Theorem pj1fval 19795

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 3479	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3	2	3ad2ant1 1151	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝐺 ∈ V)
4		fveq2 6888	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5		pj1fval.v	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
6	4, 5	eqtr4di 2819	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
7	6	pweqd 4584	. . . . . 6 ⊢ (𝑔 = 𝐺 → 𝒫 (Base‘𝑔) = 𝒫 𝐵)
8		fveq2 6888	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (LSSum‘𝑔) = (LSSum‘𝐺))
9		pj1fval.s	. . . . . . . . 9 ⊢ ⊕ = (LSSum‘𝐺)
10	8, 9	eqtr4di 2819	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (LSSum‘𝑔) = ⊕ )
11	10	oveqd 7440	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑡(LSSum‘𝑔)𝑢) = (𝑡 ⊕ 𝑢))
12		fveq2 6888	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
13		pj1fval.a	. . . . . . . . . . . 12 ⊢ + = (+g‘𝐺)
14	12, 13	eqtr4di 2819	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
15	14	oveqd 7440	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
16	15	eqeq2d 2777	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑧 = (𝑥(+g‘𝑔)𝑦) ↔ 𝑧 = (𝑥 + 𝑦)))
17	16	rexbidv 3192	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑔)𝑦) ↔ ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))
18	17	riotabidv 7382	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑔)𝑦)) = (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))
19	11, 18	mpteq12dv 5203	. . . . . 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 7498	. . . . 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 19738	. . . . 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 6902	. . . . . . 7 ⊢ 𝐵 ∈ V
23	22	pwex 5356	. . . . . 6 ⊢ 𝒫 𝐵 ∈ V
24	23, 23	mpoex 8085	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 ⊕ 𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))) ∈ V
25	20, 21, 24	fvmpt 6996	. . . 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 18	. . 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 2813	. 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 7432	. . . 4 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑡 ⊕ 𝑢) = (𝑇 ⊕ 𝑈))
29	28	adantl 487	. . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑡 = 𝑇 ∧ 𝑢 = 𝑈)) → (𝑡 ⊕ 𝑢) = (𝑇 ⊕ 𝑈))
30		simprl 783	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑡 = 𝑇 ∧ 𝑢 = 𝑈)) → 𝑡 = 𝑇)
31		simprr 785	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑡 = 𝑇 ∧ 𝑢 = 𝑈)) → 𝑢 = 𝑈)
32	31	rexeqdv 3327	. . . 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 7383	. . 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 5203	. 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 1155	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑇 ⊆ 𝐵)
36	22	elpw2 5310	. . 3 ⊢ (𝑇 ∈ 𝒫 𝐵 ↔ 𝑇 ⊆ 𝐵)
37	35, 36	sylibr 237	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑇 ∈ 𝒫 𝐵)
38		simp3 1156	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑈 ⊆ 𝐵)
39	22	elpw2 5310	. . 3 ⊢ (𝑈 ∈ 𝒫 𝐵 ↔ 𝑈 ⊆ 𝐵)
40	38, 39	sylibr 237	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑈 ∈ 𝒫 𝐵)
41		ovex 7456	. . . 4 ⊢ (𝑇 ⊕ 𝑈) ∈ V
42	41	mptex 7228	. . 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 7575	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃wrex 3092  Vcvv 3458   ⊆ wss 3908  𝒫 cpw 4567   ↦ cmpt 5197  ‘cfv 6543  ℩crio 7379  (class class class)co 7423   ∈ cmpo 7425  Basecbs 17294  +_gcplusg 17335  LSSumclsm 19735  proj₁cpj1 19736

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-pj1 19738

This theorem is used by:  pj1val  19796  pj1f  19798