Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2737 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | subrg1cl.a | . . . 4 ⊢ 1 = (1r‘𝑅) | |
3 | 1, 2 | issubrg 19800 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ 1 ∈ 𝐴))) |
4 | 3 | simprbi 500 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐴 ⊆ (Base‘𝑅) ∧ 1 ∈ 𝐴)) |
5 | 4 | simprd 499 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 ↾s cress 16784 1rcur 19516 Ringcrg 19562 SubRingcsubrg 19796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fv 6388 df-ov 7216 df-subrg 19798 |
This theorem is referenced by: subrg1 19810 subrgsubm 19813 issubrg2 19820 subrgint 19822 subsubrg 19826 primefld1cl 19851 zsssubrg 20421 issubassa2 20852 subrgpsr 20944 mplassa 20983 mplbas2 20999 ply1assa 21120 isclmp 23994 taylply2 25260 subrgchr 31210 drgextlsp 31395 evlsbagval 39985 cnsrexpcl 40693 rngunsnply 40701 |
Copyright terms: Public domain | W3C validator |