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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lnmlmic | Structured version Visualization version GIF version |
Description: Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Ref | Expression |
---|---|
lnmlmic | ⊢ (𝑅 ≃𝑚 𝑆 → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brlmic 21085 | . . 3 ⊢ (𝑅 ≃𝑚 𝑆 ↔ (𝑅 LMIso 𝑆) ≠ ∅) | |
2 | n0 4359 | . . 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 21081 | . . . . . 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 2735 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | eqid 2735 | . . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
9 | 7, 8 | lmimf1o 21080 | . . . . . . 7 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) → 𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆)) |
10 | f1ofo 6856 | . . . . . . 7 ⊢ (𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆) → 𝑎:(Base‘𝑅)–onto→(Base‘𝑆)) | |
11 | forn 6824 | . . . . . . 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 43074 | . . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMHom 𝑆) ∧ 𝑅 ∈ LNoeM ∧ ran 𝑎 = (Base‘𝑆)) → 𝑆 ∈ LNoeM) |
15 | 5, 6, 13, 14 | syl3anc 1370 | . . . 4 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑅 ∈ LNoeM) → 𝑆 ∈ LNoeM) |
16 | islmim2 21083 | . . . . . . 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 5909 | . . . . . 6 ⊢ dom 𝑎 = ran ◡𝑎 | |
21 | f1odm 6853 | . . . . . . . 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 2789 | . . . . 5 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑆 ∈ LNoeM) → ran ◡𝑎 = (Base‘𝑅)) |
25 | 7 | lnmepi 43074 | . . . . 5 ⊢ ((◡𝑎 ∈ (𝑆 LMHom 𝑅) ∧ 𝑆 ∈ LNoeM ∧ ran ◡𝑎 = (Base‘𝑅)) → 𝑅 ∈ LNoeM) |
26 | 18, 19, 24, 25 | syl3anc 1370 | . . . 4 ⊢ ((𝑎 ∈ (𝑅 LMIso 𝑆) ∧ 𝑆 ∈ LNoeM) → 𝑅 ∈ LNoeM) |
27 | 15, 26 | impbida 801 | . . 3 ⊢ (𝑎 ∈ (𝑅 LMIso 𝑆) → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM)) |
28 | 27 | exlimiv 1928 | . 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 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 class class class wbr 5148 ◡ccnv 5688 dom cdm 5689 ran crn 5690 –onto→wfo 6561 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 LMHom clmhm 21036 LMIso clmim 21037 ≃𝑚 clmic 21038 LNoeMclnm 43064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-sca 17314 df-vsca 17315 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-ghm 19244 df-mgp 20153 df-ur 20200 df-ring 20253 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lmhm 21039 df-lmim 21040 df-lmic 21041 df-lfig 43057 df-lnm 43065 |
This theorem is referenced by: pwslnmlem2 43082 |
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