maxidln1 - Mathbox for Jeff Madsen

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Theorem maxidln1 37558

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 2728	. . 3 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2728	. . 3 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
3	1, 2	maxidlnr 37556	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ ran (1^st ‘𝑅))
4		maxidlidl 37555	. . 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 37540	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝑀 ↔ 𝑀 = ran (1^st ‘𝑅)))
8	7	necon3bbid 2975	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (¬ 𝑈 ∈ 𝑀 ↔ 𝑀 ≠ ran (1^st ‘𝑅)))
9	4, 8	syldan 589	. 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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ran crn 5683 ‘cfv 6553 1^st c1st 7999 2^nd c2nd 8000 GIdcgi 30328 RingOpscrngo 37408 Idlcidl 37521 MaxIdlcmaxidl 37523

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fo 6559 df-fv 6561 df-riota 7382 df-ov 7429 df-1st 8001 df-2nd 8002 df-grpo 30331 df-gid 30332 df-ablo 30383 df-ass 37357 df-exid 37359 df-mgmOLD 37363 df-sgrOLD 37375 df-mndo 37381 df-rngo 37409 df-idl 37524 df-maxidl 37526

This theorem is referenced by: maxidln0 37559

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