maxidln1 - Mathbox for Jeff Madsen

	Mathbox for Jeff Madsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > maxidln1		Structured version Visualization version GIF version

Theorem maxidln1 36717

Description: One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.)

**Hypotheses**
Ref	Expression
maxidln1.1	⊢ 𝐻 = (2^nd ‘𝑅)
maxidln1.2	⊢ 𝑈 = (GId‘𝐻)

**Assertion**
Ref	Expression
maxidln1	⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀)

**Proof of Theorem maxidln1**
Step	Hyp	Ref	Expression
1		eqid 2731	. . 3 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2731	. . 3 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
3	1, 2	maxidlnr 36715	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ ran (1^st ‘𝑅))
4		maxidlidl 36714	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))
5		maxidln1.1	. . . . 5 ⊢ 𝐻 = (2^nd ‘𝑅)
6		maxidln1.2	. . . . 5 ⊢ 𝑈 = (GId‘𝐻)
7	1, 5, 2, 6	1idl 36699	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝑀 ↔ 𝑀 = ran (1^st ‘𝑅)))
8	7	necon3bbid 2977	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (¬ 𝑈 ∈ 𝑀 ↔ 𝑀 ≠ ran (1^st ‘𝑅)))
9	4, 8	syldan 591	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → (¬ 𝑈 ∈ 𝑀 ↔ 𝑀 ≠ ran (1^st ‘𝑅)))
10	3, 9	mpbird 256	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ran crn 5670 ‘cfv 6532 1^st c1st 7955 2^nd c2nd 7956 GIdcgi 29606 RingOpscrngo 36567 Idlcidl 36680 MaxIdlcmaxidl 36682

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-fo 6538 df-fv 6540 df-riota 7349 df-ov 7396 df-1st 7957 df-2nd 7958 df-grpo 29609 df-gid 29610 df-ablo 29661 df-ass 36516 df-exid 36518 df-mgmOLD 36522 df-sgrOLD 36534 df-mndo 36540 df-rngo 36568 df-idl 36683 df-maxidl 36685

This theorem is referenced by: maxidln0 36718

W3C validator