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Mirrors > Home > MPE Home > Th. List > sravscaOLD | Structured version Visualization version GIF version |
Description: Obsolete version of sravsca 21203 as of 12-Nov-2024. The scalar product operation 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 modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
srapart.a | ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
srapart.s | ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) |
Ref | Expression |
---|---|
sravscaOLD | ⊢ (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7464 | . . . . 5 ⊢ (𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) ∈ V | |
2 | fvex 6920 | . . . . 5 ⊢ (.r‘𝑊) ∈ V | |
3 | vscaid 17366 | . . . . . 6 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx) | |
4 | 3 | setsid 17242 | . . . . 5 ⊢ (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) ∈ V ∧ (.r‘𝑊) ∈ V) → (.r‘𝑊) = ( ·𝑠 ‘((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉))) |
5 | 1, 2, 4 | mp2an 692 | . . . 4 ⊢ (.r‘𝑊) = ( ·𝑠 ‘((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉)) |
6 | 6re 12354 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
7 | 6lt8 12457 | . . . . . . 7 ⊢ 6 < 8 | |
8 | 6, 7 | ltneii 11372 | . . . . . 6 ⊢ 6 ≠ 8 |
9 | vscandx 17365 | . . . . . . 7 ⊢ ( ·𝑠 ‘ndx) = 6 | |
10 | ipndx 17376 | . . . . . . 7 ⊢ (·𝑖‘ndx) = 8 | |
11 | 9, 10 | neeq12i 3005 | . . . . . 6 ⊢ (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ↔ 6 ≠ 8) |
12 | 8, 11 | mpbir 231 | . . . . 5 ⊢ ( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) |
13 | 3, 12 | setsnid 17243 | . . . 4 ⊢ ( ·𝑠 ‘((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉)) = ( ·𝑠 ‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
14 | 5, 13 | eqtri 2763 | . . 3 ⊢ (.r‘𝑊) = ( ·𝑠 ‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
15 | srapart.a | . . . . . 6 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) | |
16 | 15 | adantl 481 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
17 | srapart.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) | |
18 | sraval 21192 | . . . . . 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)〉)) | |
19 | 17, 18 | sylan2 593 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
20 | 16, 19 | eqtrd 2775 | . . . 4 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
21 | 20 | fveq2d 6911 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝜑) → ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉))) |
22 | 14, 21 | eqtr4id 2794 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
23 | 3 | str0 17223 | . . 3 ⊢ ∅ = ( ·𝑠 ‘∅) |
24 | fvprc 6899 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (.r‘𝑊) = ∅) | |
25 | 24 | adantr 480 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = ∅) |
26 | fv2prc 6952 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅) | |
27 | 15, 26 | sylan9eqr 2797 | . . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅) |
28 | 27 | fveq2d 6911 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘∅)) |
29 | 23, 25, 28 | 3eqtr4a 2801 | . 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
30 | 22, 29 | pm2.61ian 812 | 1 ⊢ (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 〈cop 4637 ‘cfv 6563 (class class class)co 7431 6c6 12323 8c8 12325 sSet csts 17197 ndxcnx 17227 Basecbs 17245 ↾s cress 17274 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 ·𝑖cip 17303 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-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-vsca 17315 df-ip 17316 df-sra 21190 |
This theorem is referenced by: (None) |
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