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Mirrors > Home > MPE Home > Th. List > Mathboxes > lflsc0N | Structured version Visualization version GIF version |
Description: The scalar product with the zero functional is the zero functional. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lflsc0.v | ⊢ 𝑉 = (Base‘𝑊) |
lflsc0.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lflsc0.k | ⊢ 𝐾 = (Base‘𝐷) |
lflsc0.t | ⊢ · = (.r‘𝐷) |
lflsc0.o | ⊢ 0 = (0g‘𝐷) |
lflsc0.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lflsc0.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
Ref | Expression |
---|---|
lflsc0N | ⊢ (𝜑 → ((𝑉 × { 0 }) ∘f · (𝑉 × {𝑋})) = (𝑉 × { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lflsc0.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
2 | 1 | fvexi 6684 | . . . 4 ⊢ 𝑉 ∈ V |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑉 ∈ V) |
4 | lflsc0.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
5 | lflsc0.d | . . . . . 6 ⊢ 𝐷 = (Scalar‘𝑊) | |
6 | 5 | lmodring 19642 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝐷 ∈ Ring) |
7 | 4, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Ring) |
8 | lflsc0.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐷) | |
9 | lflsc0.o | . . . . 5 ⊢ 0 = (0g‘𝐷) | |
10 | 8, 9 | ring0cl 19319 | . . . 4 ⊢ (𝐷 ∈ Ring → 0 ∈ 𝐾) |
11 | 7, 10 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ 𝐾) |
12 | lflsc0.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
13 | 3, 11, 12 | ofc12 7434 | . 2 ⊢ (𝜑 → ((𝑉 × { 0 }) ∘f · (𝑉 × {𝑋})) = (𝑉 × {( 0 · 𝑋)})) |
14 | lflsc0.t | . . . . . 6 ⊢ · = (.r‘𝐷) | |
15 | 8, 14, 9 | ringlz 19337 | . . . . 5 ⊢ ((𝐷 ∈ Ring ∧ 𝑋 ∈ 𝐾) → ( 0 · 𝑋) = 0 ) |
16 | 7, 12, 15 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ( 0 · 𝑋) = 0 ) |
17 | 16 | sneqd 4579 | . . 3 ⊢ (𝜑 → {( 0 · 𝑋)} = { 0 }) |
18 | 17 | xpeq2d 5585 | . 2 ⊢ (𝜑 → (𝑉 × {( 0 · 𝑋)}) = (𝑉 × { 0 })) |
19 | 13, 18 | eqtrd 2856 | 1 ⊢ (𝜑 → ((𝑉 × { 0 }) ∘f · (𝑉 × {𝑋})) = (𝑉 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 × cxp 5553 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 Basecbs 16483 .rcmulr 16566 Scalarcsca 16568 0gc0g 16713 Ringcrg 19297 LModclmod 19634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-mgp 19240 df-ring 19299 df-lmod 19636 |
This theorem is referenced by: (None) |
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