lnmlmic - Mathbox for Stefan O'Rear

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Theorem lnmlmic 42390

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 20913	. . 3 ⊢ (𝑅 ≃_𝑚 𝑆 ↔ (𝑅 LMIso 𝑆) ≠ ∅)
2		n0 4341	. . 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 20909	. . . . . 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 2726	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2726	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
9	7, 8	lmimf1o 20908	. . . . . . 7 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) → 𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
10		f1ofo 6833	. . . . . . 7 ⊢ (𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆) → 𝑎:(Base‘𝑅)–onto→(Base‘𝑆))
11		forn 6801	. . . . . . 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 42387	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMHom 𝑆) ∧ 𝑅 ∈ LNoeM ∧ ran 𝑎 = (Base‘𝑆)) → 𝑆 ∈ LNoeM)
15	5, 6, 13, 14	syl3anc 1368	. . . 4 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑅 ∈ LNoeM) → 𝑆 ∈ LNoeM)
16		islmim2 20911	. . . . . . 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 5888	. . . . . 6 ⊢ dom 𝑎 = ran ^◡𝑎
21		f1odm 6830	. . . . . . . 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 42387	. . . . 5 ⊢ ((^◡𝑎 ∈ (𝑆 LMHom 𝑅) ∧ 𝑆 ∈ LNoeM ∧ ran ^◡𝑎 = (Base‘𝑅)) → 𝑅 ∈ LNoeM)
26	18, 19, 24, 25	syl3anc 1368	. . . 4 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑆 ∈ LNoeM) → 𝑅 ∈ LNoeM)
27	15, 26	impbida 798	. . 3 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))
28	27	exlimiv 1925	. 2 ⊢ (∃𝑎 𝑎 ∈ (𝑅 LMIso 𝑆) → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))
29	3, 28	sylbi 216	1 ⊢ (𝑅 ≃_𝑚 𝑆 → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2934 ∅c0 4317 class class class wbr 5141 ^◡ccnv 5668 dom cdm 5669 ran crn 5670 –onto→wfo 6534 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7404 Basecbs 17150 LMHom clmhm 20864 LMIso clmim 20865 ≃_𝑚 clmic 20866 LNoeMclnm 42377

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-sca 17219 df-vsca 17220 df-0g 17393 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-grp 18863 df-minusg 18864 df-sbg 18865 df-subg 19047 df-ghm 19136 df-mgp 20037 df-ur 20084 df-ring 20137 df-lmod 20705 df-lss 20776 df-lsp 20816 df-lmhm 20867 df-lmim 20868 df-lmic 20869 df-lfig 42370 df-lnm 42378

This theorem is referenced by: pwslnmlem2 42395

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