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Theorem pj1fval 19687
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 3481 . . . . 5 (𝐺𝑉𝐺 ∈ V)
323ad2ant1 1130 . . . 4 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝐺 ∈ V)
4 fveq2 6900 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5 pj1fval.v . . . . . . . 8 𝐵 = (Base‘𝐺)
64, 5eqtr4di 2783 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
76pweqd 4623 . . . . . 6 (𝑔 = 𝐺 → 𝒫 (Base‘𝑔) = 𝒫 𝐵)
8 fveq2 6900 . . . . . . . . 9 (𝑔 = 𝐺 → (LSSum‘𝑔) = (LSSum‘𝐺))
9 pj1fval.s . . . . . . . . 9 = (LSSum‘𝐺)
108, 9eqtr4di 2783 . . . . . . . 8 (𝑔 = 𝐺 → (LSSum‘𝑔) = )
1110oveqd 7440 . . . . . . 7 (𝑔 = 𝐺 → (𝑡(LSSum‘𝑔)𝑢) = (𝑡 𝑢))
12 fveq2 6900 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
13 pj1fval.a . . . . . . . . . . . 12 + = (+g𝐺)
1412, 13eqtr4di 2783 . . . . . . . . . . 11 (𝑔 = 𝐺 → (+g𝑔) = + )
1514oveqd 7440 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑦) = (𝑥 + 𝑦))
1615eqeq2d 2736 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑧 = (𝑥(+g𝑔)𝑦) ↔ 𝑧 = (𝑥 + 𝑦)))
1716rexbidv 3168 . . . . . . . 8 (𝑔 = 𝐺 → (∃𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦) ↔ ∃𝑦𝑢 𝑧 = (𝑥 + 𝑦)))
1817riotabidv 7381 . . . . . . 7 (𝑔 = 𝐺 → (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦)) = (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))
1911, 18mpteq12dv 5243 . . . . . 6 (𝑔 = 𝐺 → (𝑧 ∈ (𝑡(LSSum‘𝑔)𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦))) = (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦))))
207, 7, 19mpoeq123dv 7499 . . . . 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 19630 . . . . 5 proj1 = (𝑔 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑔), 𝑢 ∈ 𝒫 (Base‘𝑔) ↦ (𝑧 ∈ (𝑡(LSSum‘𝑔)𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑔)𝑦)))))
225fvexi 6914 . . . . . . 7 𝐵 ∈ V
2322pwex 5383 . . . . . 6 𝒫 𝐵 ∈ V
2423, 23mpoex 8092 . . . . 5 (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))) ∈ V
2520, 21, 24fvmpt 7008 . . . 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, 26eqtrid 2777 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑃 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)))))
28 oveq12 7432 . . . 4 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑡 𝑢) = (𝑇 𝑈))
2928adantl 480 . . 3 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (𝑡 𝑢) = (𝑇 𝑈))
30 simprl 769 . . . 4 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → 𝑡 = 𝑇)
31 simprr 771 . . . . 5 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → 𝑢 = 𝑈)
3231rexeqdv 3315 . . . 4 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (∃𝑦𝑢 𝑧 = (𝑥 + 𝑦) ↔ ∃𝑦𝑈 𝑧 = (𝑥 + 𝑦)))
3330, 32riotaeqbidv 7382 . . 3 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦)) = (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦)))
3429, 33mpteq12dv 5243 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑡 = 𝑇𝑢 = 𝑈)) → (𝑧 ∈ (𝑡 𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥 + 𝑦))) = (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))))
35 simp2 1134 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑇𝐵)
3622elpw2 5351 . . 3 (𝑇 ∈ 𝒫 𝐵𝑇𝐵)
3735, 36sylibr 233 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑇 ∈ 𝒫 𝐵)
38 simp3 1135 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑈𝐵)
3922elpw2 5351 . . 3 (𝑈 ∈ 𝒫 𝐵𝑈𝐵)
4038, 39sylibr 233 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → 𝑈 ∈ 𝒫 𝐵)
41 ovex 7456 . . . 4 (𝑇 𝑈) ∈ V
4241mptex 7239 . . 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 7577 1 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wrex 3059  Vcvv 3461  wss 3946  𝒫 cpw 4606  cmpt 5235  cfv 6553  crio 7378  (class class class)co 7423  cmpo 7425  Basecbs 17208  +gcplusg 17261  LSSumclsm 19627  proj1cpj1 19628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 8002  df-2nd 8003  df-pj1 19630
This theorem is referenced by:  pj1val  19688  pj1f  19690
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