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Mirrors > Home > MPE Home > Th. List > frlmvscafval | Structured version Visualization version GIF version |
Description: Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
Ref | Expression |
---|---|
frlmvscafval.y | ⊢ 𝑌 = (𝑅 freeLMod 𝐼) |
frlmvscafval.b | ⊢ 𝐵 = (Base‘𝑌) |
frlmvscafval.k | ⊢ 𝐾 = (Base‘𝑅) |
frlmvscafval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
frlmvscafval.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
frlmvscafval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
frlmvscafval.v | ⊢ ∙ = ( ·𝑠 ‘𝑌) |
frlmvscafval.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
frlmvscafval | ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘f · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmvscafval.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
2 | frlmvscafval.y | . . . . . . . 8 ⊢ 𝑌 = (𝑅 freeLMod 𝐼) | |
3 | frlmvscafval.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑌) | |
4 | 2, 3 | frlmrcl 20446 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑅 ∈ V) |
5 | 1, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V) |
6 | frlmvscafval.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
7 | 2, 3 | frlmpws 20439 | . . . . . 6 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑊) → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
8 | 5, 6, 7 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
9 | 8 | fveq2d 6649 | . . . 4 ⊢ (𝜑 → ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
10 | frlmvscafval.v | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝑌) | |
11 | 3 | fvexi 6659 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | eqid 2798 | . . . . . 6 ⊢ (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) | |
13 | eqid 2798 | . . . . . 6 ⊢ ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) = ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) | |
14 | 12, 13 | ressvsca 16643 | . . . . 5 ⊢ (𝐵 ∈ V → ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) = ( ·𝑠 ‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
15 | 11, 14 | ax-mp 5 | . . . 4 ⊢ ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) = ( ·𝑠 ‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
16 | 9, 10, 15 | 3eqtr4g 2858 | . . 3 ⊢ (𝜑 → ∙ = ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼))) |
17 | 16 | oveqd 7152 | . 2 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼))𝑋)) |
18 | eqid 2798 | . . 3 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼) | |
19 | eqid 2798 | . . 3 ⊢ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)) | |
20 | frlmvscafval.t | . . . 4 ⊢ · = (.r‘𝑅) | |
21 | rlmvsca 19967 | . . . 4 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
22 | 20, 21 | eqtri 2821 | . . 3 ⊢ · = ( ·𝑠 ‘(ringLMod‘𝑅)) |
23 | eqid 2798 | . . 3 ⊢ (Scalar‘(ringLMod‘𝑅)) = (Scalar‘(ringLMod‘𝑅)) | |
24 | eqid 2798 | . . 3 ⊢ (Base‘(Scalar‘(ringLMod‘𝑅))) = (Base‘(Scalar‘(ringLMod‘𝑅))) | |
25 | fvexd 6660 | . . 3 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ V) | |
26 | frlmvscafval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
27 | frlmvscafval.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
28 | rlmsca 19965 | . . . . . . 7 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘(ringLMod‘𝑅))) | |
29 | 5, 28 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘(ringLMod‘𝑅))) |
30 | 29 | fveq2d 6649 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
31 | 27, 30 | syl5eq 2845 | . . . 4 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
32 | 26, 31 | eleqtrd 2892 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘(ringLMod‘𝑅)))) |
33 | 8 | fveq2d 6649 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
34 | 3, 33 | syl5eq 2845 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
35 | 12, 19 | ressbasss 16548 | . . . . 5 ⊢ (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) ⊆ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) |
36 | 34, 35 | eqsstrdi 3969 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
37 | 36, 1 | sseldd 3916 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
38 | 18, 19, 22, 13, 23, 24, 25, 6, 32, 37 | pwsvscafval 16759 | . 2 ⊢ (𝜑 → (𝐴( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼))𝑋) = ((𝐼 × {𝐴}) ∘f · 𝑋)) |
39 | 17, 38 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘f · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 × cxp 5517 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 Basecbs 16475 ↾s cress 16476 .rcmulr 16558 Scalarcsca 16560 ·𝑠 cvsca 16561 ↑s cpws 16712 ringLModcrglmod 19934 freeLMod cfrlm 20435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-prds 16713 df-pws 16715 df-sra 19937 df-rgmod 19938 df-dsmm 20421 df-frlm 20436 |
This theorem is referenced by: frlmvscaval 20457 uvcresum 20482 matvsca2 21033 matunitlindflem1 35053 matunitlindflem2 35054 frlmvscadiccat 39440 0prjspnrel 39613 zlmodzxzscm 44759 aacllem 45329 |
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