MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pj1val Structured version   Visualization version   GIF version

Theorem pj1val 19282
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
pj1val (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑇,𝑦   𝑥,𝑈,𝑦   𝑥, ,𝑦   𝑥,𝐺,𝑦   𝑥,𝑉,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑃(𝑥,𝑦)   + (𝑥,𝑦)

Proof of Theorem pj1val
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 pj1fval.v . . . 4 𝐵 = (Base‘𝐺)
2 pj1fval.a . . . 4 + = (+g𝐺)
3 pj1fval.s . . . 4 = (LSSum‘𝐺)
4 pj1fval.p . . . 4 𝑃 = (proj1𝐺)
51, 2, 3, 4pj1fval 19281 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))))
65adantr 480 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))))
7 simpr 484 . . . . 5 ((((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) ∧ 𝑧 = 𝑋) → 𝑧 = 𝑋)
87eqeq1d 2741 . . . 4 ((((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) ∧ 𝑧 = 𝑋) → (𝑧 = (𝑥 + 𝑦) ↔ 𝑋 = (𝑥 + 𝑦)))
98rexbidv 3227 . . 3 ((((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) ∧ 𝑧 = 𝑋) → (∃𝑦𝑈 𝑧 = (𝑥 + 𝑦) ↔ ∃𝑦𝑈 𝑋 = (𝑥 + 𝑦)))
109riotabidv 7227 . 2 ((((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) ∧ 𝑧 = 𝑋) → (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦)) = (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)))
11 simpr 484 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) → 𝑋 ∈ (𝑇 𝑈))
12 riotaex 7229 . . 3 (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)) ∈ V
1312a1i 11 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) → (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)) ∈ V)
146, 10, 11, 13fvmptd 6876 1 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1541  wcel 2109  wrex 3066  Vcvv 3430  wss 3891  cmpt 5161  cfv 6430  crio 7224  (class class class)co 7268  Basecbs 16893  +gcplusg 16943  LSSumclsm 19220  proj1cpj1 19221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-pj1 19223
This theorem is referenced by:  pj1id  19286
  Copyright terms: Public domain W3C validator