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Theorem pj1fval 18749
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 3510 . . . . 5 (𝐺𝑉𝐺 ∈ V)
323ad2ant1 1125 . . . 4 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝐺 ∈ V)
4 fveq2 6663 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5 pj1fval.v . . . . . . . 8 𝐵 = (Base‘𝐺)
64, 5syl6eqr 2871 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
76pweqd 4540 . . . . . 6 (𝑔 = 𝐺 → 𝒫 (Base‘𝑔) = 𝒫 𝐵)
8 fveq2 6663 . . . . . . . . 9 (𝑔 = 𝐺 → (LSSum‘𝑔) = (LSSum‘𝐺))
9 pj1fval.s . . . . . . . . 9 = (LSSum‘𝐺)
108, 9syl6eqr 2871 . . . . . . . 8 (𝑔 = 𝐺 → (LSSum‘𝑔) = )
1110oveqd 7162 . . . . . . 7 (𝑔 = 𝐺 → (𝑡(LSSum‘𝑔)𝑢) = (𝑡 𝑢))
12 fveq2 6663 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
13 pj1fval.a . . . . . . . . . . . 12 + = (+g𝐺)
1412, 13syl6eqr 2871 . . . . . . . . . . 11 (𝑔 = 𝐺 → (+g𝑔) = + )
1514oveqd 7162 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑦) = (𝑥 + 𝑦))
1615eqeq2d 2829 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑧 = (𝑥(+g𝑔)𝑦) ↔ 𝑧 = (𝑥 + 𝑦)))
1716rexbidv 3294 . . . . . . . 8 (𝑔 = 𝐺 → (∃𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦) ↔ ∃𝑦𝑢 𝑧 = (𝑥 + 𝑦)))
1817riotabidv 7105 . . . . . . 7 (𝑔 = 𝐺 → (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦)) = (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))
1911, 18mpteq12dv 5142 . . . . . 6 (𝑔 = 𝐺 → (𝑧 ∈ (𝑡(LSSum‘𝑔)𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦))) = (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦))))
207, 7, 19mpoeq123dv 7218 . . . . 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 18691 . . . . 5 proj1 = (𝑔 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑔), 𝑢 ∈ 𝒫 (Base‘𝑔) ↦ (𝑧 ∈ (𝑡(LSSum‘𝑔)𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦)))))
225fvexi 6677 . . . . . . 7 𝐵 ∈ V
2322pwex 5272 . . . . . 6 𝒫 𝐵 ∈ V
2423, 23mpoex 7766 . . . . 5 (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))) ∈ V
2520, 21, 24fvmpt 6761 . . . 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 2865 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑃 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))))
28 oveq12 7154 . . . 4 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑡 𝑢) = (𝑇 𝑈))
2928adantl 482 . . 3 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (𝑡 𝑢) = (𝑇 𝑈))
30 simprl 767 . . . 4 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → 𝑡 = 𝑇)
31 simprr 769 . . . . 5 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → 𝑢 = 𝑈)
3231rexeqdv 3414 . . . 4 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (∃𝑦𝑢 𝑧 = (𝑥 + 𝑦) ↔ ∃𝑦𝑈 𝑧 = (𝑥 + 𝑦)))
3330, 32riotaeqbidv 7106 . . 3 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)) = (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦)))
3429, 33mpteq12dv 5142 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦))) = (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))))
35 simp2 1129 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑇𝐵)
3622elpw2 5239 . . 3 (𝑇 ∈ 𝒫 𝐵𝑇𝐵)
3735, 36sylibr 235 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑇 ∈ 𝒫 𝐵)
38 simp3 1130 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑈𝐵)
3922elpw2 5239 . . 3 (𝑈 ∈ 𝒫 𝐵𝑈𝐵)
4038, 39sylibr 235 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑈 ∈ 𝒫 𝐵)
41 ovex 7178 . . . 4 (𝑇 𝑈) ∈ V
4241mptex 6977 . . 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, 43ovmpod 7291 1 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3136  Vcvv 3492  wss 3933  𝒫 cpw 4535  cmpt 5137  cfv 6348  crio 7102  (class class class)co 7145  cmpo 7147  Basecbs 16471  +gcplusg 16553  LSSumclsm 18688  proj1cpj1 18689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-pj1 18691
This theorem is referenced by:  pj1val  18750  pj1f  18752
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