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Theorem pj1fval 18023
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 3203 . . . . 5 (𝐺𝑉𝐺 ∈ V)
323ad2ant1 1080 . . . 4 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝐺 ∈ V)
4 fveq2 6150 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5 pj1fval.v . . . . . . . 8 𝐵 = (Base‘𝐺)
64, 5syl6eqr 2678 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
76pweqd 4140 . . . . . 6 (𝑔 = 𝐺 → 𝒫 (Base‘𝑔) = 𝒫 𝐵)
8 fveq2 6150 . . . . . . . . 9 (𝑔 = 𝐺 → (LSSum‘𝑔) = (LSSum‘𝐺))
9 pj1fval.s . . . . . . . . 9 = (LSSum‘𝐺)
108, 9syl6eqr 2678 . . . . . . . 8 (𝑔 = 𝐺 → (LSSum‘𝑔) = )
1110oveqd 6622 . . . . . . 7 (𝑔 = 𝐺 → (𝑡(LSSum‘𝑔)𝑢) = (𝑡 𝑢))
12 fveq2 6150 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
13 pj1fval.a . . . . . . . . . . . 12 + = (+g𝐺)
1412, 13syl6eqr 2678 . . . . . . . . . . 11 (𝑔 = 𝐺 → (+g𝑔) = + )
1514oveqd 6622 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑦) = (𝑥 + 𝑦))
1615eqeq2d 2636 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑧 = (𝑥(+g𝑔)𝑦) ↔ 𝑧 = (𝑥 + 𝑦)))
1716rexbidv 3050 . . . . . . . 8 (𝑔 = 𝐺 → (∃𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦) ↔ ∃𝑦𝑢 𝑧 = (𝑥 + 𝑦)))
1817riotabidv 6568 . . . . . . 7 (𝑔 = 𝐺 → (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦)) = (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))
1911, 18mpteq12dv 4698 . . . . . 6 (𝑔 = 𝐺 → (𝑧 ∈ (𝑡(LSSum‘𝑔)𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦))) = (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦))))
207, 7, 19mpt2eq123dv 6671 . . . . 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 17968 . . . . 5 proj1 = (𝑔 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑔), 𝑢 ∈ 𝒫 (Base‘𝑔) ↦ (𝑧 ∈ (𝑡(LSSum‘𝑔)𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦)))))
22 fvex 6160 . . . . . . . 8 (Base‘𝐺) ∈ V
235, 22eqeltri 2700 . . . . . . 7 𝐵 ∈ V
2423pwex 4813 . . . . . 6 𝒫 𝐵 ∈ V
2524, 24mpt2ex 7193 . . . . 5 (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))) ∈ V
2620, 21, 25fvmpt 6240 . . . 4 (𝐺 ∈ V → (proj1𝐺) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))))
273, 26syl 17 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (proj1𝐺) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))))
281, 27syl5eq 2672 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑃 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))))
29 oveq12 6614 . . . 4 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑡 𝑢) = (𝑇 𝑈))
3029adantl 482 . . 3 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (𝑡 𝑢) = (𝑇 𝑈))
31 simprl 793 . . . 4 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → 𝑡 = 𝑇)
32 simprr 795 . . . . 5 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → 𝑢 = 𝑈)
3332rexeqdv 3139 . . . 4 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (∃𝑦𝑢 𝑧 = (𝑥 + 𝑦) ↔ ∃𝑦𝑈 𝑧 = (𝑥 + 𝑦)))
3431, 33riotaeqbidv 6569 . . 3 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)) = (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦)))
3530, 34mpteq12dv 4698 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦))) = (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))))
36 simp2 1060 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑇𝐵)
3723elpw2 4793 . . 3 (𝑇 ∈ 𝒫 𝐵𝑇𝐵)
3836, 37sylibr 224 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑇 ∈ 𝒫 𝐵)
39 simp3 1061 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑈𝐵)
4023elpw2 4793 . . 3 (𝑈 ∈ 𝒫 𝐵𝑈𝐵)
4139, 40sylibr 224 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑈 ∈ 𝒫 𝐵)
42 ovex 6633 . . . 4 (𝑇 𝑈) ∈ V
4342mptex 6441 . . 3 (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))) ∈ V
4443a1i 11 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))) ∈ V)
4528, 35, 38, 41, 44ovmpt2d 6742 1 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  wrex 2913  Vcvv 3191  wss 3560  𝒫 cpw 4135  cmpt 4678  cfv 5850  crio 6565  (class class class)co 6605  cmpt2 6607  Basecbs 15776  +gcplusg 15857  LSSumclsm 17965  proj1cpj1 17966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-pj1 17968
This theorem is referenced by:  pj1val  18024  pj1f  18026
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