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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 482 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
3 | srapart.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) | |
4 | sraval 19176 | . . . . . 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 491 | . . . . 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 2656 | . . . 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 6195 | . . 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 6678 | . . . . 5 ⊢ (𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) ∈ V | |
9 | fvex 6201 | . . . . 5 ⊢ (.r‘𝑊) ∈ V | |
10 | vscaid 16016 | . . . . . 6 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx) | |
11 | 10 | setsid 15914 | . . . . 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 708 | . . . 4 ⊢ (.r‘𝑊) = ( ·𝑠 ‘((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉)) |
13 | 6re 11101 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
14 | 6lt8 11216 | . . . . . . 7 ⊢ 6 < 8 | |
15 | 13, 14 | ltneii 10150 | . . . . . 6 ⊢ 6 ≠ 8 |
16 | vscandx 16015 | . . . . . . 7 ⊢ ( ·𝑠 ‘ndx) = 6 | |
17 | ipndx 16022 | . . . . . . 7 ⊢ (·𝑖‘ndx) = 8 | |
18 | 16, 17 | neeq12i 2860 | . . . . . 6 ⊢ (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ↔ 6 ≠ 8) |
19 | 15, 18 | mpbir 221 | . . . . 5 ⊢ ( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) |
20 | 10, 19 | setsnid 15915 | . . . 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 2644 | . . 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 2675 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
23 | 10 | str0 15911 | . . 3 ⊢ ∅ = ( ·𝑠 ‘∅) |
24 | fvprc 6185 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (.r‘𝑊) = ∅) | |
25 | 24 | adantr 481 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = ∅) |
26 | fvprc 6185 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (subringAlg ‘𝑊) = ∅) | |
27 | 26 | fveq1d 6193 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = (∅‘𝑆)) |
28 | 0fv 6227 | . . . . . 6 ⊢ (∅‘𝑆) = ∅ | |
29 | 27, 28 | syl6eq 2672 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅) |
30 | 1, 29 | sylan9eqr 2678 | . . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅) |
31 | 30 | fveq2d 6195 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘∅)) |
32 | 23, 25, 31 | 3eqtr4a 2682 | . 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
33 | 22, 32 | pm2.61ian 831 | 1 ⊢ (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 〈cop 4183 ‘cfv 5888 (class class class)co 6650 6c6 11074 8c8 11076 ndxcnx 15854 sSet csts 15855 Basecbs 15857 ↾s cress 15858 .rcmulr 15942 Scalarcsca 15944 ·𝑠 cvsca 15945 ·𝑖cip 15946 subringAlg csra 19168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-ndx 15860 df-slot 15861 df-sets 15864 df-vsca 15958 df-ip 15959 df-sra 19172 |
This theorem is referenced by: sralmod 19187 rlmvsca 19202 sraassa 19325 sranlm 22488 |
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