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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 2823 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | 1, 2 | resssca 16652 | . . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (Scalar‘𝑊) = (Scalar‘𝑋)) |
4 | 3 | fveq2d 6676 | . . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑋))) |
5 | eqid 2823 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
6 | 1, 5 | ressvsca 16653 | . . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑋)) |
7 | eqidd 2824 | . . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑥 = 𝑥) | |
8 | eqid 2823 | . . . . 5 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
9 | 1, 8 | subrg1 19547 | . . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (1r‘𝑊) = (1r‘𝑋)) |
10 | 6, 7, 9 | oveq123d 7179 | . . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊)) = (𝑥( ·𝑠 ‘𝑋)(1r‘𝑋))) |
11 | 4, 10 | mpteq12dv 5153 | . 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 2823 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
14 | 12, 2, 13, 5, 8 | asclfval 20110 | . 2 ⊢ 𝐴 = (𝑥 ∈ (Base‘(Scalar‘𝑊)) ↦ (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))) |
15 | eqid 2823 | . . 3 ⊢ (algSc‘𝑋) = (algSc‘𝑋) | |
16 | eqid 2823 | . . 3 ⊢ (Scalar‘𝑋) = (Scalar‘𝑋) | |
17 | eqid 2823 | . . 3 ⊢ (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋)) | |
18 | eqid 2823 | . . 3 ⊢ ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑋) | |
19 | eqid 2823 | . . 3 ⊢ (1r‘𝑋) = (1r‘𝑋) | |
20 | 15, 16, 17, 18, 19 | asclfval 20110 | . 2 ⊢ (algSc‘𝑋) = (𝑥 ∈ (Base‘(Scalar‘𝑋)) ↦ (𝑥( ·𝑠 ‘𝑋)(1r‘𝑋))) |
21 | 11, 14, 20 | 3eqtr4g 2883 | 1 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 = (algSc‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 ↾s cress 16486 Scalarcsca 16570 ·𝑠 cvsca 16571 1rcur 19253 SubRingcsubrg 19533 algSccascl 20086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-subg 18278 df-mgp 19242 df-ur 19254 df-ring 19301 df-subrg 19535 df-ascl 20089 |
This theorem is referenced by: evlseu 20298 |
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