lnmlmic - Mathbox for Stefan O'Rear

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Theorem lnmlmic 43129

Description: Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)

**Assertion**
Ref	Expression
lnmlmic	⊢ (𝑅 ≃_𝑚 𝑆 → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))

Proof of Theorem lnmlmic Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brlmic 21002	. . 3 ⊢ (𝑅 ≃_𝑚 𝑆 ↔ (𝑅 LMIso 𝑆) ≠ ∅)
2		n0 4300	. . 3 ⊢ ((𝑅 LMIso 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝑅 LMIso 𝑆))
3	1, 2	bitri 275	. 2 ⊢ (𝑅 ≃_𝑚 𝑆 ↔ ∃𝑎 𝑎 ∈ (𝑅 LMIso 𝑆))
4		lmimlmhm 20998	. . . . . 6 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) → 𝑎 ∈ (𝑅 LMHom 𝑆))
5	4	adantr 480	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑅 ∈ LNoeM) → 𝑎 ∈ (𝑅 LMHom 𝑆))
6		simpr 484	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑅 ∈ LNoeM) → 𝑅 ∈ LNoeM)
7		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
9	7, 8	lmimf1o 20997	. . . . . . 7 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) → 𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
10		f1ofo 6770	. . . . . . 7 ⊢ (𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆) → 𝑎:(Base‘𝑅)–onto→(Base‘𝑆))
11		forn 6738	. . . . . . 7 ⊢ (𝑎:(Base‘𝑅)–onto→(Base‘𝑆) → ran 𝑎 = (Base‘𝑆))
12	9, 10, 11	3syl 18	. . . . . 6 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) → ran 𝑎 = (Base‘𝑆))
13	12	adantr 480	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑅 ∈ LNoeM) → ran 𝑎 = (Base‘𝑆))
14	8	lnmepi 43126	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMHom 𝑆) ∧ 𝑅 ∈ LNoeM ∧ ran 𝑎 = (Base‘𝑆)) → 𝑆 ∈ LNoeM)
15	5, 6, 13, 14	syl3anc 1373	. . . 4 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑅 ∈ LNoeM) → 𝑆 ∈ LNoeM)
16		islmim2 21000	. . . . . . 7 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) ↔ (𝑎 ∈ (𝑅 LMHom 𝑆) ∧ ^◡𝑎 ∈ (𝑆 LMHom 𝑅)))
17	16	simprbi 496	. . . . . 6 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) → ^◡𝑎 ∈ (𝑆 LMHom 𝑅))
18	17	adantr 480	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑆 ∈ LNoeM) → ^◡𝑎 ∈ (𝑆 LMHom 𝑅))
19		simpr 484	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑆 ∈ LNoeM) → 𝑆 ∈ LNoeM)
20		dfdm4 5834	. . . . . 6 ⊢ dom 𝑎 = ran ^◡𝑎
21		f1odm 6767	. . . . . . . 8 ⊢ (𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆) → dom 𝑎 = (Base‘𝑅))
22	9, 21	syl 17	. . . . . . 7 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) → dom 𝑎 = (Base‘𝑅))
23	22	adantr 480	. . . . . 6 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑆 ∈ LNoeM) → dom 𝑎 = (Base‘𝑅))
24	20, 23	eqtr3id 2780	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑆 ∈ LNoeM) → ran ^◡𝑎 = (Base‘𝑅))
25	7	lnmepi 43126	. . . . 5 ⊢ ((^◡𝑎 ∈ (𝑆 LMHom 𝑅) ∧ 𝑆 ∈ LNoeM ∧ ran ^◡𝑎 = (Base‘𝑅)) → 𝑅 ∈ LNoeM)
26	18, 19, 24, 25	syl3anc 1373	. . . 4 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑆 ∈ LNoeM) → 𝑅 ∈ LNoeM)
27	15, 26	impbida 800	. . 3 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))
28	27	exlimiv 1931	. 2 ⊢ (∃𝑎 𝑎 ∈ (𝑅 LMIso 𝑆) → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))
29	3, 28	sylbi 217	1 ⊢ (𝑅 ≃_𝑚 𝑆 → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∃wex 1780 ∈ wcel 2111 ≠ wne 2928 ∅c0 4280 class class class wbr 5089 ^◡ccnv 5613 dom cdm 5614 ran crn 5615 –onto→wfo 6479 –1-1-onto→wf1o 6480 ‘cfv 6481 (class class class)co 7346 Basecbs 17120 LMHom clmhm 20953 LMIso clmim 20954 ≃_𝑚 clmic 20955 LNoeMclnm 43116

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-sca 17177 df-vsca 17178 df-0g 17345 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-grp 18849 df-minusg 18850 df-sbg 18851 df-subg 19036 df-ghm 19125 df-mgp 20059 df-ur 20100 df-ring 20153 df-lmod 20795 df-lss 20865 df-lsp 20905 df-lmhm 20956 df-lmim 20957 df-lmic 20958 df-lfig 43109 df-lnm 43117

This theorem is referenced by: pwslnmlem2 43134

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