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Mathbox for Jeff Madsen |
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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 36444 | . . . . 5 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp) |
3 | 1 | eleq1i 2825 | . . . . . 6 ⊢ (𝐻 ∈ MndOp ↔ (2nd ‘𝑅) ∈ MndOp) |
4 | mndoismgmOLD 36379 | . . . . . . 7 ⊢ ((2nd ‘𝑅) ∈ MndOp → (2nd ‘𝑅) ∈ Magma) | |
5 | mndoisexid 36378 | . . . . . . 7 ⊢ ((2nd ‘𝑅) ∈ MndOp → (2nd ‘𝑅) ∈ ExId ) | |
6 | 4, 5 | jca 513 | . . . . . 6 ⊢ ((2nd ‘𝑅) ∈ MndOp → ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId )) |
7 | 3, 6 | sylbi 216 | . . . . 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 3930 | . . . 4 ⊢ ((2nd ‘𝑅) ∈ (Magma ∩ ExId ) ↔ ((2nd ‘𝑅) ∈ Magma ∧ (2nd ‘𝑅) ∈ ExId )) | |
10 | 8, 9 | sylibr 233 | . . 3 ⊢ (𝑅 ∈ RingOps → (2nd ‘𝑅) ∈ (Magma ∩ ExId )) |
11 | eqid 2733 | . . . 4 ⊢ ran (2nd ‘𝑅) = ran (2nd ‘𝑅) | |
12 | ring1cl.3 | . . . . 5 ⊢ 𝑈 = (GId‘𝐻) | |
13 | 1 | fveq2i 6849 | . . . . 5 ⊢ (GId‘𝐻) = (GId‘(2nd ‘𝑅)) |
14 | 12, 13 | eqtri 2761 | . . . 4 ⊢ 𝑈 = (GId‘(2nd ‘𝑅)) |
15 | 11, 14 | iorlid 36367 | . . 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 2733 | . . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
19 | eqid 2733 | . . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
20 | 18, 19 | rngorn1eq 36443 | . . 3 ⊢ (𝑅 ∈ RingOps → ran (1st ‘𝑅) = ran (2nd ‘𝑅)) |
21 | eqtr 2756 | . . . 4 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran (2nd ‘𝑅)) → 𝑋 = ran (2nd ‘𝑅)) | |
22 | 21 | eleq2d 2820 | . . 3 ⊢ ((𝑋 = ran (1st ‘𝑅) ∧ ran (1st ‘𝑅) = ran (2nd ‘𝑅)) → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ ran (2nd ‘𝑅))) |
23 | 17, 20, 22 | sylancr 588 | . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3913 ran crn 5638 ‘cfv 6500 1st c1st 7923 2nd c2nd 7924 GIdcgi 29481 ExId cexid 36353 Magmacmagm 36357 MndOpcmndo 36375 RingOpscrngo 36403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fo 6506 df-fv 6508 df-riota 7317 df-ov 7364 df-1st 7925 df-2nd 7926 df-grpo 29484 df-gid 29485 df-ablo 29536 df-ass 36352 df-exid 36354 df-mgmOLD 36358 df-sgrOLD 36370 df-mndo 36376 df-rngo 36404 |
This theorem is referenced by: rngoueqz 36449 rngonegmn1l 36450 rngonegmn1r 36451 rngoneglmul 36452 rngonegrmul 36453 isdrngo2 36467 rngohomco 36483 rngoisocnv 36490 idlnegcl 36531 1idl 36535 0rngo 36536 smprngopr 36561 prnc 36576 isfldidl 36577 isdmn3 36583 |
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