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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > primefldgen1 | Structured version Visualization version GIF version |
Description: The prime field of a division ring is the subfield generated by the multiplicative identity element. In general, we should write "prime division ring", but since most later usages are in the case where the ambient ring is commutative, we keep the term "prime field". (Contributed by Thierry Arnoux, 11-Jan-2025.) |
Ref | Expression |
---|---|
primefldgen1.b | ⊢ 𝐵 = (Base‘𝑅) |
primefldgen1.1 | ⊢ 1 = (1r‘𝑅) |
primefldgen1.r | ⊢ (𝜑 → 𝑅 ∈ DivRing) |
Ref | Expression |
---|---|
primefldgen1 | ⊢ (𝜑 → ∩ (SubDRing‘𝑅) = (𝑅 fldGen { 1 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issdrg 20813 | . . . . . . . . 9 ⊢ (𝑎 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑎 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑎) ∈ DivRing)) | |
2 | 1 | simp2bi 1146 | . . . . . . . 8 ⊢ (𝑎 ∈ (SubDRing‘𝑅) → 𝑎 ∈ (SubRing‘𝑅)) |
3 | primefldgen1.1 | . . . . . . . . 9 ⊢ 1 = (1r‘𝑅) | |
4 | 3 | subrg1cl 20610 | . . . . . . . 8 ⊢ (𝑎 ∈ (SubRing‘𝑅) → 1 ∈ 𝑎) |
5 | 2, 4 | syl 17 | . . . . . . 7 ⊢ (𝑎 ∈ (SubDRing‘𝑅) → 1 ∈ 𝑎) |
6 | 5 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝑅)) → 1 ∈ 𝑎) |
7 | 6 | snssd 4834 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝑅)) → { 1 } ⊆ 𝑎) |
8 | 7 | ralrimiva 3152 | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ (SubDRing‘𝑅){ 1 } ⊆ 𝑎) |
9 | rabid2 3478 | . . . 4 ⊢ ((SubDRing‘𝑅) = {𝑎 ∈ (SubDRing‘𝑅) ∣ { 1 } ⊆ 𝑎} ↔ ∀𝑎 ∈ (SubDRing‘𝑅){ 1 } ⊆ 𝑎) | |
10 | 8, 9 | sylibr 234 | . . 3 ⊢ (𝜑 → (SubDRing‘𝑅) = {𝑎 ∈ (SubDRing‘𝑅) ∣ { 1 } ⊆ 𝑎}) |
11 | 10 | inteqd 4975 | . 2 ⊢ (𝜑 → ∩ (SubDRing‘𝑅) = ∩ {𝑎 ∈ (SubDRing‘𝑅) ∣ { 1 } ⊆ 𝑎}) |
12 | primefldgen1.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
13 | primefldgen1.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ DivRing) | |
14 | 13 | drngringd 20761 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
15 | 12, 3 | ringidcl 20291 | . . . . 5 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵) |
16 | 14, 15 | syl 17 | . . . 4 ⊢ (𝜑 → 1 ∈ 𝐵) |
17 | 16 | snssd 4834 | . . 3 ⊢ (𝜑 → { 1 } ⊆ 𝐵) |
18 | 12, 13, 17 | fldgenval 33281 | . 2 ⊢ (𝜑 → (𝑅 fldGen { 1 }) = ∩ {𝑎 ∈ (SubDRing‘𝑅) ∣ { 1 } ⊆ 𝑎}) |
19 | 11, 18 | eqtr4d 2783 | 1 ⊢ (𝜑 → ∩ (SubDRing‘𝑅) = (𝑅 fldGen { 1 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 {crab 3443 ⊆ wss 3976 {csn 4648 ∩ cint 4970 ‘cfv 6575 (class class class)co 7450 Basecbs 17260 ↾s cress 17289 1rcur 20210 Ringcrg 20262 SubRingcsubrg 20597 DivRingcdr 20753 SubDRingcsdrg 20811 fldGen cfldgen 33279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-0g 17503 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-mgp 20164 df-ur 20211 df-ring 20264 df-subrg 20599 df-drng 20755 df-sdrg 20812 df-fldgen 33280 |
This theorem is referenced by: (None) |
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