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Mirrors > Home > MPE Home > Th. List > ressascl | Structured version Visualization version GIF version |
Description: The injection of scalars is invariant between subalgebras and superalgebras. (Contributed by Mario Carneiro, 9-Mar-2015.) |
Ref | Expression |
---|---|
ressascl.a | ⊢ 𝐴 = (algSc‘𝑊) |
ressascl.x | ⊢ 𝑋 = (𝑊 ↾s 𝑆) |
Ref | Expression |
---|---|
ressascl | ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 = (algSc‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressascl.x | . . . . 5 ⊢ 𝑋 = (𝑊 ↾s 𝑆) | |
2 | eqid 2726 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | 1, 2 | resssca 17357 | . . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (Scalar‘𝑊) = (Scalar‘𝑋)) |
4 | 3 | fveq2d 6905 | . . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑋))) |
5 | eqid 2726 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
6 | 1, 5 | ressvsca 17358 | . . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑋)) |
7 | eqidd 2727 | . . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑥 = 𝑥) | |
8 | eqid 2726 | . . . . 5 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
9 | 1, 8 | subrg1 20566 | . . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (1r‘𝑊) = (1r‘𝑋)) |
10 | 6, 7, 9 | oveq123d 7445 | . . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊)) = (𝑥( ·𝑠 ‘𝑋)(1r‘𝑋))) |
11 | 4, 10 | mpteq12dv 5244 | . 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑥 ∈ (Base‘(Scalar‘𝑊)) ↦ (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))) = (𝑥 ∈ (Base‘(Scalar‘𝑋)) ↦ (𝑥( ·𝑠 ‘𝑋)(1r‘𝑋)))) |
12 | ressascl.a | . . 3 ⊢ 𝐴 = (algSc‘𝑊) | |
13 | eqid 2726 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
14 | 12, 2, 13, 5, 8 | asclfval 21876 | . 2 ⊢ 𝐴 = (𝑥 ∈ (Base‘(Scalar‘𝑊)) ↦ (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))) |
15 | eqid 2726 | . . 3 ⊢ (algSc‘𝑋) = (algSc‘𝑋) | |
16 | eqid 2726 | . . 3 ⊢ (Scalar‘𝑋) = (Scalar‘𝑋) | |
17 | eqid 2726 | . . 3 ⊢ (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋)) | |
18 | eqid 2726 | . . 3 ⊢ ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑋) | |
19 | eqid 2726 | . . 3 ⊢ (1r‘𝑋) = (1r‘𝑋) | |
20 | 15, 16, 17, 18, 19 | asclfval 21876 | . 2 ⊢ (algSc‘𝑋) = (𝑥 ∈ (Base‘(Scalar‘𝑋)) ↦ (𝑥( ·𝑠 ‘𝑋)(1r‘𝑋))) |
21 | 11, 14, 20 | 3eqtr4g 2791 | 1 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 = (algSc‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 ↾s cress 17242 Scalarcsca 17269 ·𝑠 cvsca 17270 1rcur 20164 SubRingcsubrg 20551 algSccascl 21850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-subg 19117 df-mgp 20118 df-ur 20165 df-ring 20218 df-subrg 20553 df-ascl 21853 |
This theorem is referenced by: evlseu 22098 |
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