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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngo1cl | Structured version Visualization version GIF version |
Description: The unit 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 32904 | . . . . 5 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp) |
3 | 1 | eleq1i 2679 | . . . . . 6 ⊢ (𝐻 ∈ MndOp ↔ (2nd ‘𝑅) ∈ MndOp) |
4 | mndoismgmOLD 32839 | . . . . . . 7 ⊢ ((2nd ‘𝑅) ∈ MndOp → (2nd ‘𝑅) ∈ Magma) | |
5 | mndoisexid 32838 | . . . . . . 7 ⊢ ((2nd ‘𝑅) ∈ MndOp → (2nd ‘𝑅) ∈ ExId ) | |
6 | 4, 5 | jca 553 | . . . . . 6 ⊢ ((2nd ‘𝑅) ∈ MndOp → ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId )) |
7 | 3, 6 | sylbi 206 | . . . . 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 3758 | . . . 4 ⊢ ((2nd ‘𝑅) ∈ (Magma ∩ ExId ) ↔ ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId )) | |
10 | 8, 9 | sylibr 223 | . . 3 ⊢ (𝑅 ∈ RingOps → (2nd ‘𝑅) ∈ (Magma ∩ ExId )) |
11 | eqid 2610 | . . . 4 ⊢ ran (2nd ‘𝑅) = ran (2nd ‘𝑅) | |
12 | ring1cl.3 | . . . . 5 ⊢ 𝑈 = (GId‘𝐻) | |
13 | 1 | fveq2i 6106 | . . . . 5 ⊢ (GId‘𝐻) = (GId‘(2nd ‘𝑅)) |
14 | 12, 13 | eqtri 2632 | . . . 4 ⊢ 𝑈 = (GId‘(2nd ‘𝑅)) |
15 | 11, 14 | iorlid 32827 | . . 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 2610 | . . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
19 | eqid 2610 | . . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
20 | 18, 19 | rngorn1eq 32903 | . . 3 ⊢ (𝑅 ∈ RingOps → ran (1st ‘𝑅) = ran (2nd ‘𝑅)) |
21 | eqtr 2629 | . . . 4 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran (2nd ‘𝑅)) → 𝑋 = ran (2nd ‘𝑅)) | |
22 | 21 | eleq2d 2673 | . . 3 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran (2nd ‘𝑅)) → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ ran (2nd ‘𝑅))) |
23 | 17, 20, 22 | sylancr 694 | . 2 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ ran (2nd ‘𝑅))) |
24 | 16, 23 | mpbird 246 | 1 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ran crn 5039 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 GIdcgi 26728 ExId cexid 32813 Magmacmagm 32817 MndOpcmndo 32835 RingOpscrngo 32863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-riota 6511 df-ov 6552 df-1st 7059 df-2nd 7060 df-grpo 26731 df-gid 26732 df-ablo 26783 df-ass 32812 df-exid 32814 df-mgmOLD 32818 df-sgrOLD 32830 df-mndo 32836 df-rngo 32864 |
This theorem is referenced by: rngoueqz 32909 rngonegmn1l 32910 rngonegmn1r 32911 rngoneglmul 32912 rngonegrmul 32913 isdrngo2 32927 rngohomco 32943 rngoisocnv 32950 idlnegcl 32991 1idl 32995 0rngo 32996 smprngopr 33021 prnc 33036 isfldidl 33037 isdmn3 33043 |
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