lnmlmic - Mathbox for Stefan O'Rear

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Theorem lnmlmic 43533

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 21058	. . 3 ⊢ (𝑅 ≃_𝑚 𝑆 ↔ (𝑅 LMIso 𝑆) ≠ ∅)
2		n0 4281	. . 3 ⊢ ((𝑅 LMIso 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝑅 LMIso 𝑆))
3	1, 2	bitri 276	. 2 ⊢ (𝑅 ≃_𝑚 𝑆 ↔ ∃𝑎 𝑎 ∈ (𝑅 LMIso 𝑆))
4		lmimlmhm 21054	. . . . . 6 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) → 𝑎 ∈ (𝑅 LMHom 𝑆))
5	4	adantr 481	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑅 ∈ LNoeM) → 𝑎 ∈ (𝑅 LMHom 𝑆))
6		simpr 485	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑅 ∈ LNoeM) → 𝑅 ∈ LNoeM)
7		eqid 2739	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2739	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
9	7, 8	lmimf1o 21053	. . . . . . 7 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) → 𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
10		f1ofo 6774	. . . . . . 7 ⊢ (𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆) → 𝑎:(Base‘𝑅)–onto→(Base‘𝑆))
11		forn 6742	. . . . . . 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 481	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑅 ∈ LNoeM) → ran 𝑎 = (Base‘𝑆))
14	8	lnmepi 43530	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMHom 𝑆) ∧ 𝑅 ∈ LNoeM ∧ ran 𝑎 = (Base‘𝑆)) → 𝑆 ∈ LNoeM)
15	5, 6, 13, 14	syl3anc 1379	. . . 4 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑅 ∈ LNoeM) → 𝑆 ∈ LNoeM)
16		islmim2 21056	. . . . . . 7 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) ↔ (𝑎 ∈ (𝑅 LMHom 𝑆) ∧ ^◡𝑎 ∈ (𝑆 LMHom 𝑅)))
17	16	simprbi 498	. . . . . 6 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) → ^◡𝑎 ∈ (𝑆 LMHom 𝑅))
18	17	adantr 481	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑆 ∈ LNoeM) → ^◡𝑎 ∈ (𝑆 LMHom 𝑅))
19		simpr 485	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑆 ∈ LNoeM) → 𝑆 ∈ LNoeM)
20		dfdm4 5837	. . . . . 6 ⊢ dom 𝑎 = ran ^◡𝑎
21		f1odm 6771	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑆 ∈ LNoeM) → dom 𝑎 = (Base‘𝑅))
24	20, 23	eqtr3id 2788	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑆 ∈ LNoeM) → ran ^◡𝑎 = (Base‘𝑅))
25	7	lnmepi 43530	. . . . 5 ⊢ ((^◡𝑎 ∈ (𝑆 LMHom 𝑅) ∧ 𝑆 ∈ LNoeM ∧ ran ^◡𝑎 = (Base‘𝑅)) → 𝑅 ∈ LNoeM)
26	18, 19, 24, 25	syl3anc 1379	. . . 4 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑆 ∈ LNoeM) → 𝑅 ∈ LNoeM)
27	15, 26	impbida 806	. . 3 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))
28	27	exlimiv 1937	. 2 ⊢ (∃𝑎 𝑎 ∈ (𝑅 LMIso 𝑆) → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))
29	3, 28	sylbi 218	1 ⊢ (𝑅 ≃_𝑚 𝑆 → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∃wex 1786 ∈ wcel 2119 ≠ wne 2934 ∅c0 4261 class class class wbr 5072 ^◡ccnv 5617 dom cdm 5618 ran crn 5619 –onto→wfo 6483 –1-1-onto→wf1o 6484 ‘cfv 6485 (class class class)co 7356 Basecbs 17170 LMHom clmhm 21009 LMIso clmim 21010 ≃_𝑚 clmic 21011 LNoeMclnm 43520

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-sca 17227 df-vsca 17228 df-0g 17395 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-ghm 19179 df-mgp 20113 df-ur 20154 df-ring 20207 df-lmod 20852 df-lss 20922 df-lsp 20962 df-lmhm 21012 df-lmim 21013 df-lmic 21014 df-lfig 43513 df-lnm 43521

This theorem is referenced by: pwslnmlem2 43538

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