Theorem pj1fval 19624

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 3468	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3	2	3ad2ant1 1133	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝐺 ∈ V)
4		fveq2 6858	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5		pj1fval.v	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
6	4, 5	eqtr4di 2782	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
7	6	pweqd 4580	. . . . . 6 ⊢ (𝑔 = 𝐺 → 𝒫 (Base‘𝑔) = 𝒫 𝐵)
8		fveq2 6858	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (LSSum‘𝑔) = (LSSum‘𝐺))
9		pj1fval.s	. . . . . . . . 9 ⊢ ⊕ = (LSSum‘𝐺)
10	8, 9	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (LSSum‘𝑔) = ⊕ )
11	10	oveqd 7404	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑡(LSSum‘𝑔)𝑢) = (𝑡 ⊕ 𝑢))
12		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
13		pj1fval.a	. . . . . . . . . . . 12 ⊢ + = (+g‘𝐺)
14	12, 13	eqtr4di 2782	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
15	14	oveqd 7404	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
16	15	eqeq2d 2740	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑧 = (𝑥(+g‘𝑔)𝑦) ↔ 𝑧 = (𝑥 + 𝑦)))
17	16	rexbidv 3157	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑔)𝑦) ↔ ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))
18	17	riotabidv 7346	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑔)𝑦)) = (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))
19	11, 18	mpteq12dv 5194	. . . . . 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 7464	. . . . 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 19567	. . . . 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 6872	. . . . . . 7 ⊢ 𝐵 ∈ V
23	22	pwex 5335	. . . . . 6 ⊢ 𝒫 𝐵 ∈ V
24	23, 23	mpoex 8058	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 ⊕ 𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))) ∈ V
25	20, 21, 24	fvmpt 6968	. . . 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 2776	. 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 7396	. . . 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 770	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑡 = 𝑇 ∧ 𝑢 = 𝑈)) → 𝑡 = 𝑇)
31		simprr 772	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑡 = 𝑇 ∧ 𝑢 = 𝑈)) → 𝑢 = 𝑈)
32	31	rexeqdv 3300	. . . 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 7347	. . 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 5194	. 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 1137	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑇 ⊆ 𝐵)
36	22	elpw2 5289	. . 3 ⊢ (𝑇 ∈ 𝒫 𝐵 ↔ 𝑇 ⊆ 𝐵)
37	35, 36	sylibr 234	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑇 ∈ 𝒫 𝐵)
38		simp3 1138	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑈 ⊆ 𝐵)
39	22	elpw2 5289	. . 3 ⊢ (𝑈 ∈ 𝒫 𝐵 ↔ 𝑈 ⊆ 𝐵)
40	38, 39	sylibr 234	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑈 ∈ 𝒫 𝐵)
41		ovex 7420	. . . 4 ⊢ (𝑇 ⊕ 𝑈) ∈ V
42	41	mptex 7197	. . 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 7541	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 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3447   ⊆ wss 3914  𝒫 cpw 4563   ↦ cmpt 5188  ‘cfv 6511  ℩crio 7343  (class class class)co 7387   ∈ cmpo 7389  Basecbs 17179  +_gcplusg 17220  LSSumclsm 19564  proj₁cpj1 19565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-pj1 19567

This theorem is referenced by:  pj1val  19625  pj1f  19627