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Theorem pwslnmlem2 38165
Description: A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypotheses
Ref Expression
pwslnmlem2.a 𝐴 ∈ V
pwslnmlem2.b 𝐵 ∈ V
pwslnmlem2.x 𝑋 = (𝑊s 𝐴)
pwslnmlem2.y 𝑌 = (𝑊s 𝐵)
pwslnmlem2.z 𝑍 = (𝑊s (𝐴𝐵))
pwslnmlem2.w (𝜑𝑊 ∈ LMod)
pwslnmlem2.dj (𝜑 → (𝐴𝐵) = ∅)
pwslnmlem2.xn (𝜑𝑋 ∈ LNoeM)
pwslnmlem2.yn (𝜑𝑌 ∈ LNoeM)
Assertion
Ref Expression
pwslnmlem2 (𝜑𝑍 ∈ LNoeM)

Proof of Theorem pwslnmlem2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwslnmlem2.w . . 3 (𝜑𝑊 ∈ LMod)
2 pwslnmlem2.a . . . . 5 𝐴 ∈ V
3 pwslnmlem2.b . . . . 5 𝐵 ∈ V
42, 3unex 7121 . . . 4 (𝐴𝐵) ∈ V
54a1i 11 . . 3 (𝜑 → (𝐴𝐵) ∈ V)
6 ssun1 3919 . . . 4 𝐴 ⊆ (𝐴𝐵)
76a1i 11 . . 3 (𝜑𝐴 ⊆ (𝐴𝐵))
8 pwslnmlem2.z . . . 4 𝑍 = (𝑊s (𝐴𝐵))
9 pwslnmlem2.x . . . 4 𝑋 = (𝑊s 𝐴)
10 eqid 2760 . . . 4 (Base‘𝑍) = (Base‘𝑍)
11 eqid 2760 . . . 4 (Base‘𝑋) = (Base‘𝑋)
12 eqid 2760 . . . 4 (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) = (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))
138, 9, 10, 11, 12pwssplit3 19263 . . 3 ((𝑊 ∈ LMod ∧ (𝐴𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) ∈ (𝑍 LMHom 𝑋))
141, 5, 7, 13syl3anc 1477 . 2 (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) ∈ (𝑍 LMHom 𝑋))
15 fvex 6362 . . . . . 6 (0g𝑋) ∈ V
1612mptiniseg 5790 . . . . . 6 ((0g𝑋) ∈ V → ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (0g𝑋)})
1715, 16ax-mp 5 . . . . 5 ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (0g𝑋)}
18 lmodgrp 19072 . . . . . . . . . 10 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
19 grpmnd 17630 . . . . . . . . . 10 (𝑊 ∈ Grp → 𝑊 ∈ Mnd)
201, 18, 193syl 18 . . . . . . . . 9 (𝜑𝑊 ∈ Mnd)
21 eqid 2760 . . . . . . . . . 10 (0g𝑊) = (0g𝑊)
229, 21pws0g 17527 . . . . . . . . 9 ((𝑊 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × {(0g𝑊)}) = (0g𝑋))
2320, 2, 22sylancl 697 . . . . . . . 8 (𝜑 → (𝐴 × {(0g𝑊)}) = (0g𝑋))
2423eqcomd 2766 . . . . . . 7 (𝜑 → (0g𝑋) = (𝐴 × {(0g𝑊)}))
2524eqeq2d 2770 . . . . . 6 (𝜑 → ((𝑥𝐴) = (0g𝑋) ↔ (𝑥𝐴) = (𝐴 × {(0g𝑊)})))
2625rabbidv 3329 . . . . 5 (𝜑 → {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (0g𝑋)} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})})
2717, 26syl5eq 2806 . . . 4 (𝜑 → ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})})
2827oveq2d 6829 . . 3 (𝜑 → (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})) = (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}))
29 pwslnmlem2.yn . . . 4 (𝜑𝑌 ∈ LNoeM)
30 pwslnmlem2.dj . . . . . 6 (𝜑 → (𝐴𝐵) = ∅)
31 eqid 2760 . . . . . . 7 {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}
32 eqid 2760 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} ↦ (𝑦𝐵)) = (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} ↦ (𝑦𝐵))
33 pwslnmlem2.y . . . . . . 7 𝑌 = (𝑊s 𝐵)
34 eqid 2760 . . . . . . 7 (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) = (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})})
358, 10, 21, 31, 32, 9, 33, 34pwssplit4 38161 . . . . . 6 ((𝑊 ∈ LMod ∧ (𝐴𝐵) ∈ V ∧ (𝐴𝐵) = ∅) → (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} ↦ (𝑦𝐵)) ∈ ((𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) LMIso 𝑌))
361, 5, 30, 35syl3anc 1477 . . . . 5 (𝜑 → (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} ↦ (𝑦𝐵)) ∈ ((𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) LMIso 𝑌))
37 brlmici 19271 . . . . 5 ((𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} ↦ (𝑦𝐵)) ∈ ((𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) LMIso 𝑌) → (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) ≃𝑚 𝑌)
38 lnmlmic 38160 . . . . 5 ((𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) ≃𝑚 𝑌 → ((𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) ∈ LNoeM ↔ 𝑌 ∈ LNoeM))
3936, 37, 383syl 18 . . . 4 (𝜑 → ((𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) ∈ LNoeM ↔ 𝑌 ∈ LNoeM))
4029, 39mpbird 247 . . 3 (𝜑 → (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) ∈ LNoeM)
4128, 40eqeltrd 2839 . 2 (𝜑 → (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})) ∈ LNoeM)
428, 9, 10, 11, 12pwssplit1 19261 . . . . . . 7 ((𝑊 ∈ Mnd ∧ (𝐴𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
4320, 5, 7, 42syl3anc 1477 . . . . . 6 (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
44 forn 6279 . . . . . 6 ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)):(Base‘𝑍)–onto→(Base‘𝑋) → ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) = (Base‘𝑋))
4543, 44syl 17 . . . . 5 (𝜑 → ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) = (Base‘𝑋))
4645oveq2d 6829 . . . 4 (𝜑 → (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) = (𝑋s (Base‘𝑋)))
47 pwslnmlem2.xn . . . . 5 (𝜑𝑋 ∈ LNoeM)
4811ressid 16137 . . . . 5 (𝑋 ∈ LNoeM → (𝑋s (Base‘𝑋)) = 𝑋)
4947, 48syl 17 . . . 4 (𝜑 → (𝑋s (Base‘𝑋)) = 𝑋)
5046, 49eqtrd 2794 . . 3 (𝜑 → (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) = 𝑋)
5150, 47eqeltrd 2839 . 2 (𝜑 → (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) ∈ LNoeM)
52 eqid 2760 . . 3 (0g𝑋) = (0g𝑋)
53 eqid 2760 . . 3 ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}) = ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})
54 eqid 2760 . . 3 (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})) = (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}))
55 eqid 2760 . . 3 (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) = (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)))
5652, 53, 54, 55lmhmlnmsplit 38159 . 2 (((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) ∈ (𝑍 LMHom 𝑋) ∧ (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})) ∈ LNoeM ∧ (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) ∈ LNoeM) → 𝑍 ∈ LNoeM)
5714, 41, 51, 56syl3anc 1477 1 (𝜑𝑍 ∈ LNoeM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  cun 3713  cin 3714  wss 3715  c0 4058  {csn 4321   class class class wbr 4804  cmpt 4881   × cxp 5264  ccnv 5265  ran crn 5267  cres 5268  cima 5269  ontowfo 6047  cfv 6049  (class class class)co 6813  Basecbs 16059  s cress 16060  0gc0g 16302  s cpws 16309  Mndcmnd 17495  Grpcgrp 17623  LModclmod 19065   LMHom clmhm 19221   LMIso clmim 19222  𝑚 clmic 19223  LNoeMclnm 38147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-hom 16168  df-cco 16169  df-0g 16304  df-prds 16310  df-pws 16312  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-grp 17626  df-minusg 17627  df-sbg 17628  df-subg 17792  df-ghm 17859  df-cntz 17950  df-lsm 18251  df-cmn 18395  df-abl 18396  df-mgp 18690  df-ur 18702  df-ring 18749  df-lmod 19067  df-lss 19135  df-lsp 19174  df-lmhm 19224  df-lmim 19225  df-lmic 19226  df-lfig 38140  df-lnm 38148
This theorem is referenced by:  pwslnm  38166
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