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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 20301 | . . . . . . 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 20294 | . . . . . 6 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑊) → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
8 | 5, 6, 7 | syl2anc 696 | . . . . 5 ⊢ (𝜑 → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
9 | 8 | fveq2d 6354 | . . . 4 ⊢ (𝜑 → ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
10 | frlmvscafval.v | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝑌) | |
11 | fvex 6360 | . . . . . 6 ⊢ (Base‘𝑌) ∈ V | |
12 | 3, 11 | eqeltri 2833 | . . . . 5 ⊢ 𝐵 ∈ V |
13 | eqid 2758 | . . . . . 6 ⊢ (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) | |
14 | eqid 2758 | . . . . . 6 ⊢ ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) = ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) | |
15 | 13, 14 | ressvsca 16232 | . . . . 5 ⊢ (𝐵 ∈ V → ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) = ( ·𝑠 ‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
16 | 12, 15 | ax-mp 5 | . . . 4 ⊢ ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼)) = ( ·𝑠 ‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
17 | 9, 10, 16 | 3eqtr4g 2817 | . . 3 ⊢ (𝜑 → ∙ = ( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼))) |
18 | 17 | oveqd 6828 | . 2 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼))𝑋)) |
19 | eqid 2758 | . . 3 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼) | |
20 | eqid 2758 | . . 3 ⊢ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)) | |
21 | frlmvscafval.t | . . . 4 ⊢ · = (.r‘𝑅) | |
22 | rlmvsca 19402 | . . . 4 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
23 | 21, 22 | eqtri 2780 | . . 3 ⊢ · = ( ·𝑠 ‘(ringLMod‘𝑅)) |
24 | eqid 2758 | . . 3 ⊢ (Scalar‘(ringLMod‘𝑅)) = (Scalar‘(ringLMod‘𝑅)) | |
25 | eqid 2758 | . . 3 ⊢ (Base‘(Scalar‘(ringLMod‘𝑅))) = (Base‘(Scalar‘(ringLMod‘𝑅))) | |
26 | fvexd 6362 | . . 3 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ V) | |
27 | frlmvscafval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
28 | frlmvscafval.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
29 | rlmsca 19400 | . . . . . . 7 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘(ringLMod‘𝑅))) | |
30 | 5, 29 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘(ringLMod‘𝑅))) |
31 | 30 | fveq2d 6354 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
32 | 28, 31 | syl5eq 2804 | . . . 4 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
33 | 27, 32 | eleqtrd 2839 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘(ringLMod‘𝑅)))) |
34 | 8 | fveq2d 6354 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
35 | 3, 34 | syl5eq 2804 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
36 | 13, 20 | ressbasss 16132 | . . . . 5 ⊢ (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) ⊆ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) |
37 | 35, 36 | syl6eqss 3794 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
38 | 37, 1 | sseldd 3743 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
39 | 19, 20, 23, 14, 24, 25, 26, 6, 33, 38 | pwsvscafval 16354 | . 2 ⊢ (𝜑 → (𝐴( ·𝑠 ‘((ringLMod‘𝑅) ↑s 𝐼))𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) |
40 | 18, 39 | eqtrd 2792 | 1 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1630 ∈ wcel 2137 Vcvv 3338 {csn 4319 × cxp 5262 ‘cfv 6047 (class class class)co 6811 ∘𝑓 cof 7058 Basecbs 16057 ↾s cress 16058 .rcmulr 16142 Scalarcsca 16144 ·𝑠 cvsca 16145 ↑s cpws 16307 ringLModcrglmod 19369 freeLMod cfrlm 20290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-8 2139 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-rep 4921 ax-sep 4931 ax-nul 4939 ax-pow 4990 ax-pr 5053 ax-un 7112 ax-cnex 10182 ax-resscn 10183 ax-1cn 10184 ax-icn 10185 ax-addcl 10186 ax-addrcl 10187 ax-mulcl 10188 ax-mulrcl 10189 ax-mulcom 10190 ax-addass 10191 ax-mulass 10192 ax-distr 10193 ax-i2m1 10194 ax-1ne0 10195 ax-1rid 10196 ax-rnegex 10197 ax-rrecex 10198 ax-cnre 10199 ax-pre-lttri 10200 ax-pre-lttrn 10201 ax-pre-ltadd 10202 ax-pre-mulgt0 10203 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ne 2931 df-nel 3034 df-ral 3053 df-rex 3054 df-reu 3055 df-rab 3057 df-v 3340 df-sbc 3575 df-csb 3673 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-pss 3729 df-nul 4057 df-if 4229 df-pw 4302 df-sn 4320 df-pr 4322 df-tp 4324 df-op 4326 df-uni 4587 df-int 4626 df-iun 4672 df-br 4803 df-opab 4863 df-mpt 4880 df-tr 4903 df-id 5172 df-eprel 5177 df-po 5185 df-so 5186 df-fr 5223 df-we 5225 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-res 5276 df-ima 5277 df-pred 5839 df-ord 5885 df-on 5886 df-lim 5887 df-suc 5888 df-iota 6010 df-fun 6049 df-fn 6050 df-f 6051 df-f1 6052 df-fo 6053 df-f1o 6054 df-fv 6055 df-riota 6772 df-ov 6814 df-oprab 6815 df-mpt2 6816 df-of 7060 df-om 7229 df-1st 7331 df-2nd 7332 df-wrecs 7574 df-recs 7635 df-rdg 7673 df-1o 7727 df-oadd 7731 df-er 7909 df-map 8023 df-ixp 8073 df-en 8120 df-dom 8121 df-sdom 8122 df-fin 8123 df-sup 8511 df-pnf 10266 df-mnf 10267 df-xr 10268 df-ltxr 10269 df-le 10270 df-sub 10458 df-neg 10459 df-nn 11211 df-2 11269 df-3 11270 df-4 11271 df-5 11272 df-6 11273 df-7 11274 df-8 11275 df-9 11276 df-n0 11483 df-z 11568 df-dec 11684 df-uz 11878 df-fz 12518 df-struct 16059 df-ndx 16060 df-slot 16061 df-base 16063 df-sets 16064 df-ress 16065 df-plusg 16154 df-mulr 16155 df-sca 16157 df-vsca 16158 df-ip 16159 df-tset 16160 df-ple 16161 df-ds 16164 df-hom 16166 df-cco 16167 df-prds 16308 df-pws 16310 df-sra 19372 df-rgmod 19373 df-dsmm 20276 df-frlm 20291 |
This theorem is referenced by: frlmvscaval 20310 uvcresum 20332 matvsca2 20434 matunitlindflem1 33716 matunitlindflem2 33717 zlmodzxzscm 42643 aacllem 43058 |
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