lbslcic - Metamath Proof Explorer

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Theorem lbslcic 22020

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 1156	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝐼 ≈ 𝐵)
2		bren 8955	. . 3 ⊢ (𝐼 ≈ 𝐵 ↔ ∃𝑒 𝑒:𝐼–1-1-onto→𝐵)
3	1, 2	sylib 221	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → ∃𝑒 𝑒:𝐼–1-1-onto→𝐵)
4		eqid 2765	. . . 4 ⊢ (𝐹 freeLMod 𝐼) = (𝐹 freeLMod 𝐼)
5		eqid 2765	. . . 4 ⊢ (Base‘(𝐹 freeLMod 𝐼)) = (Base‘(𝐹 freeLMod 𝐼))
6		eqid 2765	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
7		eqid 2765	. . . 4 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
8		eqid 2765	. . . 4 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
9		eqid 2765	. . . 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 1210	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑊 ∈ LMod)
11		relen 8950	. . . . . . 7 ⊢ Rel ≈
12	11	brrelex1i 5719	. . . . . 6 ⊢ (𝐼 ≈ 𝐵 → 𝐼 ∈ V)
13	12	3ad2ant3 1153	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝐼 ∈ V)
14	13	adantr 486	. . . 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 6832	. . . . 5 ⊢ (𝑒:𝐼–1-1-onto→𝐵 → 𝑒:𝐼–onto→𝐵)
18	17	adantl 487	. . . 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 22012	. . . . . . . 8 ⊢ 𝐽 ⊆ (LIndS‘𝑊)
21	20	sseli 3934	. . . . . . 7 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ∈ (LIndS‘𝑊))
22	21	3ad2ant2 1152	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝐵 ∈ (LIndS‘𝑊))
23	22	adantr 486	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝐵 ∈ (LIndS‘𝑊))
24		f1of1 6823	. . . . . 6 ⊢ (𝑒:𝐼–1-1-onto→𝐵 → 𝑒:𝐼–1-1→𝐵)
25	24	adantl 487	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑒:𝐼–1-1→𝐵)
26		f1linds 22004	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ (LIndS‘𝑊) ∧ 𝑒:𝐼–1-1→𝐵) → 𝑒 LIndF 𝑊)
27	10, 23, 25, 26	syl3anc 1398	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑒 LIndF 𝑊)
28	6, 19, 8	lbssp 21229	. . . . . 6 ⊢ (𝐵 ∈ 𝐽 → ((LSpan‘𝑊)‘𝐵) = (Base‘𝑊))
29	28	3ad2ant2 1152	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → ((LSpan‘𝑊)‘𝐵) = (Base‘𝑊))
30	29	adantr 486	. . . 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 22019	. . 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 21217	. . 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 21219	. . 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 19	. 2 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) ∧ 𝑒:𝐼–1-1-onto→𝐵) → 𝑊 ≃_𝑚 (𝐹 freeLMod 𝐼))
35	3, 34	exlimddv 1968	1 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝑊 ≃_𝑚 (𝐹 freeLMod 𝐼))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2146 Vcvv 3457 class class class wbr 5111 ↦ cmpt 5194 ^◡ccnv 5662 –1-1→wf1 6537 –onto→wfo 6538 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7416 ∘_f cof 7678 ≈ cen 8942 Basecbs 17286 Scalarcsca 17330 ·_𝑠 cvsca 17331 Σ_g cgsu 17510 LModclmod 21010 LSpanclspn 21121 LMIso clmim 21170 ≃_𝑚 clmic 21171 LBasisclbs 21224 freeLMod cfrlm 21925 LIndF clindf 21983 LIndSclinds 21984

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9325 df-sup 9405 df-oi 9475 df-card 9937 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-z 12603 df-dec 12723 df-uz 12874 df-fz 13547 df-fzo 13695 df-seq 14051 df-hash 14380 df-struct 17224 df-sets 17241 df-slot 17259 df-ndx 17271 df-base 17287 df-ress 17308 df-plusg 17340 df-mulr 17341 df-sca 17343 df-vsca 17344 df-ip 17345 df-tset 17346 df-ple 17347 df-ds 17349 df-hom 17351 df-cco 17352 df-0g 17511 df-gsum 17512 df-prds 17517 df-pws 17519 df-mre 17655 df-mrc 17656 df-acs 17658 df-mgm 18715 df-sgrp 18798 df-mnd 18814 df-mhm 18864 df-submnd 18865 df-grp 19026 df-minusg 19027 df-sbg 19028 df-mulg 19157 df-subg 19212 df-ghm 19307 df-cntz 19410 df-cmn 19875 df-abl 19876 df-mgp 20240 df-rng 20254 df-ur 20287 df-ring 20340 df-nzr 20639 df-subrg 20698 df-lmod 21012 df-lss 21082 df-lsp 21122 df-lmhm 21172 df-lmim 21173 df-lmic 21174 df-lbs 21225 df-sra 21323 df-rgmod 21324 df-dsmm 21911 df-frlm 21926 df-uvc 21962 df-lindf 21985 df-linds 21986

This theorem is used by: lmisfree 22021 frlmisfrlm 22027

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