Theorem pj1val 19661

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
pj1val	⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑇,𝑦   𝑥,𝑈,𝑦   𝑥, ⊕ ,𝑦   𝑥,𝐺,𝑦   𝑥,𝑉,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝑃(𝑥,𝑦)   + (𝑥,𝑦)

Proof of Theorem pj1val

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pj1fval.v	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		pj1fval.a	. . . 4 ⊢ + = (+g‘𝐺)
3		pj1fval.s	. . . 4 ⊢ ⊕ = (LSSum‘𝐺)
4		pj1fval.p	. . . 4 ⊢ 𝑃 = (proj1‘𝐺)
5	1, 2, 3, 4	pj1fval 19660	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))))
6	5	adantr 480	. 2 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))))
7		simpr 484	. . . . 5 ⊢ ((((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ 𝑧 = 𝑋) → 𝑧 = 𝑋)
8	7	eqeq1d 2739	. . . 4 ⊢ ((((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ 𝑧 = 𝑋) → (𝑧 = (𝑥 + 𝑦) ↔ 𝑋 = (𝑥 + 𝑦)))
9	8	rexbidv 3162	. . 3 ⊢ ((((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ 𝑧 = 𝑋) → (∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦) ↔ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)))
10	9	riotabidv 7319	. 2 ⊢ ((((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ 𝑧 = 𝑋) → (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦)) = (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)))
11		simpr 484	. 2 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → 𝑋 ∈ (𝑇 ⊕ 𝑈))
12		riotaex 7321	. . 3 ⊢ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)) ∈ V
13	12	a1i 11	. 2 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)) ∈ V)
14	6, 10, 11, 13	fvmptd 6949	1 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890   ↦ cmpt 5167  ‘cfv 6492  ℩crio 7316  (class class class)co 7360  Basecbs 17170  +_gcplusg 17211  LSSumclsm 19600  proj₁cpj1 19601

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 5302  ax-pr 5370  ax-un 7682

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 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-pj1 19603

This theorem is referenced by:  pj1id  19665