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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.) |
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 16625 | . . . . 5 ⊢ Scalar = Slot (Scalar‘ndx) | |
2 | 5re 11716 | . . . . . . 7 ⊢ 5 ∈ ℝ | |
3 | 5lt6 11810 | . . . . . . 7 ⊢ 5 < 6 | |
4 | 2, 3 | ltneii 10745 | . . . . . 6 ⊢ 5 ≠ 6 |
5 | scandx 16624 | . . . . . . 7 ⊢ (Scalar‘ndx) = 5 | |
6 | vscandx 16626 | . . . . . . 7 ⊢ ( ·𝑠 ‘ndx) = 6 | |
7 | 5, 6 | neeq12i 3080 | . . . . . 6 ⊢ ((Scalar‘ndx) ≠ ( ·𝑠 ‘ndx) ↔ 5 ≠ 6) |
8 | 4, 7 | mpbir 233 | . . . . 5 ⊢ (Scalar‘ndx) ≠ ( ·𝑠 ‘ndx) |
9 | 1, 8 | setsnid 16531 | . . . 4 ⊢ (Scalar‘(𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉)) = (Scalar‘((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉)) |
10 | 5lt8 11823 | . . . . . . 7 ⊢ 5 < 8 | |
11 | 2, 10 | ltneii 10745 | . . . . . 6 ⊢ 5 ≠ 8 |
12 | ipndx 16633 | . . . . . . 7 ⊢ (·𝑖‘ndx) = 8 | |
13 | 5, 12 | neeq12i 3080 | . . . . . 6 ⊢ ((Scalar‘ndx) ≠ (·𝑖‘ndx) ↔ 5 ≠ 8) |
14 | 11, 13 | mpbir 233 | . . . . 5 ⊢ (Scalar‘ndx) ≠ (·𝑖‘ndx) |
15 | 1, 14 | setsnid 16531 | . . . 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)〉)) |
16 | 9, 15 | eqtri 2842 | . . 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)〉)) |
17 | ovexd 7183 | . . . 4 ⊢ (𝜑 → (𝑊 ↾s 𝑆) ∈ V) | |
18 | 1 | setsid 16530 | . . . 4 ⊢ ((𝑊 ∈ V ∧ (𝑊 ↾s 𝑆) ∈ V) → (𝑊 ↾s 𝑆) = (Scalar‘(𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉))) |
19 | 17, 18 | sylan2 594 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝑊 ↾s 𝑆) = (Scalar‘(𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉))) |
20 | srapart.a | . . . . . 6 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) | |
21 | 20 | adantl 484 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
22 | srapart.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) | |
23 | sraval 19940 | . . . . . 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)〉)) | |
24 | 22, 23 | 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)〉)) |
25 | 21, 24 | eqtrd 2854 | . . . 4 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
26 | 25 | fveq2d 6667 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (Scalar‘𝐴) = (Scalar‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉))) |
27 | 16, 19, 26 | 3eqtr4a 2880 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) |
28 | 1 | str0 16527 | . . 3 ⊢ ∅ = (Scalar‘∅) |
29 | reldmress 16542 | . . . . 5 ⊢ Rel dom ↾s | |
30 | 29 | ovprc1 7187 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝑆) = ∅) |
31 | 30 | adantr 483 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝑊 ↾s 𝑆) = ∅) |
32 | fv2prc 6703 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅) | |
33 | 20, 32 | sylan9eqr 2876 | . . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅) |
34 | 33 | fveq2d 6667 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (Scalar‘𝐴) = (Scalar‘∅)) |
35 | 28, 31, 34 | 3eqtr4a 2880 | . 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) |
36 | 27, 35 | pm2.61ian 810 | 1 ⊢ (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 Vcvv 3493 ⊆ wss 3934 ∅c0 4289 〈cop 4565 ‘cfv 6348 (class class class)co 7148 5c5 11687 6c6 11688 8c8 11690 ndxcnx 16472 sSet csts 16473 Basecbs 16475 ↾s cress 16476 .rcmulr 16558 Scalarcsca 16560 ·𝑠 cvsca 16561 ·𝑖cip 16562 subringAlg csra 19932 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-ndx 16478 df-slot 16479 df-sets 16482 df-ress 16483 df-sca 16573 df-vsca 16574 df-ip 16575 df-sra 19936 |
This theorem is referenced by: sralmod 19951 rlmsca 19964 rlmsca2 19965 sraassa 20091 frlmip 20914 sranlm 23285 srabn 23955 rrxprds 23984 sralvec 30983 drgext0gsca 30987 drgextlsp 30989 fedgmullem1 31018 fedgmullem2 31019 fedgmul 31020 extdg1id 31046 ccfldsrarelvec 31049 ccfldextdgrr 31050 |
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