Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > lcoval | Structured version Visualization version GIF version |
Description: The value of a linear combination. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.) |
Ref | Expression |
---|---|
lcoop.b | ⊢ 𝐵 = (Base‘𝑀) |
lcoop.s | ⊢ 𝑆 = (Scalar‘𝑀) |
lcoop.r | ⊢ 𝑅 = (Base‘𝑆) |
Ref | Expression |
---|---|
lcoval | ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ (𝐶 ∈ 𝐵 ∧ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcoop.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
2 | lcoop.s | . . . 4 ⊢ 𝑆 = (Scalar‘𝑀) | |
3 | lcoop.r | . . . 4 ⊢ 𝑅 = (Base‘𝑆) | |
4 | 1, 2, 3 | lcoop 44394 | . . 3 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))}) |
5 | 4 | eleq2d 2895 | . 2 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ 𝐶 ∈ {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})) |
6 | eqeq1 2822 | . . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐 = (𝑠( linC ‘𝑀)𝑉) ↔ 𝐶 = (𝑠( linC ‘𝑀)𝑉))) | |
7 | 6 | anbi2d 628 | . . . 4 ⊢ (𝑐 = 𝐶 → ((𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉)) ↔ (𝑠 finSupp (0g‘𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))) |
8 | 7 | rexbidv 3294 | . . 3 ⊢ (𝑐 = 𝐶 → (∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉)) ↔ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))) |
9 | 8 | elrab 3677 | . 2 ⊢ (𝐶 ∈ {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))} ↔ (𝐶 ∈ 𝐵 ∧ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))) |
10 | 5, 9 | syl6bb 288 | 1 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ (𝐶 ∈ 𝐵 ∧ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 {crab 3139 𝒫 cpw 4535 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 finSupp cfsupp 8821 Basecbs 16471 Scalarcsca 16556 0gc0g 16701 linC clinc 44387 LinCo clinco 44388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-lco 44390 |
This theorem is referenced by: lcoel0 44411 lincsumcl 44414 lincscmcl 44415 lincolss 44417 ellcoellss 44418 lcoss 44419 lindslinindsimp1 44440 lindslinindsimp2 44446 |
Copyright terms: Public domain | W3C validator |