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Theorem oppglsm 18264
Description: The subspace sum operation in the opposite group. (Contributed by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
oppglsm.o 𝑂 = (oppg𝐺)
oppglsm.p = (LSSum‘𝐺)
Assertion
Ref Expression
oppglsm (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇)

Proof of Theorem oppglsm
Dummy variables 𝑢 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2771 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2771 . . . . . . . 8 (+g𝐺) = (+g𝐺)
3 oppglsm.p . . . . . . . 8 = (LSSum‘𝐺)
41, 2, 3lsmfval 18260 . . . . . . 7 (𝐺 ∈ V → = (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))))
54tposeqd 7511 . . . . . 6 (𝐺 ∈ V → tpos = tpos (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))))
6 eqid 2771 . . . . . . . . . . . . 13 (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
76reldmmpt2 6922 . . . . . . . . . . . 12 Rel dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
86mpt2fun 6913 . . . . . . . . . . . . 13 Fun (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
9 funforn 6264 . . . . . . . . . . . . 13 (Fun (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) ↔ (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)):dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))–onto→ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)))
108, 9mpbi 220 . . . . . . . . . . . 12 (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)):dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))–onto→ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
11 tposfo2 7531 . . . . . . . . . . . 12 (Rel dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) → ((𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)):dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))–onto→ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) → tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)):dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))–onto→ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))))
127, 10, 11mp2 9 . . . . . . . . . . 11 tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)):dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))–onto→ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
13 forn 6260 . . . . . . . . . . 11 (tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)):dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))–onto→ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) → ran tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)))
1412, 13ax-mp 5 . . . . . . . . . 10 ran tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
15 oppglsm.o . . . . . . . . . . . . . . . 16 𝑂 = (oppg𝐺)
16 eqid 2771 . . . . . . . . . . . . . . . 16 (+g𝑂) = (+g𝑂)
172, 15, 16oppgplus 17986 . . . . . . . . . . . . . . 15 (𝑥(+g𝑂)𝑦) = (𝑦(+g𝐺)𝑥)
1817eqcomi 2780 . . . . . . . . . . . . . 14 (𝑦(+g𝐺)𝑥) = (𝑥(+g𝑂)𝑦)
1918a1i 11 . . . . . . . . . . . . 13 ((𝑦𝑢𝑥𝑡) → (𝑦(+g𝐺)𝑥) = (𝑥(+g𝑂)𝑦))
2019mpt2eq3ia 6871 . . . . . . . . . . . 12 (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = (𝑦𝑢, 𝑥𝑡 ↦ (𝑥(+g𝑂)𝑦))
2120tposmpt2 7545 . . . . . . . . . . 11 tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))
2221rneqi 5489 . . . . . . . . . 10 ran tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))
2314, 22eqtr3i 2795 . . . . . . . . 9 ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))
2423a1i 11 . . . . . . . 8 ((𝑢 ∈ 𝒫 (Base‘𝐺) ∧ 𝑡 ∈ 𝒫 (Base‘𝐺)) → ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)))
2524mpt2eq3ia 6871 . . . . . . 7 (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))) = (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)))
2625tposmpt2 7545 . . . . . 6 tpos (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)))
275, 26syl6eq 2821 . . . . 5 (𝐺 ∈ V → tpos = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))))
2815fvexi 6345 . . . . . 6 𝑂 ∈ V
2915, 1oppgbas 17988 . . . . . . 7 (Base‘𝐺) = (Base‘𝑂)
30 eqid 2771 . . . . . . 7 (LSSum‘𝑂) = (LSSum‘𝑂)
3129, 16, 30lsmfval 18260 . . . . . 6 (𝑂 ∈ V → (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))))
3228, 31ax-mp 5 . . . . 5 (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)))
3327, 32syl6reqr 2824 . . . 4 (𝐺 ∈ V → (LSSum‘𝑂) = tpos )
3433oveqd 6813 . . 3 (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇tpos 𝑈))
35 ovtpos 7523 . . 3 (𝑇tpos 𝑈) = (𝑈 𝑇)
3634, 35syl6eq 2821 . 2 (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇))
37 eqid 2771 . . . . . . 7 (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)
38 0ex 4925 . . . . . . 7 ∅ ∈ V
39 eqidd 2772 . . . . . . 7 ((𝑡 = 𝑇𝑢 = 𝑈) → ∅ = ∅)
4037, 38, 39elovmpt2 7030 . . . . . 6 (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ↔ (𝑇 ∈ 𝒫 (Base‘𝐺) ∧ 𝑈 ∈ 𝒫 (Base‘𝐺) ∧ 𝑥 ∈ ∅))
4140simp3bi 1141 . . . . 5 (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) → 𝑥 ∈ ∅)
4241ssriv 3756 . . . 4 (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ⊆ ∅
43 ss0 4119 . . . 4 ((𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ⊆ ∅ → (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) = ∅)
4442, 43ax-mp 5 . . 3 (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) = ∅
45 elpwi 4308 . . . . . . . . . . . . 13 (𝑡 ∈ 𝒫 (Base‘𝐺) → 𝑡 ⊆ (Base‘𝐺))
46453ad2ant2 1128 . . . . . . . . . . . 12 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ (Base‘𝐺))
47 fvprc 6327 . . . . . . . . . . . . 13 𝐺 ∈ V → (Base‘𝐺) = ∅)
48473ad2ant1 1127 . . . . . . . . . . . 12 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (Base‘𝐺) = ∅)
4946, 48sseqtrd 3790 . . . . . . . . . . 11 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ ∅)
50 ss0 4119 . . . . . . . . . . 11 (𝑡 ⊆ ∅ → 𝑡 = ∅)
5149, 50syl 17 . . . . . . . . . 10 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 = ∅)
52 eqid 2771 . . . . . . . . . 10 𝑢 = 𝑢
53 mpt2eq12 6866 . . . . . . . . . 10 ((𝑡 = ∅ ∧ 𝑢 = 𝑢) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = (𝑥 ∈ ∅, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)))
5451, 52, 53sylancl 574 . . . . . . . . 9 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = (𝑥 ∈ ∅, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)))
55 mpt20 6876 . . . . . . . . 9 (𝑥 ∈ ∅, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = ∅
5654, 55syl6eq 2821 . . . . . . . 8 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = ∅)
5756rneqd 5490 . . . . . . 7 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = ran ∅)
58 rn0 5514 . . . . . . 7 ran ∅ = ∅
5957, 58syl6eq 2821 . . . . . 6 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = ∅)
6059mpt2eq3dva 6870 . . . . 5 𝐺 ∈ V → (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅))
6132, 60syl5eq 2817 . . . 4 𝐺 ∈ V → (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅))
6261oveqd 6813 . . 3 𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈))
63 fvprc 6327 . . . . . 6 𝐺 ∈ V → (LSSum‘𝐺) = ∅)
643, 63syl5eq 2817 . . . . 5 𝐺 ∈ V → = ∅)
6564oveqd 6813 . . . 4 𝐺 ∈ V → (𝑈 𝑇) = (𝑈𝑇))
66 0ov 6831 . . . 4 (𝑈𝑇) = ∅
6765, 66syl6eq 2821 . . 3 𝐺 ∈ V → (𝑈 𝑇) = ∅)
6844, 62, 673eqtr4a 2831 . 2 𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇))
6936, 68pm2.61i 176 1 (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  wss 3723  c0 4063  𝒫 cpw 4298  ccnv 5249  dom cdm 5250  ran crn 5251  Rel wrel 5255  Fun wfun 6024  ontowfo 6028  cfv 6030  (class class class)co 6796  cmpt2 6798  tpos ctpos 7507  Basecbs 16064  +gcplusg 16149  oppgcoppg 17982  LSSumclsm 18256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-tpos 7508  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-plusg 16162  df-oppg 17983  df-lsm 18258
This theorem is referenced by:  lsmmod2  18296  lsmdisj2r  18305
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