Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rgspncl | Structured version Visualization version GIF version |
Description: The ring-span of a set is a subring. (Contributed by Stefan O'Rear, 7-Dec-2014.) |
Ref | Expression |
---|---|
rgspnval.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
rgspnval.b | ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) |
rgspnval.ss | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
rgspnval.n | ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅)) |
rgspnval.sp | ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴)) |
Ref | Expression |
---|---|
rgspncl | ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rgspnval.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | rgspnval.b | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) | |
3 | rgspnval.ss | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
4 | rgspnval.n | . . 3 ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅)) | |
5 | rgspnval.sp | . . 3 ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴)) | |
6 | 1, 2, 3, 4, 5 | rgspnval 37738 | . 2 ⊢ (𝜑 → 𝑈 = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡}) |
7 | ssrab2 3687 | . . 3 ⊢ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ⊆ (SubRing‘𝑅) | |
8 | eqid 2622 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 8 | subrgid 18782 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅)) |
10 | 1, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑅) ∈ (SubRing‘𝑅)) |
11 | 2, 10 | eqeltrd 2701 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅)) |
12 | sseq2 3627 | . . . . . 6 ⊢ (𝑡 = 𝐵 → (𝐴 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝐵)) | |
13 | 12 | rspcev 3309 | . . . . 5 ⊢ ((𝐵 ∈ (SubRing‘𝑅) ∧ 𝐴 ⊆ 𝐵) → ∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡) |
14 | 11, 3, 13 | syl2anc 693 | . . . 4 ⊢ (𝜑 → ∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡) |
15 | rabn0 3958 | . . . 4 ⊢ ({𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ≠ ∅ ↔ ∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡) | |
16 | 14, 15 | sylibr 224 | . . 3 ⊢ (𝜑 → {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ≠ ∅) |
17 | subrgint 18802 | . . 3 ⊢ (({𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ⊆ (SubRing‘𝑅) ∧ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ≠ ∅) → ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ∈ (SubRing‘𝑅)) | |
18 | 7, 16, 17 | sylancr 695 | . 2 ⊢ (𝜑 → ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ∈ (SubRing‘𝑅)) |
19 | 6, 18 | eqeltrd 2701 | 1 ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 {crab 2916 ⊆ wss 3574 ∅c0 3915 ∩ cint 4475 ‘cfv 5888 Basecbs 15857 Ringcrg 18547 SubRingcsubrg 18776 RingSpancrgspn 18777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-subg 17591 df-mgp 18490 df-ur 18502 df-ring 18549 df-subrg 18778 df-rgspn 18779 |
This theorem is referenced by: rngunsnply 37743 |
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