Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 | ovex 7224 | . . . . 5 ⊢ (𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) ∈ V | |
2 | fvex 6708 | . . . . 5 ⊢ (.r‘𝑊) ∈ V | |
3 | vscaid 16819 | . . . . . 6 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx) | |
4 | 3 | setsid 16719 | . . . . 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 11885 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
7 | 6lt8 11988 | . . . . . . 7 ⊢ 6 < 8 | |
8 | 6, 7 | ltneii 10910 | . . . . . 6 ⊢ 6 ≠ 8 |
9 | vscandx 16818 | . . . . . . 7 ⊢ ( ·𝑠 ‘ndx) = 6 | |
10 | ipndx 16825 | . . . . . . 7 ⊢ (·𝑖‘ndx) = 8 | |
11 | 9, 10 | neeq12i 2998 | . . . . . 6 ⊢ (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ↔ 6 ≠ 8) |
12 | 8, 11 | mpbir 234 | . . . . 5 ⊢ ( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) |
13 | 3, 12 | setsnid 16720 | . . . 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 2759 | . . 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 485 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
17 | srapart.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) | |
18 | sraval 20167 | . . . . . 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 596 | . . . . 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 2771 | . . . 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 6699 | . . 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 2790 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
23 | 3 | str0 16717 | . . 3 ⊢ ∅ = ( ·𝑠 ‘∅) |
24 | fvprc 6687 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (.r‘𝑊) = ∅) | |
25 | 24 | adantr 484 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = ∅) |
26 | fv2prc 6735 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅) | |
27 | 15, 26 | sylan9eqr 2793 | . . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅) |
28 | 27 | fveq2d 6699 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘∅)) |
29 | 23, 25, 28 | 3eqtr4a 2797 | . 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 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 Vcvv 3398 ⊆ wss 3853 ∅c0 4223 〈cop 4533 ‘cfv 6358 (class class class)co 7191 6c6 11854 8c8 11856 ndxcnx 16663 sSet csts 16664 Basecbs 16666 ↾s cress 16667 .rcmulr 16750 Scalarcsca 16752 ·𝑠 cvsca 16753 ·𝑖cip 16754 subringAlg csra 20159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-ndx 16669 df-slot 16670 df-sets 16673 df-vsca 16766 df-ip 16767 df-sra 20163 |
This theorem is referenced by: sralmod 20178 rlmvsca 20193 sraassa 20783 sranlm 23536 drgextvsca 31346 drgextlsp 31349 fedgmullem1 31378 extdg1id 31406 ccfldsrarelvec 31409 ccfldextdgrr 31410 |
Copyright terms: Public domain | W3C validator |