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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 35217 | . . . . 5 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp) |
3 | 1 | eleq1i 2906 | . . . . . 6 ⊢ (𝐻 ∈ MndOp ↔ (2nd ‘𝑅) ∈ MndOp) |
4 | mndoismgmOLD 35152 | . . . . . . 7 ⊢ ((2nd ‘𝑅) ∈ MndOp → (2nd ‘𝑅) ∈ Magma) | |
5 | mndoisexid 35151 | . . . . . . 7 ⊢ ((2nd ‘𝑅) ∈ MndOp → (2nd ‘𝑅) ∈ ExId ) | |
6 | 4, 5 | jca 514 | . . . . . 6 ⊢ ((2nd ‘𝑅) ∈ MndOp → ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId )) |
7 | 3, 6 | sylbi 219 | . . . . 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 4172 | . . . 4 ⊢ ((2nd ‘𝑅) ∈ (Magma ∩ ExId ) ↔ ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId )) | |
10 | 8, 9 | sylibr 236 | . . 3 ⊢ (𝑅 ∈ RingOps → (2nd ‘𝑅) ∈ (Magma ∩ ExId )) |
11 | eqid 2824 | . . . 4 ⊢ ran (2nd ‘𝑅) = ran (2nd ‘𝑅) | |
12 | ring1cl.3 | . . . . 5 ⊢ 𝑈 = (GId‘𝐻) | |
13 | 1 | fveq2i 6676 | . . . . 5 ⊢ (GId‘𝐻) = (GId‘(2nd ‘𝑅)) |
14 | 12, 13 | eqtri 2847 | . . . 4 ⊢ 𝑈 = (GId‘(2nd ‘𝑅)) |
15 | 11, 14 | iorlid 35140 | . . 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 2824 | . . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
19 | eqid 2824 | . . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
20 | 18, 19 | rngorn1eq 35216 | . . 3 ⊢ (𝑅 ∈ RingOps → ran (1st ‘𝑅) = ran (2nd ‘𝑅)) |
21 | eqtr 2844 | . . . 4 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran (2nd ‘𝑅)) → 𝑋 = ran (2nd ‘𝑅)) | |
22 | 21 | eleq2d 2901 | . . 3 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran (2nd ‘𝑅)) → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ ran (2nd ‘𝑅))) |
23 | 17, 20, 22 | sylancr 589 | . 2 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ ran (2nd ‘𝑅))) |
24 | 16, 23 | mpbird 259 | 1 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∩ cin 3938 ran crn 5559 ‘cfv 6358 1st c1st 7690 2nd c2nd 7691 GIdcgi 28270 ExId cexid 35126 Magmacmagm 35130 MndOpcmndo 35148 RingOpscrngo 35176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 df-riota 7117 df-ov 7162 df-1st 7692 df-2nd 7693 df-grpo 28273 df-gid 28274 df-ablo 28325 df-ass 35125 df-exid 35127 df-mgmOLD 35131 df-sgrOLD 35143 df-mndo 35149 df-rngo 35177 |
This theorem is referenced by: rngoueqz 35222 rngonegmn1l 35223 rngonegmn1r 35224 rngoneglmul 35225 rngonegrmul 35226 isdrngo2 35240 rngohomco 35256 rngoisocnv 35263 idlnegcl 35304 1idl 35308 0rngo 35309 smprngopr 35334 prnc 35349 isfldidl 35350 isdmn3 35356 |
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