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 39646 | . 2 ⊢ (𝜑 → 𝑈 = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡}) |
7 | ssrab2 4053 | . . 3 ⊢ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ⊆ (SubRing‘𝑅) | |
8 | eqid 2818 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 8 | subrgid 19466 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅)) |
10 | 1, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑅) ∈ (SubRing‘𝑅)) |
11 | 2, 10 | eqeltrd 2910 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅)) |
12 | sseq2 3990 | . . . . . 6 ⊢ (𝑡 = 𝐵 → (𝐴 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝐵)) | |
13 | 12 | rspcev 3620 | . . . . 5 ⊢ ((𝐵 ∈ (SubRing‘𝑅) ∧ 𝐴 ⊆ 𝐵) → ∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡) |
14 | 11, 3, 13 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡) |
15 | rabn0 4336 | . . . 4 ⊢ ({𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ≠ ∅ ↔ ∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡) | |
16 | 14, 15 | sylibr 235 | . . 3 ⊢ (𝜑 → {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ≠ ∅) |
17 | subrgint 19486 | . . 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 587 | . 2 ⊢ (𝜑 → ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ∈ (SubRing‘𝑅)) |
19 | 6, 18 | eqeltrd 2910 | 1 ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∃wrex 3136 {crab 3139 ⊆ wss 3933 ∅c0 4288 ∩ cint 4867 ‘cfv 6348 Basecbs 16471 Ringcrg 19226 SubRingcsubrg 19460 RingSpancrgspn 19461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-subg 18214 df-mgp 19169 df-ur 19181 df-ring 19228 df-subrg 19462 df-rgspn 19463 |
This theorem is referenced by: rngunsnply 39651 |
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