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| Mirrors > Home > MPE Home > Th. List > rrxvsca | Structured version Visualization version GIF version | ||
| Description: The scalar product over generalized Euclidean spaces is the componentwise real number multiplication. (Contributed by Thierry Arnoux, 18-Jan-2023.) |
| Ref | Expression |
|---|---|
| rrxval.r | ⊢ 𝐻 = (ℝ^‘𝐼) |
| rrxbase.b | ⊢ 𝐵 = (Base‘𝐻) |
| rrxvsca.r | ⊢ ∙ = ( ·𝑠 ‘𝐻) |
| rrxvsca.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
| rrxvsca.j | ⊢ (𝜑 → 𝐽 ∈ 𝐼) |
| rrxvsca.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| rrxvsca.x | ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻)) |
| Ref | Expression |
|---|---|
| rrxvsca | ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rrxvsca.r | . . . . 5 ⊢ ∙ = ( ·𝑠 ‘𝐻) | |
| 2 | rrxvsca.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
| 3 | rrxval.r | . . . . . . . 8 ⊢ 𝐻 = (ℝ^‘𝐼) | |
| 4 | 3 | rrxval 25363 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
| 5 | 2, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
| 6 | 5 | fveq2d 6836 | . . . . 5 ⊢ (𝜑 → ( ·𝑠 ‘𝐻) = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
| 7 | 1, 6 | eqtrid 2784 | . . . 4 ⊢ (𝜑 → ∙ = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
| 8 | 7 | oveqd 7375 | . . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑋)) |
| 9 | 8 | fveq1d 6834 | . 2 ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = ((𝐴( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑋)‘𝐽)) |
| 10 | eqid 2737 | . . 3 ⊢ (ℝfld freeLMod 𝐼) = (ℝfld freeLMod 𝐼) | |
| 11 | eqid 2737 | . . 3 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(ℝfld freeLMod 𝐼)) | |
| 12 | rebase 21594 | . . 3 ⊢ ℝ = (Base‘ℝfld) | |
| 13 | rrxvsca.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 14 | rrxvsca.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻)) | |
| 15 | 5 | fveq2d 6836 | . . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
| 16 | eqid 2737 | . . . . . 6 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)) | |
| 17 | 16, 11 | tcphbas 25195 | . . . . 5 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
| 18 | 15, 17 | eqtr4di 2790 | . . . 4 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(ℝfld freeLMod 𝐼))) |
| 19 | 14, 18 | eleqtrd 2839 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
| 20 | rrxvsca.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼) | |
| 21 | eqid 2737 | . . . . 5 ⊢ ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) = ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) | |
| 22 | 16, 21 | tcphvsca 25200 | . . . 4 ⊢ ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
| 23 | 22 | eqcomi 2746 | . . 3 ⊢ ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) = ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) |
| 24 | remulr 21599 | . . 3 ⊢ · = (.r‘ℝfld) | |
| 25 | 10, 11, 12, 2, 13, 19, 20, 23, 24 | frlmvscaval 21756 | . 2 ⊢ (𝜑 → ((𝐴( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) |
| 26 | 9, 25 | eqtrd 2772 | 1 ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ‘cfv 6490 (class class class)co 7358 ℝcr 11026 · cmul 11032 Basecbs 17168 ·𝑠 cvsca 17213 ℝfldcrefld 21592 freeLMod cfrlm 21734 toℂPreHilctcph 25143 ℝ^crrx 25359 |
| 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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-mulf 11107 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-er 8634 df-map 8766 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-sup 9346 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-rp 12932 df-fz 13451 df-seq 13953 df-exp 14013 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-0g 17393 df-prds 17399 df-pws 17401 df-sra 21158 df-rgmod 21159 df-cnfld 21343 df-refld 21593 df-dsmm 21720 df-frlm 21735 df-tng 24558 df-tcph 25145 df-rrx 25361 |
| This theorem is referenced by: (None) |
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