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Mathbox for Norm Megill |
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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 6447 | . . . 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 19227 | . . . . 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 18923 | . . . 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 7182 | . 2 ⊢ (𝜑 → ((𝑉 × { 0 }) ∘𝑓 · (𝑉 × {𝑋})) = (𝑉 × {( 0 · 𝑋)})) |
14 | lflsc0.t | . . . . . 6 ⊢ · = (.r‘𝐷) | |
15 | 8, 14, 9 | ringlz 18941 | . . . . 5 ⊢ ((𝐷 ∈ Ring ∧ 𝑋 ∈ 𝐾) → ( 0 · 𝑋) = 0 ) |
16 | 7, 12, 15 | syl2anc 581 | . . . 4 ⊢ (𝜑 → ( 0 · 𝑋) = 0 ) |
17 | 16 | sneqd 4409 | . . 3 ⊢ (𝜑 → {( 0 · 𝑋)} = { 0 }) |
18 | 17 | xpeq2d 5372 | . 2 ⊢ (𝜑 → (𝑉 × {( 0 · 𝑋)}) = (𝑉 × { 0 })) |
19 | 13, 18 | eqtrd 2861 | 1 ⊢ (𝜑 → ((𝑉 × { 0 }) ∘𝑓 · (𝑉 × {𝑋})) = (𝑉 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 Vcvv 3414 {csn 4397 × cxp 5340 ‘cfv 6123 (class class class)co 6905 ∘𝑓 cof 7155 Basecbs 16222 .rcmulr 16306 Scalarcsca 16308 0gc0g 16453 Ringcrg 18901 LModclmod 19219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-plusg 16318 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-minusg 17780 df-mgp 18844 df-ring 18903 df-lmod 19221 |
This theorem is referenced by: (None) |
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