Theorem pwslnmlem2 40834

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 = (𝑊 ↑s 𝐴)
pwslnmlem2.y	⊢ 𝑌 = (𝑊 ↑s 𝐵)
pwslnmlem2.z	⊢ 𝑍 = (𝑊 ↑s (𝐴 ∪ 𝐵))
pwslnmlem2.w	⊢ (𝜑 → 𝑊 ∈ LMod)
pwslnmlem2.dj	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
pwslnmlem2.xn	⊢ (𝜑 → 𝑋 ∈ LNoeM)
pwslnmlem2.yn	⊢ (𝜑 → 𝑌 ∈ LNoeM)

Assertion

Ref	Expression
pwslnmlem2	⊢ (𝜑 → 𝑍 ∈ LNoeM)

Proof of Theorem pwslnmlem2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwslnmlem2.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ LMod)
2		pwslnmlem2.a	. . . . 5 ⊢ 𝐴 ∈ V
3		pwslnmlem2.b	. . . . 5 ⊢ 𝐵 ∈ V
4	2, 3	unex 7574	. . . 4 ⊢ (𝐴 ∪ 𝐵) ∈ V
5	4	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
6		ssun1 4102	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
7	6	a1i 11	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (𝐴 ∪ 𝐵))
8		pwslnmlem2.z	. . . 4 ⊢ 𝑍 = (𝑊 ↑s (𝐴 ∪ 𝐵))
9		pwslnmlem2.x	. . . 4 ⊢ 𝑋 = (𝑊 ↑s 𝐴)
10		eqid 2738	. . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍)
11		eqid 2738	. . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋)
12		eqid 2738	. . . 4 ⊢ (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) = (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))
13	8, 9, 10, 11, 12	pwssplit3 20238	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) ∈ (𝑍 LMHom 𝑋))
14	1, 5, 7, 13	syl3anc 1369	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) ∈ (𝑍 LMHom 𝑋))
15		fvex 6769	. . . . . 6 ⊢ (0g‘𝑋) ∈ V
16	12	mptiniseg 6131	. . . . . 6 ⊢ ((0g‘𝑋) ∈ V → (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (0g‘𝑋)})
17	15, 16	ax-mp 5	. . . . 5 ⊢ (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (0g‘𝑋)}
18		lmodgrp 20045	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
19		grpmnd 18499	. . . . . . . . . 10 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Mnd)
20	1, 18, 19	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Mnd)
21		eqid 2738	. . . . . . . . . 10 ⊢ (0g‘𝑊) = (0g‘𝑊)
22	9, 21	pws0g 18336	. . . . . . . . 9 ⊢ ((𝑊 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × {(0g‘𝑊)}) = (0g‘𝑋))
23	20, 2, 22	sylancl 585	. . . . . . . 8 ⊢ (𝜑 → (𝐴 × {(0g‘𝑊)}) = (0g‘𝑋))
24	23	eqcomd 2744	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑋) = (𝐴 × {(0g‘𝑊)}))
25	24	eqeq2d 2749	. . . . . 6 ⊢ (𝜑 → ((𝑥 ↾ 𝐴) = (0g‘𝑋) ↔ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})))
26	25	rabbidv 3404	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (0g‘𝑋)} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})})
27	17, 26	syl5eq 2791	. . . 4 ⊢ (𝜑 → (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})})
28	27	oveq2d 7271	. . 3 ⊢ (𝜑 → (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})) = (𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}))
29		pwslnmlem2.yn	. . . 4 ⊢ (𝜑 → 𝑌 ∈ LNoeM)
30		pwslnmlem2.dj	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
31		eqid 2738	. . . . . . 7 ⊢ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}
32		eqid 2738	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})} ↦ (𝑦 ↾ 𝐵)) = (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})} ↦ (𝑦 ↾ 𝐵))
33		pwslnmlem2.y	. . . . . . 7 ⊢ 𝑌 = (𝑊 ↑s 𝐵)
34		eqid 2738	. . . . . . 7 ⊢ (𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) = (𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})})
35	8, 10, 21, 31, 32, 9, 33, 34	pwssplit4 40830	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ V ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})} ↦ (𝑦 ↾ 𝐵)) ∈ ((𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) LMIso 𝑌))
36	1, 5, 30, 35	syl3anc 1369	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})} ↦ (𝑦 ↾ 𝐵)) ∈ ((𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) LMIso 𝑌))
37		brlmici 20246	. . . . 5 ⊢ ((𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})} ↦ (𝑦 ↾ 𝐵)) ∈ ((𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) LMIso 𝑌) → (𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) ≃𝑚 𝑌)
38		lnmlmic 40829	. . . . 5 ⊢ ((𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) ≃𝑚 𝑌 → ((𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) ∈ LNoeM ↔ 𝑌 ∈ LNoeM))
39	36, 37, 38	3syl 18	. . . 4 ⊢ (𝜑 → ((𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) ∈ LNoeM ↔ 𝑌 ∈ LNoeM))
40	29, 39	mpbird 256	. . 3 ⊢ (𝜑 → (𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) ∈ LNoeM)
41	28, 40	eqeltrd 2839	. 2 ⊢ (𝜑 → (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})) ∈ LNoeM)
42	8, 9, 10, 11, 12	pwssplit1 20236	. . . . . . 7 ⊢ ((𝑊 ∈ Mnd ∧ (𝐴 ∪ 𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
43	20, 5, 7, 42	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
44		forn 6675	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)):(Base‘𝑍)–onto→(Base‘𝑋) → ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) = (Base‘𝑋))
45	43, 44	syl 17	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) = (Base‘𝑋))
46	45	oveq2d 7271	. . . 4 ⊢ (𝜑 → (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) = (𝑋 ↾s (Base‘𝑋)))
47		pwslnmlem2.xn	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ LNoeM)
48	11	ressid 16880	. . . . 5 ⊢ (𝑋 ∈ LNoeM → (𝑋 ↾s (Base‘𝑋)) = 𝑋)
49	47, 48	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 ↾s (Base‘𝑋)) = 𝑋)
50	46, 49	eqtrd 2778	. . 3 ⊢ (𝜑 → (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) = 𝑋)
51	50, 47	eqeltrd 2839	. 2 ⊢ (𝜑 → (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) ∈ LNoeM)
52		eqid 2738	. . 3 ⊢ (0g‘𝑋) = (0g‘𝑋)
53		eqid 2738	. . 3 ⊢ (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}) = (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})
54		eqid 2738	. . 3 ⊢ (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})) = (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}))
55		eqid 2738	. . 3 ⊢ (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) = (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)))
56	52, 53, 54, 55	lmhmlnmsplit 40828	. 2 ⊢ (((𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) ∈ (𝑍 LMHom 𝑋) ∧ (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})) ∈ LNoeM ∧ (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) ∈ LNoeM) → 𝑍 ∈ LNoeM)
57	14, 41, 51, 56	syl3anc 1369	1 ⊢ (𝜑 → 𝑍 ∈ LNoeM)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578  ^◡ccnv 5579  ran crn 5581   ↾ cres 5582   “ cima 5583  –onto→wfo 6416  ‘cfv 6418  (class class class)co 7255  Basecbs 16840   ↾_s cress 16867  0_gc0g 17067   ↑_s cpws 17074  Mndcmnd 18300  Grpcgrp 18492  LModclmod 20038   LMHom clmhm 20196   LMIso clmim 20197   ≃_𝑚 clmic 20198  LNoeMclnm 40816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-hom 16912  df-cco 16913  df-0g 17069  df-prds 17075  df-pws 17077  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-ghm 18747  df-cntz 18838  df-lsm 19156  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-lmod 20040  df-lss 20109  df-lsp 20149  df-lmhm 20199  df-lmim 20200  df-lmic 20201  df-lfig 40809  df-lnm 40817

This theorem is referenced by:  pwslnm  40835