Theorem idl0cl 35799

Description: An ideal contains 0. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
idl0cl.1	⊢ 𝐺 = (1st ‘𝑅)
idl0cl.2	⊢ 𝑍 = (GId‘𝐺)

Assertion

Ref	Expression
idl0cl	⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑍 ∈ 𝐼)

Proof of Theorem idl0cl

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

Step	Hyp	Ref	Expression
1		idl0cl.1	. . . 4 ⊢ 𝐺 = (1st ‘𝑅)
2		eqid 2738	. . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
3		eqid 2738	. . . 4 ⊢ ran 𝐺 = ran 𝐺
4		idl0cl.2	. . . 4 ⊢ 𝑍 = (GId‘𝐺)
5	1, 2, 3, 4	isidl 35795	. . 3 ⊢ (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ ran 𝐺 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼)))))
6	5	biimpa 480	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 ⊆ ran 𝐺 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼))))
7	6	simp2d 1144	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑍 ∈ 𝐼)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∀wral 3053   ⊆ wss 3843  ran crn 5526  ‘cfv 6339  (class class class)co 7170  1^st c1st 7712  2^nd c2nd 7713  GIdcgi 28425  RingOpscrngo 35675  Idlcidl 35788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7173  df-idl 35791

This theorem is referenced by:  divrngidl  35809  intidl  35810  unichnidl  35812  maxidln0  35826