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| Mirrors > Home > MPE Home > Th. List > Mathboxes > drgext0gsca | Structured version Visualization version GIF version | ||
| Description: The additive neutral element of the scalar field of a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.) | 
| Ref | Expression | 
|---|---|
| drgext.b | ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) | 
| drgext.1 | ⊢ (𝜑 → 𝐸 ∈ DivRing) | 
| drgext.2 | ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) | 
| Ref | Expression | 
|---|---|
| drgext0gsca | ⊢ (𝜑 → (0g‘𝐵) = (0g‘(Scalar‘𝐵))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | drgext.1 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ DivRing) | |
| 2 | drngring 20737 | . . . 4 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring) | |
| 3 | ringmnd 20241 | . . . 4 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Mnd) | |
| 4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐸 ∈ Mnd) | 
| 5 | drgext.2 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) | |
| 6 | subrgsubg 20578 | . . . 4 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ∈ (SubGrp‘𝐸)) | |
| 7 | eqid 2736 | . . . . 5 ⊢ (0g‘𝐸) = (0g‘𝐸) | |
| 8 | 7 | subg0cl 19153 | . . . 4 ⊢ (𝑈 ∈ (SubGrp‘𝐸) → (0g‘𝐸) ∈ 𝑈) | 
| 9 | 5, 6, 8 | 3syl 18 | . . 3 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝑈) | 
| 10 | eqid 2736 | . . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
| 11 | 10 | subrgss 20573 | . . . 4 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸)) | 
| 12 | 5, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸)) | 
| 13 | eqid 2736 | . . . 4 ⊢ (𝐸 ↾s 𝑈) = (𝐸 ↾s 𝑈) | |
| 14 | 13, 10, 7 | ress0g 18776 | . . 3 ⊢ ((𝐸 ∈ Mnd ∧ (0g‘𝐸) ∈ 𝑈 ∧ 𝑈 ⊆ (Base‘𝐸)) → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝑈))) | 
| 15 | 4, 9, 12, 14 | syl3anc 1372 | . 2 ⊢ (𝜑 → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝑈))) | 
| 16 | drgext.b | . . 3 ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) | |
| 17 | 16, 1, 5 | drgext0g 33641 | . 2 ⊢ (𝜑 → (0g‘𝐸) = (0g‘𝐵)) | 
| 18 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 = ((subringAlg ‘𝐸)‘𝑈)) | 
| 19 | 18, 12 | srasca 21184 | . . 3 ⊢ (𝜑 → (𝐸 ↾s 𝑈) = (Scalar‘𝐵)) | 
| 20 | 19 | fveq2d 6909 | . 2 ⊢ (𝜑 → (0g‘(𝐸 ↾s 𝑈)) = (0g‘(Scalar‘𝐵))) | 
| 21 | 15, 17, 20 | 3eqtr3d 2784 | 1 ⊢ (𝜑 → (0g‘𝐵) = (0g‘(Scalar‘𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 ↾s cress 17275 Scalarcsca 17301 0gc0g 17485 Mndcmnd 18748 SubGrpcsubg 19139 Ringcrg 20231 SubRingcsubrg 20570 DivRingcdr 20730 subringAlg csra 21171 | 
| 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-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-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-sca 17314 df-vsca 17315 df-ip 17316 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-subg 19142 df-ring 20233 df-subrg 20571 df-drng 20732 df-sra 21173 | 
| This theorem is referenced by: fedgmullem2 33682 | 
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