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 21766

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 1138	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝐼 ≈ 𝐵)
2		bren 8889	. . 3 ⊢ (𝐼 ≈ 𝐵 ↔ ∃𝑒 𝑒:𝐼–1-1-onto→𝐵)
3	1, 2	sylib 218	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → ∃𝑒 𝑒:𝐼–1-1-onto→𝐵)
4		eqid 2729	. . . 4 ⊢ (𝐹 freeLMod 𝐼) = (𝐹 freeLMod 𝐼)
5		eqid 2729	. . . 4 ⊢ (Base‘(𝐹 freeLMod 𝐼)) = (Base‘(𝐹 freeLMod 𝐼))
6		eqid 2729	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
7		eqid 2729	. . . 4 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
8		eqid 2729	. . . 4 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
9		eqid 2729	. . . 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 1192	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑊 ∈ LMod)
11		relen 8884	. . . . . . 7 ⊢ Rel ≈
12	11	brrelex1i 5679	. . . . . 6 ⊢ (𝐼 ≈ 𝐵 → 𝐼 ∈ V)
13	12	3ad2ant3 1135	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝐼 ∈ V)
14	13	adantr 480	. . . 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 6775	. . . . 5 ⊢ (𝑒:𝐼–1-1-onto→𝐵 → 𝑒:𝐼–onto→𝐵)
18	17	adantl 481	. . . 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 21758	. . . . . . . 8 ⊢ 𝐽 ⊆ (LIndS‘𝑊)
21	20	sseli 3933	. . . . . . 7 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ∈ (LIndS‘𝑊))
22	21	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝐵 ∈ (LIndS‘𝑊))
23	22	adantr 480	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝐵 ∈ (LIndS‘𝑊))
24		f1of1 6767	. . . . . 6 ⊢ (𝑒:𝐼–1-1-onto→𝐵 → 𝑒:𝐼–1-1→𝐵)
25	24	adantl 481	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑒:𝐼–1-1→𝐵)
26		f1linds 21750	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ (LIndS‘𝑊) ∧ 𝑒:𝐼–1-1→𝐵) → 𝑒 LIndF 𝑊)
27	10, 23, 25, 26	syl3anc 1373	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑒 LIndF 𝑊)
28	6, 19, 8	lbssp 21001	. . . . . 6 ⊢ (𝐵 ∈ 𝐽 → ((LSpan‘𝑊)‘𝐵) = (Base‘𝑊))
29	28	3ad2ant2 1134	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → ((LSpan‘𝑊)‘𝐵) = (Base‘𝑊))
30	29	adantr 480	. . . 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 21765	. . 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 20989	. . 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 20991	. . 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 1935	1 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝑊 ≃_𝑚 (𝐹 freeLMod 𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∃wex 1779 ∈ wcel 2109 Vcvv 3438 class class class wbr 5095 ↦ cmpt 5176 ^◡ccnv 5622 –1-1→wf1 6483 –onto→wfo 6484 –1-1-onto→wf1o 6485 ‘cfv 6486 (class class class)co 7353 ∘_f cof 7615 ≈ cen 8876 Basecbs 17138 Scalarcsca 17182 ·_𝑠 cvsca 17183 Σ_g cgsu 17362 LModclmod 20781 LSpanclspn 20892 LMIso clmim 20942 ≃_𝑚 clmic 20943 LBasisclbs 20996 freeLMod cfrlm 21671 LIndF clindf 21729 LIndSclinds 21730

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-sup 9351 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-fz 13429 df-fzo 13576 df-seq 13927 df-hash 14256 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-hom 17203 df-cco 17204 df-0g 17363 df-gsum 17364 df-prds 17369 df-pws 17371 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-mhm 18675 df-submnd 18676 df-grp 18833 df-minusg 18834 df-sbg 18835 df-mulg 18965 df-subg 19020 df-ghm 19110 df-cntz 19214 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-nzr 20416 df-subrg 20473 df-lmod 20783 df-lss 20853 df-lsp 20893 df-lmhm 20944 df-lmim 20945 df-lmic 20946 df-lbs 20997 df-sra 21095 df-rgmod 21096 df-dsmm 21657 df-frlm 21672 df-uvc 21708 df-lindf 21731 df-linds 21732

This theorem is referenced by: lmisfree 21767 frlmisfrlm 21773

W3C validator