Theorem pwslnmlem2 43050

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 7779	. . . 4 ⊢ (𝐴 ∪ 𝐵) ∈ V
5	4	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
6		ssun1 4201	. . . 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 2740	. . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍)
11		eqid 2740	. . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋)
12		eqid 2740	. . . 4 ⊢ (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) = (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))
13	8, 9, 10, 11, 12	pwssplit3 21083	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) ∈ (𝑍 LMHom 𝑋))
14	1, 5, 7, 13	syl3anc 1371	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) ∈ (𝑍 LMHom 𝑋))
15		fvex 6933	. . . . . 6 ⊢ (0g‘𝑋) ∈ V
16	12	mptiniseg 6270	. . . . . 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 20887	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
19		grpmnd 18980	. . . . . . . . . 10 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Mnd)
20	1, 18, 19	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Mnd)
21		eqid 2740	. . . . . . . . . 10 ⊢ (0g‘𝑊) = (0g‘𝑊)
22	9, 21	pws0g 18808	. . . . . . . . 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 2746	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑋) = (𝐴 × {(0g‘𝑊)}))
25	24	eqeq2d 2751	. . . . . 6 ⊢ (𝜑 → ((𝑥 ↾ 𝐴) = (0g‘𝑋) ↔ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})))
26	25	rabbidv 3451	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (0g‘𝑋)} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})})
27	17, 26	eqtrid 2792	. . . 4 ⊢ (𝜑 → (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})})
28	27	oveq2d 7464	. . 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 2740	. . . . . . 7 ⊢ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}
32		eqid 2740	. . . . . . 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 2740	. . . . . . 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 43046	. . . . . 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 1371	. . . . 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 21091	. . . . 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 43045	. . . . 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 257	. . 3 ⊢ (𝜑 → (𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) ∈ LNoeM)
41	28, 40	eqeltrd 2844	. 2 ⊢ (𝜑 → (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})) ∈ LNoeM)
42	8, 9, 10, 11, 12	pwssplit1 21081	. . . . . . 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 1371	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
44		forn 6837	. . . . . 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 7464	. . . 4 ⊢ (𝜑 → (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) = (𝑋 ↾s (Base‘𝑋)))
47		pwslnmlem2.xn	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ LNoeM)
48	11	ressid 17303	. . . . 5 ⊢ (𝑋 ∈ LNoeM → (𝑋 ↾s (Base‘𝑋)) = 𝑋)
49	47, 48	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 ↾s (Base‘𝑋)) = 𝑋)
50	46, 49	eqtrd 2780	. . 3 ⊢ (𝜑 → (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) = 𝑋)
51	50, 47	eqeltrd 2844	. 2 ⊢ (𝜑 → (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) ∈ LNoeM)
52		eqid 2740	. . 3 ⊢ (0g‘𝑋) = (0g‘𝑋)
53		eqid 2740	. . 3 ⊢ (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}) = (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})
54		eqid 2740	. . 3 ⊢ (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})) = (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}))
55		eqid 2740	. . 3 ⊢ (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) = (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)))
56	52, 53, 54, 55	lmhmlnmsplit 43044	. 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 1371	1 ⊢ (𝜑 → 𝑍 ∈ LNoeM)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  {crab 3443  Vcvv 3488   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  {csn 4648   class class class wbr 5166   ↦ cmpt 5249   × cxp 5698  ^◡ccnv 5699  ran crn 5701   ↾ cres 5702   “ cima 5703  –onto→wfo 6571  ‘cfv 6573  (class class class)co 7448  Basecbs 17258   ↾_s cress 17287  0_gc0g 17499   ↑_s cpws 17506  Mndcmnd 18772  Grpcgrp 18973  LModclmod 20880   LMHom clmhm 21041   LMIso clmim 21042   ≃_𝑚 clmic 21043  LNoeMclnm 43032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-hom 17335  df-cco 17336  df-0g 17501  df-prds 17507  df-pws 17509  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-subg 19163  df-ghm 19253  df-cntz 19357  df-lsm 19678  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-lmod 20882  df-lss 20953  df-lsp 20993  df-lmhm 21044  df-lmim 21045  df-lmic 21046  df-lfig 43025  df-lnm 43033

This theorem is referenced by:  pwslnm  43051