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Theorem oppglsm 18763
 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 6663 . . . . . 6 𝑂 ∈ V
3 eqid 2801 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
41, 3oppgbas 18475 . . . . . . 7 (Base‘𝐺) = (Base‘𝑂)
5 eqid 2801 . . . . . . 7 (+g𝑂) = (+g𝑂)
6 eqid 2801 . . . . . . 7 (LSSum‘𝑂) = (LSSum‘𝑂)
74, 5, 6lsmfval 18759 . . . . . 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 2801 . . . . . . . 8 (+g𝐺) = (+g𝐺)
10 oppglsm.p . . . . . . . 8 = (LSSum‘𝐺)
113, 9, 10lsmfval 18759 . . . . . . 7 (𝐺 ∈ V → = (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))))
1211tposeqd 7882 . . . . . 6 (𝐺 ∈ V → tpos = tpos (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))))
13 eqid 2801 . . . . . . . . . . . . 13 (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
1413reldmmpo 7268 . . . . . . . . . . . 12 Rel dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
1513mpofun 7259 . . . . . . . . . . . . 13 Fun (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
16 funforn 6576 . . . . . . . . . . . . 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 233 . . . . . . . . . . . 12 (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)):dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))–onto→ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
18 tposfo2 7902 . . . . . . . . . . . 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 6572 . . . . . . . . . . 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 18473 . . . . . . . . . . . . . . 15 (𝑥(+g𝑂)𝑦) = (𝑦(+g𝐺)𝑥)
2322eqcomi 2810 . . . . . . . . . . . . . 14 (𝑦(+g𝐺)𝑥) = (𝑥(+g𝑂)𝑦)
2423a1i 11 . . . . . . . . . . . . 13 ((𝑦𝑢𝑥𝑡) → (𝑦(+g𝐺)𝑥) = (𝑥(+g𝑂)𝑦))
2524mpoeq3ia 7215 . . . . . . . . . . . 12 (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = (𝑦𝑢, 𝑥𝑡 ↦ (𝑥(+g𝑂)𝑦))
2625tposmpo 7916 . . . . . . . . . . 11 tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))
2726rneqi 5775 . . . . . . . . . 10 ran tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))
2821, 27eqtr3i 2826 . . . . . . . . 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 7215 . . . . . . 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 7916 . . . . . 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 2852 . . . . 5 (𝐺 ∈ V → tpos = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))))
338, 32eqtr4id 2855 . . . 4 (𝐺 ∈ V → (LSSum‘𝑂) = tpos )
3433oveqd 7156 . . 3 (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇tpos 𝑈))
35 ovtpos 7894 . . 3 (𝑇tpos 𝑈) = (𝑈 𝑇)
3634, 35eqtrdi 2852 . 2 (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇))
37 eqid 2801 . . . . . . 7 (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)
38 0ex 5178 . . . . . . 7 ∅ ∈ V
39 eqidd 2802 . . . . . . 7 ((𝑡 = 𝑇𝑢 = 𝑈) → ∅ = ∅)
4037, 38, 39elovmpo 7374 . . . . . 6 (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ↔ (𝑇 ∈ 𝒫 (Base‘𝐺) ∧ 𝑈 ∈ 𝒫 (Base‘𝐺) ∧ 𝑥 ∈ ∅))
4140simp3bi 1144 . . . . 5 (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) → 𝑥 ∈ ∅)
4241ssriv 3922 . . . 4 (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ⊆ ∅
43 ss0 4309 . . . 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 4509 . . . . . . . . . . . . 13 (𝑡 ∈ 𝒫 (Base‘𝐺) → 𝑡 ⊆ (Base‘𝐺))
46453ad2ant2 1131 . . . . . . . . . . . 12 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ (Base‘𝐺))
47 fvprc 6642 . . . . . . . . . . . . 13 𝐺 ∈ V → (Base‘𝐺) = ∅)
48473ad2ant1 1130 . . . . . . . . . . . 12 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (Base‘𝐺) = ∅)
4946, 48sseqtrd 3958 . . . . . . . . . . 11 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ ∅)
50 ss0 4309 . . . . . . . . . . 11 (𝑡 ⊆ ∅ → 𝑡 = ∅)
5149, 50syl 17 . . . . . . . . . 10 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 = ∅)
5251orcd 870 . . . . . . . . 9 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (𝑡 = ∅ ∨ 𝑢 = ∅))
53 0mpo0 7220 . . . . . . . . 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 5776 . . . . . . 7 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = ran ∅)
56 rn0 5764 . . . . . . 7 ran ∅ = ∅
5755, 56eqtrdi 2852 . . . . . 6 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = ∅)
5857mpoeq3dva 7214 . . . . 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, 58syl5eq 2848 . . . 4 𝐺 ∈ V → (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅))
6059oveqd 7156 . . 3 𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈))
61 fvprc 6642 . . . . . 6 𝐺 ∈ V → (LSSum‘𝐺) = ∅)
6210, 61syl5eq 2848 . . . . 5 𝐺 ∈ V → = ∅)
6362oveqd 7156 . . . 4 𝐺 ∈ V → (𝑈 𝑇) = (𝑈𝑇))
64 0ov 7176 . . . 4 (𝑈𝑇) = ∅
6563, 64eqtrdi 2852 . . 3 𝐺 ∈ V → (𝑈 𝑇) = ∅)
6644, 60, 653eqtr4a 2862 . 2 𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇))
6736, 66pm2.61i 185 1 (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  Vcvv 3444   ⊆ wss 3884  ∅c0 4246  𝒫 cpw 4500  ◡ccnv 5522  dom cdm 5523  ran crn 5524  Rel wrel 5528  Fun wfun 6322  –onto→wfo 6326  ‘cfv 6328  (class class class)co 7139   ∈ cmpo 7141  tpos ctpos 7878  Basecbs 16479  +gcplusg 16561  oppgcoppg 18469  LSSumclsm 18755 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-tpos 7879  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-plusg 16574  df-oppg 18470  df-lsm 18757 This theorem is referenced by:  lsmmod2  18798  lsmdisj2r  18807
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