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Mirrors > Home > MPE Home > Th. List > srasca | Structured version Visualization version GIF version |
Description: The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.) |
Ref | Expression |
---|---|
srapart.a | ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
srapart.s | ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) |
Ref | Expression |
---|---|
srasca | ⊢ (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scaid 17290 | . . . . 5 ⊢ Scalar = Slot (Scalar‘ndx) | |
2 | vscandxnscandx 17299 | . . . . . 6 ⊢ ( ·𝑠 ‘ndx) ≠ (Scalar‘ndx) | |
3 | 2 | necomi 2991 | . . . . 5 ⊢ (Scalar‘ndx) ≠ ( ·𝑠 ‘ndx) |
4 | 1, 3 | setsnid 17172 | . . . 4 ⊢ (Scalar‘(𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉)) = (Scalar‘((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉)) |
5 | slotsdifipndx 17310 | . . . . . 6 ⊢ (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·𝑖‘ndx)) | |
6 | 5 | simpri 485 | . . . . 5 ⊢ (Scalar‘ndx) ≠ (·𝑖‘ndx) |
7 | 1, 6 | setsnid 17172 | . . . 4 ⊢ (Scalar‘((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉)) = (Scalar‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
8 | 4, 7 | eqtri 2756 | . . 3 ⊢ (Scalar‘(𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉)) = (Scalar‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
9 | ovexd 7450 | . . . 4 ⊢ (𝜑 → (𝑊 ↾s 𝑆) ∈ V) | |
10 | 1 | setsid 17171 | . . . 4 ⊢ ((𝑊 ∈ V ∧ (𝑊 ↾s 𝑆) ∈ V) → (𝑊 ↾s 𝑆) = (Scalar‘(𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉))) |
11 | 9, 10 | sylan2 592 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝑊 ↾s 𝑆) = (Scalar‘(𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉))) |
12 | srapart.a | . . . . . 6 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) | |
13 | 12 | adantl 481 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
14 | srapart.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) | |
15 | sraval 21054 | . . . . . 6 ⊢ ((𝑊 ∈ V ∧ 𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) | |
16 | 14, 15 | sylan2 592 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
17 | 13, 16 | eqtrd 2768 | . . . 4 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
18 | 17 | fveq2d 6896 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (Scalar‘𝐴) = (Scalar‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉))) |
19 | 8, 11, 18 | 3eqtr4a 2794 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) |
20 | 1 | str0 17152 | . . 3 ⊢ ∅ = (Scalar‘∅) |
21 | reldmress 17205 | . . . . 5 ⊢ Rel dom ↾s | |
22 | 21 | ovprc1 7454 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝑆) = ∅) |
23 | 22 | adantr 480 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝑊 ↾s 𝑆) = ∅) |
24 | fv2prc 6937 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅) | |
25 | 12, 24 | sylan9eqr 2790 | . . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅) |
26 | 25 | fveq2d 6896 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (Scalar‘𝐴) = (Scalar‘∅)) |
27 | 20, 23, 26 | 3eqtr4a 2794 | . 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) |
28 | 19, 27 | pm2.61ian 811 | 1 ⊢ (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 Vcvv 3470 ⊆ wss 3945 ∅c0 4319 〈cop 4631 ‘cfv 6543 (class class class)co 7415 sSet csts 17126 ndxcnx 17156 Basecbs 17174 ↾s cress 17203 .rcmulr 17228 Scalarcsca 17230 ·𝑠 cvsca 17231 ·𝑖cip 17232 subringAlg csra 21050 |
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-rep 5280 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-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-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 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-sets 17127 df-slot 17145 df-ndx 17157 df-ress 17204 df-sca 17243 df-vsca 17244 df-ip 17245 df-sra 21052 |
This theorem is referenced by: sralmod 21074 rlmsca 21085 rlmsca2 21086 frlmip 21706 sraassab 21795 sraassaOLD 21797 sranlm 24595 srabn 25282 rrxprds 25311 sralvec 33276 drgext0gsca 33282 drgextlsp 33284 fedgmullem1 33318 fedgmullem2 33319 fedgmul 33320 extdg1id 33346 ccfldsrarelvec 33350 ccfldextdgrr 33351 evls1maplmhm 33365 |
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