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Mirrors > Home > MPE Home > Th. List > sravsca | Structured version Visualization version GIF version |
Description: 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.) |
Ref | Expression |
---|---|
srapart.a | ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
srapart.s | ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) |
Ref | Expression |
---|---|
sravsca | ⊢ (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srapart.a | . . . . . 6 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) | |
2 | 1 | adantl 484 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
3 | srapart.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) | |
4 | sraval 19947 | . . . . . 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)〉)) | |
5 | 3, 4 | sylan2 594 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
6 | 2, 5 | eqtrd 2856 | . . . 4 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
7 | 6 | fveq2d 6673 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝜑) → ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉))) |
8 | ovex 7188 | . . . . 5 ⊢ (𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) ∈ V | |
9 | fvex 6682 | . . . . 5 ⊢ (.r‘𝑊) ∈ V | |
10 | vscaid 16634 | . . . . . 6 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx) | |
11 | 10 | setsid 16537 | . . . . 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)〉))) |
12 | 8, 9, 11 | mp2an 690 | . . . 4 ⊢ (.r‘𝑊) = ( ·𝑠 ‘((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉)) |
13 | 6re 11726 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
14 | 6lt8 11829 | . . . . . . 7 ⊢ 6 < 8 | |
15 | 13, 14 | ltneii 10752 | . . . . . 6 ⊢ 6 ≠ 8 |
16 | vscandx 16633 | . . . . . . 7 ⊢ ( ·𝑠 ‘ndx) = 6 | |
17 | ipndx 16640 | . . . . . . 7 ⊢ (·𝑖‘ndx) = 8 | |
18 | 16, 17 | neeq12i 3082 | . . . . . 6 ⊢ (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ↔ 6 ≠ 8) |
19 | 15, 18 | mpbir 233 | . . . . 5 ⊢ ( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) |
20 | 10, 19 | setsnid 16538 | . . . 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)〉)) |
21 | 12, 20 | eqtri 2844 | . . 3 ⊢ (.r‘𝑊) = ( ·𝑠 ‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
22 | 7, 21 | syl6reqr 2875 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
23 | 10 | str0 16534 | . . 3 ⊢ ∅ = ( ·𝑠 ‘∅) |
24 | fvprc 6662 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (.r‘𝑊) = ∅) | |
25 | 24 | adantr 483 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = ∅) |
26 | fv2prc 6709 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅) | |
27 | 1, 26 | sylan9eqr 2878 | . . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅) |
28 | 27 | fveq2d 6673 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘∅)) |
29 | 23, 25, 28 | 3eqtr4a 2882 | . 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
30 | 22, 29 | pm2.61ian 810 | 1 ⊢ (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ⊆ wss 3935 ∅c0 4290 〈cop 4572 ‘cfv 6354 (class class class)co 7155 6c6 11695 8c8 11697 ndxcnx 16479 sSet csts 16480 Basecbs 16482 ↾s cress 16483 .rcmulr 16565 Scalarcsca 16567 ·𝑠 cvsca 16568 ·𝑖cip 16569 subringAlg csra 19939 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-ndx 16485 df-slot 16486 df-sets 16489 df-vsca 16581 df-ip 16582 df-sra 19943 |
This theorem is referenced by: sralmod 19958 rlmvsca 19973 sraassa 20098 sranlm 23292 drgextvsca 30993 drgextlsp 30996 fedgmullem1 31025 extdg1id 31053 ccfldsrarelvec 31056 ccfldextdgrr 31057 |
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