Theorem pj1fval 19658

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	⊢ + = (+g‘𝐺)
pj1fval.s	⊢ ⊕ = (LSSum‘𝐺)
pj1fval.p	⊢ 𝑃 = (proj1‘𝐺)

Assertion

Ref	Expression
pj1fval	⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 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.

Step	Hyp	Ref	Expression
1		pj1fval.p	. . 3 ⊢ 𝑃 = (proj1‘𝐺)
2		elex 3451	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3	2	3ad2ant1 1134	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝐺 ∈ V)
4		fveq2 6832	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5		pj1fval.v	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
6	4, 5	eqtr4di 2790	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
7	6	pweqd 4559	. . . . . 6 ⊢ (𝑔 = 𝐺 → 𝒫 (Base‘𝑔) = 𝒫 𝐵)
8		fveq2 6832	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (LSSum‘𝑔) = (LSSum‘𝐺))
9		pj1fval.s	. . . . . . . . 9 ⊢ ⊕ = (LSSum‘𝐺)
10	8, 9	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (LSSum‘𝑔) = ⊕ )
11	10	oveqd 7375	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑡(LSSum‘𝑔)𝑢) = (𝑡 ⊕ 𝑢))
12		fveq2 6832	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
13		pj1fval.a	. . . . . . . . . . . 12 ⊢ + = (+g‘𝐺)
14	12, 13	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
15	14	oveqd 7375	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
16	15	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑧 = (𝑥(+g‘𝑔)𝑦) ↔ 𝑧 = (𝑥 + 𝑦)))
17	16	rexbidv 3162	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑔)𝑦) ↔ ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))
18	17	riotabidv 7317	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑔)𝑦)) = (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))
19	11, 18	mpteq12dv 5173	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑧 ∈ (𝑡(LSSum‘𝑔)𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑔)𝑦))) = (𝑧 ∈ (𝑡 ⊕ 𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦))))
20	7, 7, 19	mpoeq123dv 7433	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑡 ∈ 𝒫 (Base‘𝑔), 𝑢 ∈ 𝒫 (Base‘𝑔) ↦ (𝑧 ∈ (𝑡(LSSum‘𝑔)𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑔)𝑦)))) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 ⊕ 𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))))
21		df-pj1 19601	. . . . 5 ⊢ proj1 = (𝑔 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑔), 𝑢 ∈ 𝒫 (Base‘𝑔) ↦ (𝑧 ∈ (𝑡(LSSum‘𝑔)𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑔)𝑦)))))
22	5	fvexi 6846	. . . . . . 7 ⊢ 𝐵 ∈ V
23	22	pwex 5315	. . . . . 6 ⊢ 𝒫 𝐵 ∈ V
24	23, 23	mpoex 8023	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 ⊕ 𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))) ∈ V
25	20, 21, 24	fvmpt 6939	. . . 4 ⊢ (𝐺 ∈ V → (proj1‘𝐺) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 ⊕ 𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))))
26	3, 25	syl 17	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (proj1‘𝐺) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 ⊕ 𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))))
27	1, 26	eqtrid 2784	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑃 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 ⊕ 𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))))
28		oveq12 7367	. . . 4 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑡 ⊕ 𝑢) = (𝑇 ⊕ 𝑈))
29	28	adantl 481	. . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑡 = 𝑇 ∧ 𝑢 = 𝑈)) → (𝑡 ⊕ 𝑢) = (𝑇 ⊕ 𝑈))
30		simprl 771	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑡 = 𝑇 ∧ 𝑢 = 𝑈)) → 𝑡 = 𝑇)
31		simprr 773	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑡 = 𝑇 ∧ 𝑢 = 𝑈)) → 𝑢 = 𝑈)
32	31	rexeqdv 3297	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑡 = 𝑇 ∧ 𝑢 = 𝑈)) → (∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦) ↔ ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦)))
33	30, 32	riotaeqbidv 7318	. . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑡 = 𝑇 ∧ 𝑢 = 𝑈)) → (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)) = (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦)))
34	29, 33	mpteq12dv 5173	. 2 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑡 = 𝑇 ∧ 𝑢 = 𝑈)) → (𝑧 ∈ (𝑡 ⊕ 𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦))) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))))
35		simp2 1138	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑇 ⊆ 𝐵)
36	22	elpw2 5269	. . 3 ⊢ (𝑇 ∈ 𝒫 𝐵 ↔ 𝑇 ⊆ 𝐵)
37	35, 36	sylibr 234	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑇 ∈ 𝒫 𝐵)
38		simp3 1139	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑈 ⊆ 𝐵)
39	22	elpw2 5269	. . 3 ⊢ (𝑈 ∈ 𝒫 𝐵 ↔ 𝑈 ⊆ 𝐵)
40	38, 39	sylibr 234	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑈 ∈ 𝒫 𝐵)
41		ovex 7391	. . . 4 ⊢ (𝑇 ⊕ 𝑈) ∈ V
42	41	mptex 7169	. . 3 ⊢ (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))) ∈ V
43	42	a1i 11	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))) ∈ V)
44	27, 34, 37, 40, 43	ovmpod 7510	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890  𝒫 cpw 4542   ↦ cmpt 5167  ‘cfv 6490  ℩crio 7314  (class class class)co 7358   ∈ cmpo 7360  Basecbs 17168  +_gcplusg 17209  LSSumclsm 19598  proj₁cpj1 19599

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-pj1 19601

This theorem is referenced by:  pj1val  19659  pj1f  19661