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Mirrors > Home > MPE Home > Th. List > pwssca | Structured version Visualization version GIF version |
Description: The ring of scalars of a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
pwssca.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pwssca.s | ⊢ 𝑆 = (Scalar‘𝑅) |
Ref | Expression |
---|---|
pwssca | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑆 = (Scalar‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 ⊢ (𝑆Xs(𝐼 × {𝑅})) = (𝑆Xs(𝐼 × {𝑅})) | |
2 | pwssca.s | . . . . 5 ⊢ 𝑆 = (Scalar‘𝑅) | |
3 | 2 | fvexi 6906 | . . . 4 ⊢ 𝑆 ∈ V |
4 | 3 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑆 ∈ V) |
5 | simpr 484 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐼 ∈ 𝑊) | |
6 | snex 5428 | . . . 4 ⊢ {𝑅} ∈ V | |
7 | xpexg 7747 | . . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ {𝑅} ∈ V) → (𝐼 × {𝑅}) ∈ V) | |
8 | 5, 6, 7 | sylancl 585 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝐼 × {𝑅}) ∈ V) |
9 | 1, 4, 8 | prdssca 17432 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑆 = (Scalar‘(𝑆Xs(𝐼 × {𝑅})))) |
10 | pwssca.y | . . . 4 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
11 | 10, 2 | pwsval 17462 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝑆Xs(𝐼 × {𝑅}))) |
12 | 11 | fveq2d 6896 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (Scalar‘𝑌) = (Scalar‘(𝑆Xs(𝐼 × {𝑅})))) |
13 | 9, 12 | eqtr4d 2771 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑆 = (Scalar‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3470 {csn 4625 × cxp 5671 ‘cfv 6543 (class class class)co 7415 Scalarcsca 17230 Xscprds 17421 ↑s cpws 17422 |
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 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-ixp 8911 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-sup 9460 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-struct 17110 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-hom 17251 df-cco 17252 df-prds 17423 df-pws 17425 |
This theorem is referenced by: pwsdiaglmhm 20936 pwssplit3 20940 frlmsca 21681 |
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