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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  ontowfo 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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