Theorem pj1fval 18107

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 3212	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3	2	3ad2ant1 1082	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝐺 ∈ V)
4		fveq2 6191	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5		pj1fval.v	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
6	4, 5	syl6eqr 2674	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
7	6	pweqd 4163	. . . . . 6 ⊢ (𝑔 = 𝐺 → 𝒫 (Base‘𝑔) = 𝒫 𝐵)
8		fveq2 6191	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (LSSum‘𝑔) = (LSSum‘𝐺))
9		pj1fval.s	. . . . . . . . 9 ⊢ ⊕ = (LSSum‘𝐺)
10	8, 9	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (LSSum‘𝑔) = ⊕ )
11	10	oveqd 6667	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑡(LSSum‘𝑔)𝑢) = (𝑡 ⊕ 𝑢))
12		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
13		pj1fval.a	. . . . . . . . . . . 12 ⊢ + = (+g‘𝐺)
14	12, 13	syl6eqr 2674	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
15	14	oveqd 6667	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
16	15	eqeq2d 2632	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑧 = (𝑥(+g‘𝑔)𝑦) ↔ 𝑧 = (𝑥 + 𝑦)))
17	16	rexbidv 3052	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑔)𝑦) ↔ ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))
18	17	riotabidv 6613	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑔)𝑦)) = (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))
19	11, 18	mpteq12dv 4733	. . . . . 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	mpt2eq123dv 6717	. . . . 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 18052	. . . . 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		fvex 6201	. . . . . . . 8 ⊢ (Base‘𝐺) ∈ V
23	5, 22	eqeltri 2697	. . . . . . 7 ⊢ 𝐵 ∈ V
24	23	pwex 4848	. . . . . 6 ⊢ 𝒫 𝐵 ∈ V
25	24, 24	mpt2ex 7247	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 ⊕ 𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))) ∈ V
26	20, 21, 25	fvmpt 6282	. . . 4 ⊢ (𝐺 ∈ V → (proj1‘𝐺) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 ⊕ 𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))))
27	3, 26	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)))))
28	1, 27	syl5eq 2668	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑃 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ (𝑧 ∈ (𝑡 ⊕ 𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦)))))
29		oveq12 6659	. . . 4 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑡 ⊕ 𝑢) = (𝑇 ⊕ 𝑈))
30	29	adantl 482	. . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑡 = 𝑇 ∧ 𝑢 = 𝑈)) → (𝑡 ⊕ 𝑢) = (𝑇 ⊕ 𝑈))
31		simprl 794	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑡 = 𝑇 ∧ 𝑢 = 𝑈)) → 𝑡 = 𝑇)
32		simprr 796	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑡 = 𝑇 ∧ 𝑢 = 𝑈)) → 𝑢 = 𝑈)
33	32	rexeqdv 3145	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑡 = 𝑇 ∧ 𝑢 = 𝑈)) → (∃𝑦 ∈ 𝑢 𝑧 = (𝑥 + 𝑦) ↔ ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦)))
34	31, 33	riotaeqbidv 6614	. . 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)))
35	30, 34	mpteq12dv 4733	. 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))))
36		simp2 1062	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑇 ⊆ 𝐵)
37	23	elpw2 4828	. . 3 ⊢ (𝑇 ∈ 𝒫 𝐵 ↔ 𝑇 ⊆ 𝐵)
38	36, 37	sylibr 224	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑇 ∈ 𝒫 𝐵)
39		simp3 1063	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑈 ⊆ 𝐵)
40	23	elpw2 4828	. . 3 ⊢ (𝑈 ∈ 𝒫 𝐵 ↔ 𝑈 ⊆ 𝐵)
41	39, 40	sylibr 224	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → 𝑈 ∈ 𝒫 𝐵)
42		ovex 6678	. . . 4 ⊢ (𝑇 ⊕ 𝑈) ∈ V
43	42	mptex 6486	. . 3 ⊢ (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))) ∈ V
44	43	a1i 11	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))) ∈ V)
45	28, 35, 38, 41, 44	ovmpt2d 6788	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  𝒫 cpw 4158   ↦ cmpt 4729  ‘cfv 5888  ℩crio 6610  (class class class)co 6650   ↦ cmpt2 6652  Basecbs 15857  +_gcplusg 15941  LSSumclsm 18049  proj₁cpj1 18050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-pj1 18052

This theorem is referenced by:  pj1val  18108  pj1f  18110