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Mirrors > Home > MPE Home > Th. List > uvcval | Structured version Visualization version GIF version |
Description: Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
Ref | Expression |
---|---|
uvcfval.u | ⊢ 𝑈 = (𝑅 unitVec 𝐼) |
uvcfval.o | ⊢ 1 = (1r‘𝑅) |
uvcfval.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
uvcval | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uvcfval.u | . . . . 5 ⊢ 𝑈 = (𝑅 unitVec 𝐼) | |
2 | uvcfval.o | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
3 | uvcfval.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
4 | 1, 2, 3 | uvcfval 20123 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑈 = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))) |
5 | 4 | fveq1d 6193 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑈‘𝐽) = ((𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽)) |
6 | 5 | 3adant3 1081 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) = ((𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽)) |
7 | simp3 1063 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 𝐽 ∈ 𝐼) | |
8 | mptexg 6484 | . . . 4 ⊢ (𝐼 ∈ 𝑊 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V) | |
9 | 8 | 3ad2ant2 1083 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V) |
10 | eqeq2 2633 | . . . . . 6 ⊢ (𝑗 = 𝐽 → (𝑘 = 𝑗 ↔ 𝑘 = 𝐽)) | |
11 | 10 | ifbid 4108 | . . . . 5 ⊢ (𝑗 = 𝐽 → if(𝑘 = 𝑗, 1 , 0 ) = if(𝑘 = 𝐽, 1 , 0 )) |
12 | 11 | mpteq2dv 4745 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))) |
13 | eqid 2622 | . . . 4 ⊢ (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) | |
14 | 12, 13 | fvmptg 6280 | . . 3 ⊢ ((𝐽 ∈ 𝐼 ∧ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V) → ((𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))) |
15 | 7, 9, 14 | syl2anc 693 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))) |
16 | 6, 15 | eqtrd 2656 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ifcif 4086 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 0gc0g 16100 1rcur 18501 unitVec cuvc 20121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-uvc 20122 |
This theorem is referenced by: uvcvval 20125 |
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