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Mirrors > Home > MPE Home > Th. List > subrg1cl | Structured version Visualization version GIF version |
Description: A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
subrg1cl.a | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
subrg1cl | ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | subrg1cl.a | . . . 4 ⊢ 1 = (1r‘𝑅) | |
3 | 1, 2 | issubrg 20351 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ 1 ∈ 𝐴))) |
4 | 3 | simprbi 498 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐴 ⊆ (Base‘𝑅) ∧ 1 ∈ 𝐴)) |
5 | 4 | simprd 497 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3947 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 ↾s cress 17169 1rcur 19996 Ringcrg 20047 SubRingcsubrg 20347 |
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 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 |
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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7407 df-subrg 20349 |
This theorem is referenced by: subrg1 20361 subrgsubm 20364 issubrg2 20371 subrgint 20374 subsubrg 20378 primefld1cl 20411 zsssubrg 20988 issubassa2 21428 subrgpsr 21521 mplassa 21563 mplbas2 21579 ply1assa 21705 isclmp 24595 taylply2 25862 0ringsubrg 32353 subrgchr 32361 fldgensdrg 32373 primefldgen1 32380 asclply1subcl 32607 drgextlsp 32627 evls1maprhm 32703 evlsmaprhm 41092 cnsrexpcl 41840 rngunsnply 41848 |
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