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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rngo1cl | Structured version Visualization version GIF version | ||
| Description: The unity element of a ring belongs to the base set. (Contributed by FL, 12-Feb-2010.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| ring1cl.1 | ⊢ 𝑋 = ran (1st ‘𝑅) | 
| ring1cl.2 | ⊢ 𝐻 = (2nd ‘𝑅) | 
| ring1cl.3 | ⊢ 𝑈 = (GId‘𝐻) | 
| Ref | Expression | 
|---|---|
| rngo1cl | ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ring1cl.2 | . . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅) | |
| 2 | 1 | rngomndo 37943 | . . . . 5 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp) | 
| 3 | 1 | eleq1i 2831 | . . . . . 6 ⊢ (𝐻 ∈ MndOp ↔ (2nd ‘𝑅) ∈ MndOp) | 
| 4 | mndoismgmOLD 37878 | . . . . . . 7 ⊢ ((2nd ‘𝑅) ∈ MndOp → (2nd ‘𝑅) ∈ Magma) | |
| 5 | mndoisexid 37877 | . . . . . . 7 ⊢ ((2nd ‘𝑅) ∈ MndOp → (2nd ‘𝑅) ∈ ExId ) | |
| 6 | 4, 5 | jca 511 | . . . . . 6 ⊢ ((2nd ‘𝑅) ∈ MndOp → ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId )) | 
| 7 | 3, 6 | sylbi 217 | . . . . 5 ⊢ (𝐻 ∈ MndOp → ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId )) | 
| 8 | 2, 7 | syl 17 | . . . 4 ⊢ (𝑅 ∈ RingOps → ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId )) | 
| 9 | elin 3966 | . . . 4 ⊢ ((2nd ‘𝑅) ∈ (Magma ∩ ExId ) ↔ ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId )) | |
| 10 | 8, 9 | sylibr 234 | . . 3 ⊢ (𝑅 ∈ RingOps → (2nd ‘𝑅) ∈ (Magma ∩ ExId )) | 
| 11 | eqid 2736 | . . . 4 ⊢ ran (2nd ‘𝑅) = ran (2nd ‘𝑅) | |
| 12 | ring1cl.3 | . . . . 5 ⊢ 𝑈 = (GId‘𝐻) | |
| 13 | 1 | fveq2i 6908 | . . . . 5 ⊢ (GId‘𝐻) = (GId‘(2nd ‘𝑅)) | 
| 14 | 12, 13 | eqtri 2764 | . . . 4 ⊢ 𝑈 = (GId‘(2nd ‘𝑅)) | 
| 15 | 11, 14 | iorlid 37866 | . . 3 ⊢ ((2nd ‘𝑅) ∈ (Magma ∩ ExId ) → 𝑈 ∈ ran (2nd ‘𝑅)) | 
| 16 | 10, 15 | syl 17 | . 2 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ ran (2nd ‘𝑅)) | 
| 17 | ring1cl.1 | . . 3 ⊢ 𝑋 = ran (1st ‘𝑅) | |
| 18 | eqid 2736 | . . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
| 19 | eqid 2736 | . . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
| 20 | 18, 19 | rngorn1eq 37942 | . . 3 ⊢ (𝑅 ∈ RingOps → ran (1st ‘𝑅) = ran (2nd ‘𝑅)) | 
| 21 | eqtr 2759 | . . . 4 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran (2nd ‘𝑅)) → 𝑋 = ran (2nd ‘𝑅)) | |
| 22 | 21 | eleq2d 2826 | . . 3 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran (2nd ‘𝑅)) → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ ran (2nd ‘𝑅))) | 
| 23 | 17, 20, 22 | sylancr 587 | . 2 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ ran (2nd ‘𝑅))) | 
| 24 | 16, 23 | mpbird 257 | 1 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∩ cin 3949 ran crn 5685 ‘cfv 6560 1st c1st 8013 2nd c2nd 8014 GIdcgi 30510 ExId cexid 37852 Magmacmagm 37856 MndOpcmndo 37874 RingOpscrngo 37902 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fo 6566 df-fv 6568 df-riota 7389 df-ov 7435 df-1st 8015 df-2nd 8016 df-grpo 30513 df-gid 30514 df-ablo 30565 df-ass 37851 df-exid 37853 df-mgmOLD 37857 df-sgrOLD 37869 df-mndo 37875 df-rngo 37903 | 
| This theorem is referenced by: rngoueqz 37948 rngonegmn1l 37949 rngonegmn1r 37950 rngoneglmul 37951 rngonegrmul 37952 isdrngo2 37966 rngohomco 37982 rngoisocnv 37989 idlnegcl 38030 1idl 38034 0rngo 38035 smprngopr 38060 prnc 38075 isfldidl 38076 isdmn3 38082 | 
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