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Theorem pj1val 18821
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 18820 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))))
65adantr 484 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))))
7 simpr 488 . . . . 5 ((((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) ∧ 𝑧 = 𝑋) → 𝑧 = 𝑋)
87eqeq1d 2826 . . . 4 ((((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) ∧ 𝑧 = 𝑋) → (𝑧 = (𝑥 + 𝑦) ↔ 𝑋 = (𝑥 + 𝑦)))
98rexbidv 3289 . . 3 ((((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) ∧ 𝑧 = 𝑋) → (∃𝑦𝑈 𝑧 = (𝑥 + 𝑦) ↔ ∃𝑦𝑈 𝑋 = (𝑥 + 𝑦)))
109riotabidv 7109 . 2 ((((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) ∧ 𝑧 = 𝑋) → (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦)) = (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)))
11 simpr 488 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) → 𝑋 ∈ (𝑇 𝑈))
12 riotaex 7111 . . 3 (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)) ∈ V
1312a1i 11 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) → (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)) ∈ V)
146, 10, 11, 13fvmptd 6766 1 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wrex 3134  Vcvv 3480  wss 3919  cmpt 5132  cfv 6343  crio 7106  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  LSSumclsm 18759  proj1cpj1 18760
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-pj1 18762
This theorem is referenced by:  pj1id  18825
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