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Theorem oppglsm 19597
Description: The subspace sum operation in the opposite group. (Contributed by Mario Carneiro, 19-Apr-2016.) (Proof shortened by AV, 2-Mar-2024.)
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 oppglsm.o . . . . . . 7 𝑂 = (oppg𝐺)
21fvexi 6911 . . . . . 6 𝑂 ∈ V
3 eqid 2728 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
41, 3oppgbas 19303 . . . . . . 7 (Base‘𝐺) = (Base‘𝑂)
5 eqid 2728 . . . . . . 7 (+g𝑂) = (+g𝑂)
6 eqid 2728 . . . . . . 7 (LSSum‘𝑂) = (LSSum‘𝑂)
74, 5, 6lsmfval 19593 . . . . . 6 (𝑂 ∈ V → (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))))
82, 7ax-mp 5 . . . . 5 (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)))
9 eqid 2728 . . . . . . . 8 (+g𝐺) = (+g𝐺)
10 oppglsm.p . . . . . . . 8 = (LSSum‘𝐺)
113, 9, 10lsmfval 19593 . . . . . . 7 (𝐺 ∈ V → = (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))))
1211tposeqd 8235 . . . . . 6 (𝐺 ∈ V → tpos = tpos (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))))
13 eqid 2728 . . . . . . . . . . . . 13 (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
1413reldmmpo 7555 . . . . . . . . . . . 12 Rel dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
1513mpofun 7544 . . . . . . . . . . . . 13 Fun (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
16 funforn 6818 . . . . . . . . . . . . 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)))
1715, 16mpbi 229 . . . . . . . . . . . 12 (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)):dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))–onto→ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
18 tposfo2 8255 . . . . . . . . . . . 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))))
1914, 17, 18mp2 9 . . . . . . . . . . 11 tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)):dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))–onto→ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
20 forn 6814 . . . . . . . . . . 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)))
2119, 20ax-mp 5 . . . . . . . . . 10 ran tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
229, 1, 5oppgplus 19300 . . . . . . . . . . . . . . 15 (𝑥(+g𝑂)𝑦) = (𝑦(+g𝐺)𝑥)
2322eqcomi 2737 . . . . . . . . . . . . . 14 (𝑦(+g𝐺)𝑥) = (𝑥(+g𝑂)𝑦)
2423a1i 11 . . . . . . . . . . . . 13 ((𝑦𝑢𝑥𝑡) → (𝑦(+g𝐺)𝑥) = (𝑥(+g𝑂)𝑦))
2524mpoeq3ia 7498 . . . . . . . . . . . 12 (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = (𝑦𝑢, 𝑥𝑡 ↦ (𝑥(+g𝑂)𝑦))
2625tposmpo 8269 . . . . . . . . . . 11 tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))
2726rneqi 5939 . . . . . . . . . 10 ran tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))
2821, 27eqtr3i 2758 . . . . . . . . 9 ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))
2928a1i 11 . . . . . . . 8 ((𝑢 ∈ 𝒫 (Base‘𝐺) ∧ 𝑡 ∈ 𝒫 (Base‘𝐺)) → ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)))
3029mpoeq3ia 7498 . . . . . . 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)))
3130tposmpo 8269 . . . . . 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)))
3212, 31eqtrdi 2784 . . . . 5 (𝐺 ∈ V → tpos = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))))
338, 32eqtr4id 2787 . . . 4 (𝐺 ∈ V → (LSSum‘𝑂) = tpos )
3433oveqd 7437 . . 3 (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇tpos 𝑈))
35 ovtpos 8247 . . 3 (𝑇tpos 𝑈) = (𝑈 𝑇)
3634, 35eqtrdi 2784 . 2 (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇))
37 eqid 2728 . . . . . . 7 (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)
38 0ex 5307 . . . . . . 7 ∅ ∈ V
39 eqidd 2729 . . . . . . 7 ((𝑡 = 𝑇𝑢 = 𝑈) → ∅ = ∅)
4037, 38, 39elovmpo 7666 . . . . . 6 (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ↔ (𝑇 ∈ 𝒫 (Base‘𝐺) ∧ 𝑈 ∈ 𝒫 (Base‘𝐺) ∧ 𝑥 ∈ ∅))
4140simp3bi 1145 . . . . 5 (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) → 𝑥 ∈ ∅)
4241ssriv 3984 . . . 4 (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ⊆ ∅
43 ss0 4399 . . . 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 4610 . . . . . . . . . . . . 13 (𝑡 ∈ 𝒫 (Base‘𝐺) → 𝑡 ⊆ (Base‘𝐺))
46453ad2ant2 1132 . . . . . . . . . . . 12 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ (Base‘𝐺))
47 fvprc 6889 . . . . . . . . . . . . 13 𝐺 ∈ V → (Base‘𝐺) = ∅)
48473ad2ant1 1131 . . . . . . . . . . . 12 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (Base‘𝐺) = ∅)
4946, 48sseqtrd 4020 . . . . . . . . . . 11 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ ∅)
50 ss0 4399 . . . . . . . . . . 11 (𝑡 ⊆ ∅ → 𝑡 = ∅)
5149, 50syl 17 . . . . . . . . . 10 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 = ∅)
5251orcd 872 . . . . . . . . 9 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (𝑡 = ∅ ∨ 𝑢 = ∅))
53 0mpo0 7503 . . . . . . . . 9 ((𝑡 = ∅ ∨ 𝑢 = ∅) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = ∅)
5452, 53syl 17 . . . . . . . 8 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = ∅)
5554rneqd 5940 . . . . . . 7 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = ran ∅)
56 rn0 5928 . . . . . . 7 ran ∅ = ∅
5755, 56eqtrdi 2784 . . . . . 6 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = ∅)
5857mpoeq3dva 7497 . . . . 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) ↦ ∅))
598, 58eqtrid 2780 . . . 4 𝐺 ∈ V → (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅))
6059oveqd 7437 . . 3 𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈))
61 fvprc 6889 . . . . . 6 𝐺 ∈ V → (LSSum‘𝐺) = ∅)
6210, 61eqtrid 2780 . . . . 5 𝐺 ∈ V → = ∅)
6362oveqd 7437 . . . 4 𝐺 ∈ V → (𝑈 𝑇) = (𝑈𝑇))
64 0ov 7457 . . . 4 (𝑈𝑇) = ∅
6563, 64eqtrdi 2784 . . 3 𝐺 ∈ V → (𝑈 𝑇) = ∅)
6644, 60, 653eqtr4a 2794 . 2 𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇))
6736, 66pm2.61i 182 1 (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wo 846  w3a 1085   = wceq 1534  wcel 2099  Vcvv 3471  wss 3947  c0 4323  𝒫 cpw 4603  ccnv 5677  dom cdm 5678  ran crn 5679  Rel wrel 5683  Fun wfun 6542  ontowfo 6546  cfv 6548  (class class class)co 7420  cmpo 7422  tpos ctpos 8231  Basecbs 17180  +gcplusg 17233  oppgcoppg 19296  LSSumclsm 19589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-tpos 8232  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-plusg 17246  df-oppg 19297  df-lsm 19591
This theorem is referenced by:  lsmmod2  19631  lsmdisj2r  19640  lsmsnorb2  33114
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