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| Mirrors > Home > MPE Home > Th. List > rgspnval | Structured version Visualization version GIF version | ||
| Description: Value of the ring-span of a set of elements in a ring. (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 | 
|---|---|
| rgspnval | ⊢ (𝜑 → 𝑈 = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rgspnval.sp | . 2 ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴)) | |
| 2 | rgspnval.n | . . 3 ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅)) | |
| 3 | 2 | fveq1d 6907 | . 2 ⊢ (𝜑 → (𝑁‘𝐴) = ((RingSpan‘𝑅)‘𝐴)) | 
| 4 | rgspnval.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
| 5 | elex 3500 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V) | |
| 6 | fveq2 6905 | . . . . . . . 8 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅)) | |
| 7 | 6 | pweqd 4616 | . . . . . . 7 ⊢ (𝑎 = 𝑅 → 𝒫 (Base‘𝑎) = 𝒫 (Base‘𝑅)) | 
| 8 | fveq2 6905 | . . . . . . . . 9 ⊢ (𝑎 = 𝑅 → (SubRing‘𝑎) = (SubRing‘𝑅)) | |
| 9 | rabeq 3450 | . . . . . . . . 9 ⊢ ((SubRing‘𝑎) = (SubRing‘𝑅) → {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}) | |
| 10 | 8, 9 | syl 17 | . . . . . . . 8 ⊢ (𝑎 = 𝑅 → {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}) | 
| 11 | 10 | inteqd 4950 | . . . . . . 7 ⊢ (𝑎 = 𝑅 → ∩ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡} = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}) | 
| 12 | 7, 11 | mpteq12dv 5232 | . . . . . 6 ⊢ (𝑎 = 𝑅 → (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ ∩ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})) | 
| 13 | df-rgspn 20612 | . . . . . 6 ⊢ RingSpan = (𝑎 ∈ V ↦ (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ ∩ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡})) | |
| 14 | fvex 6918 | . . . . . . . 8 ⊢ (Base‘𝑅) ∈ V | |
| 15 | 14 | pwex 5379 | . . . . . . 7 ⊢ 𝒫 (Base‘𝑅) ∈ V | 
| 16 | 15 | mptex 7244 | . . . . . 6 ⊢ (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}) ∈ V | 
| 17 | 12, 13, 16 | fvmpt 7015 | . . . . 5 ⊢ (𝑅 ∈ V → (RingSpan‘𝑅) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})) | 
| 18 | 4, 5, 17 | 3syl 18 | . . . 4 ⊢ (𝜑 → (RingSpan‘𝑅) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})) | 
| 19 | 18 | fveq1d 6907 | . . 3 ⊢ (𝜑 → ((RingSpan‘𝑅)‘𝐴) = ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})‘𝐴)) | 
| 20 | eqid 2736 | . . . 4 ⊢ (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}) | |
| 21 | sseq1 4008 | . . . . . 6 ⊢ (𝑏 = 𝐴 → (𝑏 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝑡)) | |
| 22 | 21 | rabbidv 3443 | . . . . 5 ⊢ (𝑏 = 𝐴 → {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡}) | 
| 23 | 22 | inteqd 4950 | . . . 4 ⊢ (𝑏 = 𝐴 → ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡} = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡}) | 
| 24 | rgspnval.ss | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
| 25 | rgspnval.b | . . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) | |
| 26 | 24, 25 | sseqtrd 4019 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (Base‘𝑅)) | 
| 27 | 14 | elpw2 5333 | . . . . 5 ⊢ (𝐴 ∈ 𝒫 (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑅)) | 
| 28 | 26, 27 | sylibr 234 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (Base‘𝑅)) | 
| 29 | eqid 2736 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 30 | 29 | subrgid 20574 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅)) | 
| 31 | 4, 30 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ∈ (SubRing‘𝑅)) | 
| 32 | 25, 31 | eqeltrd 2840 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅)) | 
| 33 | sseq2 4009 | . . . . . . 7 ⊢ (𝑡 = 𝐵 → (𝐴 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝐵)) | |
| 34 | 33 | rspcev 3621 | . . . . . 6 ⊢ ((𝐵 ∈ (SubRing‘𝑅) ∧ 𝐴 ⊆ 𝐵) → ∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡) | 
| 35 | 32, 24, 34 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡) | 
| 36 | intexrab 5346 | . . . . 5 ⊢ (∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡 ↔ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ∈ V) | |
| 37 | 35, 36 | sylib 218 | . . . 4 ⊢ (𝜑 → ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ∈ V) | 
| 38 | 20, 23, 28, 37 | fvmptd3 7038 | . . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})‘𝐴) = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡}) | 
| 39 | 19, 38 | eqtrd 2776 | . 2 ⊢ (𝜑 → ((RingSpan‘𝑅)‘𝐴) = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡}) | 
| 40 | 1, 3, 39 | 3eqtrd 2780 | 1 ⊢ (𝜑 → 𝑈 = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 {crab 3435 Vcvv 3479 ⊆ wss 3950 𝒫 cpw 4599 ∩ cint 4945 ↦ cmpt 5224 ‘cfv 6560 Basecbs 17248 Ringcrg 20231 SubRingcsubrg 20570 RingSpancrgspn 20611 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mgp 20139 df-ur 20180 df-ring 20233 df-subrg 20571 df-rgspn 20612 | 
| This theorem is referenced by: rgspncl 20614 rgspnssid 20615 rgspnmin 20616 elrgspnlem4 33250 | 
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