Theorem shintcli 29108

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 488	. . . 4 ⊢ 𝐴 ≠ ∅
3		n0 4312	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
4		intss1 4893	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑧)
5	1	simpli 486	. . . . . . . . 9 ⊢ 𝐴 ⊆ Sℋ
6	5	sseli 3965	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ Sℋ )
7		shss 28989	. . . . . . . 8 ⊢ (𝑧 ∈ Sℋ → 𝑧 ⊆ ℋ)
8	6, 7	syl 17	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ⊆ ℋ)
9	4, 8	sstrd 3979	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → ∩ 𝐴 ⊆ ℋ)
10	9	exlimiv 1931	. . . . 5 ⊢ (∃𝑧 𝑧 ∈ 𝐴 → ∩ 𝐴 ⊆ ℋ)
11	3, 10	sylbi 219	. . . 4 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ℋ)
12	2, 11	ax-mp 5	. . 3 ⊢ ∩ 𝐴 ⊆ ℋ
13		ax-hv0cl 28782	. . . . . 6 ⊢ 0ℎ ∈ ℋ
14	13	elexi 3515	. . . . 5 ⊢ 0ℎ ∈ V
15	14	elint2 4885	. . . 4 ⊢ (0ℎ ∈ ∩ 𝐴 ↔ ∀𝑧 ∈ 𝐴 0ℎ ∈ 𝑧)
16		sh0 28995	. . . . 5 ⊢ (𝑧 ∈ Sℋ → 0ℎ ∈ 𝑧)
17	6, 16	syl 17	. . . 4 ⊢ (𝑧 ∈ 𝐴 → 0ℎ ∈ 𝑧)
18	15, 17	mprgbir 3155	. . 3 ⊢ 0ℎ ∈ ∩ 𝐴
19	12, 18	pm3.2i 473	. 2 ⊢ (∩ 𝐴 ⊆ ℋ ∧ 0ℎ ∈ ∩ 𝐴)
20		elinti 4887	. . . . . . . . 9 ⊢ (𝑥 ∈ ∩ 𝐴 → (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝑧))
21	20	com12 32	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → (𝑥 ∈ ∩ 𝐴 → 𝑥 ∈ 𝑧))
22		elinti 4887	. . . . . . . . 9 ⊢ (𝑦 ∈ ∩ 𝐴 → (𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧))
23	22	com12 32	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → (𝑦 ∈ ∩ 𝐴 → 𝑦 ∈ 𝑧))
24		shaddcl 28996	. . . . . . . . . 10 ⊢ ((𝑧 ∈ Sℋ ∧ 𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧) → (𝑥 +ℎ 𝑦) ∈ 𝑧)
25	6, 24	syl3an1 1159	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧) → (𝑥 +ℎ 𝑦) ∈ 𝑧)
26	25	3expib 1118	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧) → (𝑥 +ℎ 𝑦) ∈ 𝑧))
27	21, 23, 26	syl2and 609	. . . . . . 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 3183	. . . . 5 ⊢ ((𝑥 ∈ ∩ 𝐴 ∧ 𝑦 ∈ ∩ 𝐴) → ∀𝑧 ∈ 𝐴 (𝑥 +ℎ 𝑦) ∈ 𝑧)
30		ovex 7191	. . . . . 6 ⊢ (𝑥 +ℎ 𝑦) ∈ V
31	30	elint2 4885	. . . . 5 ⊢ ((𝑥 +ℎ 𝑦) ∈ ∩ 𝐴 ↔ ∀𝑧 ∈ 𝐴 (𝑥 +ℎ 𝑦) ∈ 𝑧)
32	29, 31	sylibr 236	. . . 4 ⊢ ((𝑥 ∈ ∩ 𝐴 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑥 +ℎ 𝑦) ∈ ∩ 𝐴)
33	32	rgen2 3205	. . 3 ⊢ ∀𝑥 ∈ ∩ 𝐴∀𝑦 ∈ ∩ 𝐴(𝑥 +ℎ 𝑦) ∈ ∩ 𝐴
34		shmulcl 28997	. . . . . . . . . 10 ⊢ ((𝑧 ∈ Sℋ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑧) → (𝑥 ·ℎ 𝑦) ∈ 𝑧)
35	6, 34	syl3an1 1159	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑧) → (𝑥 ·ℎ 𝑦) ∈ 𝑧)
36	35	3expib 1118	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑧) → (𝑥 ·ℎ 𝑦) ∈ 𝑧))
37	23, 36	sylan2d 606	. . . . . . 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 3183	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ∩ 𝐴) → ∀𝑧 ∈ 𝐴 (𝑥 ·ℎ 𝑦) ∈ 𝑧)
40		ovex 7191	. . . . . 6 ⊢ (𝑥 ·ℎ 𝑦) ∈ V
41	40	elint2 4885	. . . . 5 ⊢ ((𝑥 ·ℎ 𝑦) ∈ ∩ 𝐴 ↔ ∀𝑧 ∈ 𝐴 (𝑥 ·ℎ 𝑦) ∈ 𝑧)
42	39, 41	sylibr 236	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ∩ 𝐴) → (𝑥 ·ℎ 𝑦) ∈ ∩ 𝐴)
43	42	rgen2 3205	. . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ∩ 𝐴(𝑥 ·ℎ 𝑦) ∈ ∩ 𝐴
44	33, 43	pm3.2i 473	. 2 ⊢ (∀𝑥 ∈ ∩ 𝐴∀𝑦 ∈ ∩ 𝐴(𝑥 +ℎ 𝑦) ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ∩ 𝐴(𝑥 ·ℎ 𝑦) ∈ ∩ 𝐴)
45		issh2 28988	. 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 709	1 ⊢ ∩ 𝐴 ∈ Sℋ

Syntax hints:   ∧ wa 398  ∃wex 1780   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140   ⊆ wss 3938  ∅c0 4293  ∩ cint 4878  (class class class)co 7158  ℂcc 10537   ℋchba 28698   +_ℎ cva 28699   ·_ℎ csm 28700  0_ℎc0v 28703   S_ℋ csh 28707

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-hilex 28778  ax-hfvadd 28779  ax-hv0cl 28782  ax-hfvmul 28784

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-sh 28986

This theorem is referenced by:  shintcl  29109  chintcli  29110  shincli  29141