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Theorem pj1val 18154
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 18153 . . 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 476 . . . . 5 ((((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) ∧ 𝑧 = 𝑋) → 𝑧 = 𝑋)
87eqeq1d 2653 . . . 4 ((((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) ∧ 𝑧 = 𝑋) → (𝑧 = (𝑥 + 𝑦) ↔ 𝑋 = (𝑥 + 𝑦)))
98rexbidv 3081 . . 3 ((((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) ∧ 𝑧 = 𝑋) → (∃𝑦𝑈 𝑧 = (𝑥 + 𝑦) ↔ ∃𝑦𝑈 𝑋 = (𝑥 + 𝑦)))
109riotabidv 6653 . 2 ((((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) ∧ 𝑧 = 𝑋) → (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦)) = (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)))
11 simpr 476 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) → 𝑋 ∈ (𝑇 𝑈))
12 riotaex 6655 . . 3 (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)) ∈ V
1312a1i 11 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) → (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)) ∈ V)
146, 10, 11, 13fvmptd 6327 1 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  Vcvv 3231  wss 3607  cmpt 4762  cfv 5926  crio 6650  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  LSSumclsm 18095  proj1cpj1 18096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-pj1 18098
This theorem is referenced by:  pj1id  18158
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