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Mirrors > Home > MPE Home > Th. List > dip0l | Structured version Visualization version GIF version |
Description: Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dip0r.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
dip0r.5 | ⊢ 𝑍 = (0vec‘𝑈) |
dip0r.7 | ⊢ 𝑃 = (·𝑖OLD‘𝑈) |
Ref | Expression |
---|---|
dip0l | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑍𝑃𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dip0r.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | dip0r.5 | . . . . 5 ⊢ 𝑍 = (0vec‘𝑈) | |
3 | 1, 2 | nvzcl 28975 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) |
4 | 3 | adantr 480 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋) |
5 | dip0r.7 | . . . 4 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
6 | 1, 5 | dipcj 29055 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (∗‘(𝐴𝑃𝑍)) = (𝑍𝑃𝐴)) |
7 | 4, 6 | mpd3an3 1460 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (∗‘(𝐴𝑃𝑍)) = (𝑍𝑃𝐴)) |
8 | 1, 2, 5 | dip0r 29058 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝑍) = 0) |
9 | 8 | fveq2d 6772 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (∗‘(𝐴𝑃𝑍)) = (∗‘0)) |
10 | cj0 14850 | . . 3 ⊢ (∗‘0) = 0 | |
11 | 9, 10 | eqtrdi 2795 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (∗‘(𝐴𝑃𝑍)) = 0) |
12 | 7, 11 | eqtr3d 2781 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑍𝑃𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ‘cfv 6430 (class class class)co 7268 0cc0 10855 ∗ccj 14788 NrmCVeccnv 28925 BaseSetcba 28927 0veccn0v 28929 ·𝑖OLDcdip 29041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-inf2 9360 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-sup 9162 df-oi 9230 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-n0 12217 df-z 12303 df-uz 12565 df-rp 12713 df-fz 13222 df-fzo 13365 df-seq 13703 df-exp 13764 df-hash 14026 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-clim 15178 df-sum 15379 df-grpo 28834 df-gid 28835 df-ginv 28836 df-ablo 28886 df-vc 28900 df-nv 28933 df-va 28936 df-ba 28937 df-sm 28938 df-0v 28939 df-nmcv 28941 df-dip 29042 |
This theorem is referenced by: ip2i 29169 ipasslem1 29172 ipasslem2 29173 |
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