lttrn 10224 ax-pre-ltadd 10225 ax-pre-mulgt0 10226

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 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-uni 4590 df-int 4629 df-iun 4675 df-iin 4676 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-se 5227 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-pred 5842 df-ord 5888 df-on 5889 df-lim 5890 df-suc 5891 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-isom 6059 df-riota 6776 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-of 7064 df-om 7233 df-1st 7335 df-2nd 7336 df-supp 7466 df-wrecs 7578 df-recs 7639 df-rdg 7677 df-1o 7731 df-oadd 7735 df-er 7914 df-map 8028 df-ixp 8078 df-en 8125 df-dom 8126 df-sdom 8127 df-fin 8128 df-fsupp 8444 df-sup 8516 df-oi 8583 df-card 8976 df-pnf 10289 df-mnf 10290 df-xr 10291 df-ltxr 10292 df-le 10293 df-sub 10481 df-neg 10482 df-nn 11234 df-2 11292 df-3 11293 df-4 11294 df-5 11295 df-6 11296 df-7 11297 df-8 11298 df-9 11299 df-n0 11506 df-z 11591 df-dec 11707 df-uz 11901 df-fz 12541 df-fzo 12681 df-seq 13017 df-hash 13333 df-struct 16082 df-ndx 16083 df-slot 16084 df-base 16086 df-sets 16087 df-ress 16088 df-plusg 16177 df-mulr 16178 df-sca 16180 df-vsca 16181 df-ip 16182 df-tset 16183 df-ple 16184 df-ds 16187 df-hom 16189 df-cco 16190 df-0g 16325 df-gsum 16326 df-prds 16331 df-pws 16333 df-mre 16469 df-mrc 16470 df-acs 16472 df-mgm 17464 df-sgrp 17506 df-mnd 17517 df-mhm 17557 df-submnd 17558 df-grp 17647 df-minusg 17648 df-sbg 17649 df-mulg 17763 df-subg 17813 df-ghm 17880 df-cntz 17971 df-cmn 18416 df-abl 18417 df-mgp 18711 df-ur 18723 df-ring 18770 df-subrg 19001 df-lmod 19088 df-lss 19156 df-lsp 19195 df-lmhm 19245 df-lmim 19246 df-lmic 19247 df-lbs 19298 df-sra 19395 df-rgmod 19396 df-nzr 19481 df-dsmm 20299 df-frlm 20314 df-uvc 20345 df-lindf 20368 df-linds 20369

This theorem is referenced by: lmisfree 20404 frlmisfrlm 20410

W3C validator