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Mathbox for Thierry Arnoux |
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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 20753 | . . . 4 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring) | |
3 | ringmnd 20261 | . . . 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 20594 | . . . 4 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ∈ (SubGrp‘𝐸)) | |
7 | eqid 2735 | . . . . 5 ⊢ (0g‘𝐸) = (0g‘𝐸) | |
8 | 7 | subg0cl 19165 | . . . 4 ⊢ (𝑈 ∈ (SubGrp‘𝐸) → (0g‘𝐸) ∈ 𝑈) |
9 | 5, 6, 8 | 3syl 18 | . . 3 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝑈) |
10 | eqid 2735 | . . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
11 | 10 | subrgss 20589 | . . . 4 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸)) |
12 | 5, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸)) |
13 | eqid 2735 | . . . 4 ⊢ (𝐸 ↾s 𝑈) = (𝐸 ↾s 𝑈) | |
14 | 13, 10, 7 | ress0g 18788 | . . 3 ⊢ ((𝐸 ∈ Mnd ∧ (0g‘𝐸) ∈ 𝑈 ∧ 𝑈 ⊆ (Base‘𝐸)) → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝑈))) |
15 | 4, 9, 12, 14 | syl3anc 1370 | . 2 ⊢ (𝜑 → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝑈))) |
16 | drgext.b | . . 3 ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) | |
17 | 16, 1, 5 | drgext0g 33619 | . 2 ⊢ (𝜑 → (0g‘𝐸) = (0g‘𝐵)) |
18 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 = ((subringAlg ‘𝐸)‘𝑈)) |
19 | 18, 12 | srasca 21201 | . . 3 ⊢ (𝜑 → (𝐸 ↾s 𝑈) = (Scalar‘𝐵)) |
20 | 19 | fveq2d 6911 | . 2 ⊢ (𝜑 → (0g‘(𝐸 ↾s 𝑈)) = (0g‘(Scalar‘𝐵))) |
21 | 15, 17, 20 | 3eqtr3d 2783 | 1 ⊢ (𝜑 → (0g‘𝐵) = (0g‘(Scalar‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 Scalarcsca 17301 0gc0g 17486 Mndcmnd 18760 SubGrpcsubg 19151 Ringcrg 20251 SubRingcsubrg 20586 DivRingcdr 20746 subringAlg csra 21188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-sca 17314 df-vsca 17315 df-ip 17316 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-subg 19154 df-ring 20253 df-subrg 20587 df-drng 20748 df-sra 21190 |
This theorem is referenced by: fedgmullem2 33658 |
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