Theorem shintcli 29979

Description: Closure of intersection of a nonempty subset of S_ℋ. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

Hypothesis

Ref	Expression
shintcl.1	⊢ (𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅)

Assertion

Ref	Expression
shintcli	⊢ ∩ 𝐴 ∈ Sℋ

Proof of Theorem shintcli

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		shintcl.1	. . . . 5 ⊢ (𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅)
2	1	simpri 486	. . . 4 ⊢ 𝐴 ≠ ∅
3		n0 4293	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
4		intss1 4911	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑧)
5	1	simpli 484	. . . . . . . . 9 ⊢ 𝐴 ⊆ Sℋ
6	5	sseli 3928	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ Sℋ )
7		shss 29860	. . . . . . . 8 ⊢ (𝑧 ∈ Sℋ → 𝑧 ⊆ ℋ)
8	6, 7	syl 17	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ⊆ ℋ)
9	4, 8	sstrd 3942	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → ∩ 𝐴 ⊆ ℋ)
10	9	exlimiv 1932	. . . . 5 ⊢ (∃𝑧 𝑧 ∈ 𝐴 → ∩ 𝐴 ⊆ ℋ)
11	3, 10	sylbi 216	. . . 4 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ℋ)
12	2, 11	ax-mp 5	. . 3 ⊢ ∩ 𝐴 ⊆ ℋ
13		ax-hv0cl 29653	. . . . . 6 ⊢ 0ℎ ∈ ℋ
14	13	elexi 3460	. . . . 5 ⊢ 0ℎ ∈ V
15	14	elint2 4901	. . . 4 ⊢ (0ℎ ∈ ∩ 𝐴 ↔ ∀𝑧 ∈ 𝐴 0ℎ ∈ 𝑧)
16		sh0 29866	. . . . 5 ⊢ (𝑧 ∈ Sℋ → 0ℎ ∈ 𝑧)
17	6, 16	syl 17	. . . 4 ⊢ (𝑧 ∈ 𝐴 → 0ℎ ∈ 𝑧)
18	15, 17	mprgbir 3068	. . 3 ⊢ 0ℎ ∈ ∩ 𝐴
19	12, 18	pm3.2i 471	. 2 ⊢ (∩ 𝐴 ⊆ ℋ ∧ 0ℎ ∈ ∩ 𝐴)
20		elinti 4903	. . . . . . . . 9 ⊢ (𝑥 ∈ ∩ 𝐴 → (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝑧))
21	20	com12 32	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → (𝑥 ∈ ∩ 𝐴 → 𝑥 ∈ 𝑧))
22		elinti 4903	. . . . . . . . 9 ⊢ (𝑦 ∈ ∩ 𝐴 → (𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧))
23	22	com12 32	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → (𝑦 ∈ ∩ 𝐴 → 𝑦 ∈ 𝑧))
24		shaddcl 29867	. . . . . . . . . 10 ⊢ ((𝑧 ∈ Sℋ ∧ 𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧) → (𝑥 +ℎ 𝑦) ∈ 𝑧)
25	6, 24	syl3an1 1162	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧) → (𝑥 +ℎ 𝑦) ∈ 𝑧)
26	25	3expib 1121	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧) → (𝑥 +ℎ 𝑦) ∈ 𝑧))
27	21, 23, 26	syl2and 608	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ ∩ 𝐴 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑥 +ℎ 𝑦) ∈ 𝑧))
28	27	com12 32	. . . . . 6 ⊢ ((𝑥 ∈ ∩ 𝐴 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑧 ∈ 𝐴 → (𝑥 +ℎ 𝑦) ∈ 𝑧))
29	28	ralrimiv 3138	. . . . 5 ⊢ ((𝑥 ∈ ∩ 𝐴 ∧ 𝑦 ∈ ∩ 𝐴) → ∀𝑧 ∈ 𝐴 (𝑥 +ℎ 𝑦) ∈ 𝑧)
30		ovex 7370	. . . . . 6 ⊢ (𝑥 +ℎ 𝑦) ∈ V
31	30	elint2 4901	. . . . 5 ⊢ ((𝑥 +ℎ 𝑦) ∈ ∩ 𝐴 ↔ ∀𝑧 ∈ 𝐴 (𝑥 +ℎ 𝑦) ∈ 𝑧)
32	29, 31	sylibr 233	. . . 4 ⊢ ((𝑥 ∈ ∩ 𝐴 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑥 +ℎ 𝑦) ∈ ∩ 𝐴)
33	32	rgen2 3190	. . 3 ⊢ ∀𝑥 ∈ ∩ 𝐴∀𝑦 ∈ ∩ 𝐴(𝑥 +ℎ 𝑦) ∈ ∩ 𝐴
34		shmulcl 29868	. . . . . . . . . 10 ⊢ ((𝑧 ∈ Sℋ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑧) → (𝑥 ·ℎ 𝑦) ∈ 𝑧)
35	6, 34	syl3an1 1162	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑧) → (𝑥 ·ℎ 𝑦) ∈ 𝑧)
36	35	3expib 1121	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑧) → (𝑥 ·ℎ 𝑦) ∈ 𝑧))
37	23, 36	sylan2d 605	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ∩ 𝐴) → (𝑥 ·ℎ 𝑦) ∈ 𝑧))
38	37	com12 32	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ∩ 𝐴) → (𝑧 ∈ 𝐴 → (𝑥 ·ℎ 𝑦) ∈ 𝑧))
39	38	ralrimiv 3138	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ∩ 𝐴) → ∀𝑧 ∈ 𝐴 (𝑥 ·ℎ 𝑦) ∈ 𝑧)
40		ovex 7370	. . . . . 6 ⊢ (𝑥 ·ℎ 𝑦) ∈ V
41	40	elint2 4901	. . . . 5 ⊢ ((𝑥 ·ℎ 𝑦) ∈ ∩ 𝐴 ↔ ∀𝑧 ∈ 𝐴 (𝑥 ·ℎ 𝑦) ∈ 𝑧)
42	39, 41	sylibr 233	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ∩ 𝐴) → (𝑥 ·ℎ 𝑦) ∈ ∩ 𝐴)
43	42	rgen2 3190	. . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ∩ 𝐴(𝑥 ·ℎ 𝑦) ∈ ∩ 𝐴
44	33, 43	pm3.2i 471	. 2 ⊢ (∀𝑥 ∈ ∩ 𝐴∀𝑦 ∈ ∩ 𝐴(𝑥 +ℎ 𝑦) ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ∩ 𝐴(𝑥 ·ℎ 𝑦) ∈ ∩ 𝐴)
45		issh2 29859	. 2 ⊢ (∩ 𝐴 ∈ Sℋ ↔ ((∩ 𝐴 ⊆ ℋ ∧ 0ℎ ∈ ∩ 𝐴) ∧ (∀𝑥 ∈ ∩ 𝐴∀𝑦 ∈ ∩ 𝐴(𝑥 +ℎ 𝑦) ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ∩ 𝐴(𝑥 ·ℎ 𝑦) ∈ ∩ 𝐴)))
46	19, 44, 45	mpbir2an 708	1 ⊢ ∩ 𝐴 ∈ Sℋ

Syntax hints:   ∧ wa 396  ∃wex 1780   ∈ wcel 2105   ≠ wne 2940  ∀wral 3061   ⊆ wss 3898  ∅c0 4269  ∩ cint 4894  (class class class)co 7337  ℂcc 10970   ℋchba 29569   +_ℎ cva 29570   ·_ℎ csm 29571  0_ℎc0v 29574   S_ℋ csh 29578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372  ax-hilex 29649  ax-hfvadd 29650  ax-hv0cl 29653  ax-hfvmul 29655

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-fv 6487  df-ov 7340  df-sh 29857

This theorem is referenced by:  shintcl  29980  chintcli  29981  shincli  30012