MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pj1fval Structured version   Visualization version   GIF version

Theorem pj1fval 18302
Description: The left projection function (for a direct product of group subspaces). (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
pj1fval.v 𝐵 = (Base‘𝐺)
pj1fval.a + = (+g𝐺)
pj1fval.s = (LSSum‘𝐺)
pj1fval.p 𝑃 = (proj1𝐺)
Assertion
Ref Expression
pj1fval ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))))
Distinct variable groups:   𝑧, +   𝑥,𝑦,𝑧,𝐵   𝑥,𝑇,𝑦,𝑧   𝑥,𝑈,𝑦,𝑧   𝑥, ,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝑥,𝑉,𝑦,𝑧
Allowed substitution hints:   𝑃(𝑥,𝑦,𝑧)   + (𝑥,𝑦)

Proof of Theorem pj1fval
Dummy variables 𝑡 𝑔 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pj1fval.p . . 3 𝑃 = (proj1𝐺)
2 elex 3402 . . . . 5 (𝐺𝑉𝐺 ∈ V)
323ad2ant1 1156 . . . 4 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝐺 ∈ V)
4 fveq2 6402 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5 pj1fval.v . . . . . . . 8 𝐵 = (Base‘𝐺)
64, 5syl6eqr 2854 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
76pweqd 4350 . . . . . 6 (𝑔 = 𝐺 → 𝒫 (Base‘𝑔) = 𝒫 𝐵)
8 fveq2 6402 . . . . . . . . 9 (𝑔 = 𝐺 → (LSSum‘𝑔) = (LSSum‘𝐺))
9 pj1fval.s . . . . . . . . 9 = (LSSum‘𝐺)
108, 9syl6eqr 2854 . . . . . . . 8 (𝑔 = 𝐺 → (LSSum‘𝑔) = )
1110oveqd 6885 . . . . . . 7 (𝑔 = 𝐺 → (𝑡(LSSum‘𝑔)𝑢) = (𝑡 𝑢))
12 fveq2 6402 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
13 pj1fval.a . . . . . . . . . . . 12 + = (+g𝐺)
1412, 13syl6eqr 2854 . . . . . . . . . . 11 (𝑔 = 𝐺 → (+g𝑔) = + )
1514oveqd 6885 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑦) = (𝑥 + 𝑦))
1615eqeq2d 2812 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑧 = (𝑥(+g𝑔)𝑦) ↔ 𝑧 = (𝑥 + 𝑦)))
1716rexbidv 3236 . . . . . . . 8 (𝑔 = 𝐺 → (∃𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦) ↔ ∃𝑦𝑢 𝑧 = (𝑥 + 𝑦)))
1817riotabidv 6831 . . . . . . 7 (𝑔 = 𝐺 → (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦)) = (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))
1911, 18mpteq12dv 4920 . . . . . 6 (𝑔 = 𝐺 → (𝑧 ∈ (𝑡(LSSum‘𝑔)𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦))) = (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦))))
207, 7, 19mpt2eq123dv 6941 . . . . 5 (𝑔 = 𝐺 → (𝑡 ∈ 𝒫 (Base‘𝑔), 𝑢 ∈ 𝒫 (Base‘𝑔) ↦ (𝑧 ∈ (𝑡(LSSum‘𝑔)𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦)))) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))))
21 df-pj1 18247 . . . . 5 proj1 = (𝑔 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑔), 𝑢 ∈ 𝒫 (Base‘𝑔) ↦ (𝑧 ∈ (𝑡(LSSum‘𝑔)𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦)))))
225fvexi 6416 . . . . . . 7 𝐵 ∈ V
2322pwex 5044 . . . . . 6 𝒫 𝐵 ∈ V
2423, 23mpt2ex 7474 . . . . 5 (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))) ∈ V
2520, 21, 24fvmpt 6497 . . . 4 (𝐺 ∈ V → (proj1𝐺) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))))
263, 25syl 17 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (proj1𝐺) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))))
271, 26syl5eq 2848 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑃 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))))
28 oveq12 6877 . . . 4 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑡 𝑢) = (𝑇 𝑈))
2928adantl 469 . . 3 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (𝑡 𝑢) = (𝑇 𝑈))
30 simprl 778 . . . 4 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → 𝑡 = 𝑇)
31 simprr 780 . . . . 5 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → 𝑢 = 𝑈)
3231rexeqdv 3330 . . . 4 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (∃𝑦𝑢 𝑧 = (𝑥 + 𝑦) ↔ ∃𝑦𝑈 𝑧 = (𝑥 + 𝑦)))
3330, 32riotaeqbidv 6832 . . 3 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)) = (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦)))
3429, 33mpteq12dv 4920 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦))) = (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))))
35 simp2 1160 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑇𝐵)
3622elpw2 5014 . . 3 (𝑇 ∈ 𝒫 𝐵𝑇𝐵)
3735, 36sylibr 225 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑇 ∈ 𝒫 𝐵)
38 simp3 1161 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑈𝐵)
3922elpw2 5014 . . 3 (𝑈 ∈ 𝒫 𝐵𝑈𝐵)
4038, 39sylibr 225 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑈 ∈ 𝒫 𝐵)
41 ovex 6900 . . . 4 (𝑇 𝑈) ∈ V
4241mptex 6705 . . 3 (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))) ∈ V
4342a1i 11 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))) ∈ V)
4427, 34, 37, 40, 43ovmpt2d 7012 1 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2155  wrex 3093  Vcvv 3387  wss 3763  𝒫 cpw 4345  cmpt 4916  cfv 6095  crio 6828  (class class class)co 6868  cmpt2 6870  Basecbs 16062  +gcplusg 16147  LSSumclsm 18244  proj1cpj1 18245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-id 5213  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-riota 6829  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-1st 7392  df-2nd 7393  df-pj1 18247
This theorem is referenced by:  pj1val  18303  pj1f  18305
  Copyright terms: Public domain W3C validator