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 21881

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 1150	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝐼 ≈ 𝐵)
2		bren 8931	. . 3 ⊢ (𝐼 ≈ 𝐵 ↔ ∃𝑒 𝑒:𝐼–1-1-onto→𝐵)
3	1, 2	sylib 220	. 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 1204	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑊 ∈ LMod)
11		relen 8926	. . . . . . 7 ⊢ Rel ≈
12	11	brrelex1i 5699	. . . . . 6 ⊢ (𝐼 ≈ 𝐵 → 𝐼 ∈ V)
13	12	3ad2ant3 1147	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝐼 ∈ V)
14	13	adantr 484	. . . 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 6809	. . . . 5 ⊢ (𝑒:𝐼–1-1-onto→𝐵 → 𝑒:𝐼–onto→𝐵)
18	17	adantl 485	. . . 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 21873	. . . . . . . 8 ⊢ 𝐽 ⊆ (LIndS‘𝑊)
21	20	sseli 3930	. . . . . . 7 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ∈ (LIndS‘𝑊))
22	21	3ad2ant2 1146	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝐵 ∈ (LIndS‘𝑊))
23	22	adantr 484	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝐵 ∈ (LIndS‘𝑊))
24		f1of1 6800	. . . . . 6 ⊢ (𝑒:𝐼–1-1-onto→𝐵 → 𝑒:𝐼–1-1→𝐵)
25	24	adantl 485	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑒:𝐼–1-1→𝐵)
26		f1linds 21865	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ (LIndS‘𝑊) ∧ 𝑒:𝐼–1-1→𝐵) → 𝑒 LIndF 𝑊)
27	10, 23, 25, 26	syl3anc 1389	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑒 LIndF 𝑊)
28	6, 19, 8	lbssp 21134	. . . . . 6 ⊢ (𝐵 ∈ 𝐽 → ((LSpan‘𝑊)‘𝐵) = (Base‘𝑊))
29	28	3ad2ant2 1146	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → ((LSpan‘𝑊)‘𝐵) = (Base‘𝑊))
30	29	adantr 484	. . . 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 21880	. . 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 21122	. . 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 21124	. . 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 1954	1 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝑊 ≃_𝑚 (𝐹 freeLMod 𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∃wex 1798 ∈ wcel 2141 Vcvv 3453 class class class wbr 5097 ↦ cmpt 5178 ^◡ccnv 5642 –1-1→wf1 6513 –onto→wfo 6514 –1-1-onto→wf1o 6515 ‘cfv 6516 (class class class)co 7391 ∘_f cof 7653 ≈ cen 8918 Basecbs 17236 Scalarcsca 17280 ·_𝑠 cvsca 17281 Σ_g cgsu 17460 LModclmod 20915 LSpanclspn 21026 LMIso clmim 21075 ≃_𝑚 clmic 21076 LBasisclbs 21129 freeLMod cfrlm 21786 LIndF clindf 21844 LIndSclinds 21845

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-of 7655 df-om 7842 df-1st 7965 df-2nd 7966 df-supp 8135 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-er 8672 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9302 df-sup 9382 df-oi 9452 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-z 12563 df-dec 12683 df-uz 12834 df-fz 13507 df-fzo 13654 df-seq 14009 df-hash 14338 df-struct 17174 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-sca 17293 df-vsca 17294 df-ip 17295 df-tset 17296 df-ple 17297 df-ds 17299 df-hom 17301 df-cco 17302 df-0g 17461 df-gsum 17462 df-prds 17467 df-pws 17469 df-mre 17605 df-mrc 17606 df-acs 17608 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18969 df-minusg 18970 df-sbg 18971 df-mulg 19101 df-subg 19156 df-ghm 19245 df-cntz 19348 df-cmn 19813 df-abl 19814 df-mgp 20178 df-rng 20190 df-ur 20219 df-ring 20272 df-nzr 20550 df-subrg 20607 df-lmod 20917 df-lss 20987 df-lsp 21027 df-lmhm 21077 df-lmim 21078 df-lmic 21079 df-lbs 21130 df-sra 21228 df-rgmod 21229 df-dsmm 21772 df-frlm 21787 df-uvc 21823 df-lindf 21846 df-linds 21847

This theorem is referenced by: lmisfree 21882 frlmisfrlm 21888

W3C validator