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Theorem pwslnmlem2 41449
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 7685 . . . 4 (𝐴𝐵) ∈ V
54a1i 11 . . 3 (𝜑 → (𝐴𝐵) ∈ V)
6 ssun1 4137 . . . 4 𝐴 ⊆ (𝐴𝐵)
76a1i 11 . . 3 (𝜑𝐴 ⊆ (𝐴𝐵))
8 pwslnmlem2.z . . . 4 𝑍 = (𝑊s (𝐴𝐵))
9 pwslnmlem2.x . . . 4 𝑋 = (𝑊s 𝐴)
10 eqid 2737 . . . 4 (Base‘𝑍) = (Base‘𝑍)
11 eqid 2737 . . . 4 (Base‘𝑋) = (Base‘𝑋)
12 eqid 2737 . . . 4 (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) = (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))
138, 9, 10, 11, 12pwssplit3 20538 . . 3 ((𝑊 ∈ LMod ∧ (𝐴𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) ∈ (𝑍 LMHom 𝑋))
141, 5, 7, 13syl3anc 1372 . 2 (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) ∈ (𝑍 LMHom 𝑋))
15 fvex 6860 . . . . . 6 (0g𝑋) ∈ V
1612mptiniseg 6196 . . . . . 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 20345 . . . . . . . . . 10 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
19 grpmnd 18762 . . . . . . . . . 10 (𝑊 ∈ Grp → 𝑊 ∈ Mnd)
201, 18, 193syl 18 . . . . . . . . 9 (𝜑𝑊 ∈ Mnd)
21 eqid 2737 . . . . . . . . . 10 (0g𝑊) = (0g𝑊)
229, 21pws0g 18599 . . . . . . . . 9 ((𝑊 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × {(0g𝑊)}) = (0g𝑋))
2320, 2, 22sylancl 587 . . . . . . . 8 (𝜑 → (𝐴 × {(0g𝑊)}) = (0g𝑋))
2423eqcomd 2743 . . . . . . 7 (𝜑 → (0g𝑋) = (𝐴 × {(0g𝑊)}))
2524eqeq2d 2748 . . . . . 6 (𝜑 → ((𝑥𝐴) = (0g𝑋) ↔ (𝑥𝐴) = (𝐴 × {(0g𝑊)})))
2625rabbidv 3418 . . . . 5 (𝜑 → {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (0g𝑋)} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})})
2717, 26eqtrid 2789 . . . 4 (𝜑 → ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})})
2827oveq2d 7378 . . 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 2737 . . . . . . 7 {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}
32 eqid 2737 . . . . . . 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 2737 . . . . . . 7 (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) = (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})})
358, 10, 21, 31, 32, 9, 33, 34pwssplit4 41445 . . . . . 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 1372 . . . . 5 (𝜑 → (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} ↦ (𝑦𝐵)) ∈ ((𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) LMIso 𝑌))
37 brlmici 20546 . . . . 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 41444 . . . . 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 257 . . 3 (𝜑 → (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) ∈ LNoeM)
4128, 40eqeltrd 2838 . 2 (𝜑 → (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})) ∈ LNoeM)
428, 9, 10, 11, 12pwssplit1 20536 . . . . . . 7 ((𝑊 ∈ Mnd ∧ (𝐴𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
4320, 5, 7, 42syl3anc 1372 . . . . . 6 (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
44 forn 6764 . . . . . 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 7378 . . . 4 (𝜑 → (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) = (𝑋s (Base‘𝑋)))
47 pwslnmlem2.xn . . . . 5 (𝜑𝑋 ∈ LNoeM)
4811ressid 17132 . . . . 5 (𝑋 ∈ LNoeM → (𝑋s (Base‘𝑋)) = 𝑋)
4947, 48syl 17 . . . 4 (𝜑 → (𝑋s (Base‘𝑋)) = 𝑋)
5046, 49eqtrd 2777 . . 3 (𝜑 → (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) = 𝑋)
5150, 47eqeltrd 2838 . 2 (𝜑 → (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) ∈ LNoeM)
52 eqid 2737 . . 3 (0g𝑋) = (0g𝑋)
53 eqid 2737 . . 3 ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}) = ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})
54 eqid 2737 . . 3 (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})) = (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}))
55 eqid 2737 . . 3 (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) = (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)))
5652, 53, 54, 55lmhmlnmsplit 41443 . 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 1372 1 (𝜑𝑍 ∈ LNoeM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  {crab 3410  Vcvv 3448  cun 3913  cin 3914  wss 3915  c0 4287  {csn 4591   class class class wbr 5110  cmpt 5193   × cxp 5636  ccnv 5637  ran crn 5639  cres 5640  cima 5641  ontowfo 6499  cfv 6501  (class class class)co 7362  Basecbs 17090  s cress 17119  0gc0g 17328  s cpws 17335  Mndcmnd 18563  Grpcgrp 18755  LModclmod 20338   LMHom clmhm 20496   LMIso clmim 20497  𝑚 clmic 20498  LNoeMclnm 41431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9385  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-z 12507  df-dec 12626  df-uz 12771  df-fz 13432  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-mulr 17154  df-sca 17156  df-vsca 17157  df-ip 17158  df-tset 17159  df-ple 17160  df-ds 17162  df-hom 17164  df-cco 17165  df-0g 17330  df-prds 17336  df-pws 17338  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-submnd 18609  df-grp 18758  df-minusg 18759  df-sbg 18760  df-subg 18932  df-ghm 19013  df-cntz 19104  df-lsm 19425  df-cmn 19571  df-abl 19572  df-mgp 19904  df-ur 19921  df-ring 19973  df-lmod 20340  df-lss 20409  df-lsp 20449  df-lmhm 20499  df-lmim 20500  df-lmic 20501  df-lfig 41424  df-lnm 41432
This theorem is referenced by:  pwslnm  41450
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