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Mirrors > Home > HSE Home > Th. List > normpyc | Structured version Visualization version GIF version |
Description: Corollary to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
normpyc | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 → (normℎ‘𝐴) ≤ (normℎ‘(𝐴 +ℎ 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | normcl 31055 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ) | |
2 | 1 | resqcld 14138 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) ∈ ℝ) |
3 | 2 | recnd 11283 | . . . . . . . 8 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) ∈ ℂ) |
4 | 3 | addridd 11455 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → (((normℎ‘𝐴)↑2) + 0) = ((normℎ‘𝐴)↑2)) |
5 | 4 | adantr 479 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((normℎ‘𝐴)↑2) + 0) = ((normℎ‘𝐴)↑2)) |
6 | normcl 31055 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℋ → (normℎ‘𝐵) ∈ ℝ) | |
7 | 6 | sqge0d 14150 | . . . . . . . 8 ⊢ (𝐵 ∈ ℋ → 0 ≤ ((normℎ‘𝐵)↑2)) |
8 | 7 | adantl 480 | . . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → 0 ≤ ((normℎ‘𝐵)↑2)) |
9 | 6 | resqcld 14138 | . . . . . . . 8 ⊢ (𝐵 ∈ ℋ → ((normℎ‘𝐵)↑2) ∈ ℝ) |
10 | 0re 11257 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
11 | leadd2 11724 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ ((normℎ‘𝐵)↑2) ∈ ℝ ∧ ((normℎ‘𝐴)↑2) ∈ ℝ) → (0 ≤ ((normℎ‘𝐵)↑2) ↔ (((normℎ‘𝐴)↑2) + 0) ≤ (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2)))) | |
12 | 10, 11 | mp3an1 1445 | . . . . . . . 8 ⊢ ((((normℎ‘𝐵)↑2) ∈ ℝ ∧ ((normℎ‘𝐴)↑2) ∈ ℝ) → (0 ≤ ((normℎ‘𝐵)↑2) ↔ (((normℎ‘𝐴)↑2) + 0) ≤ (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2)))) |
13 | 9, 2, 12 | syl2anr 595 | . . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (0 ≤ ((normℎ‘𝐵)↑2) ↔ (((normℎ‘𝐴)↑2) + 0) ≤ (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2)))) |
14 | 8, 13 | mpbid 231 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((normℎ‘𝐴)↑2) + 0) ≤ (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2))) |
15 | 5, 14 | eqbrtrrd 5169 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((normℎ‘𝐴)↑2) ≤ (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2))) |
16 | 15 | adantr 479 | . . . 4 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐴 ·ih 𝐵) = 0) → ((normℎ‘𝐴)↑2) ≤ (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2))) |
17 | normpyth 31075 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 → ((normℎ‘(𝐴 +ℎ 𝐵))↑2) = (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2)))) | |
18 | 17 | imp 405 | . . . 4 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐴 ·ih 𝐵) = 0) → ((normℎ‘(𝐴 +ℎ 𝐵))↑2) = (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2))) |
19 | 16, 18 | breqtrrd 5173 | . . 3 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐴 ·ih 𝐵) = 0) → ((normℎ‘𝐴)↑2) ≤ ((normℎ‘(𝐴 +ℎ 𝐵))↑2)) |
20 | 19 | ex 411 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 → ((normℎ‘𝐴)↑2) ≤ ((normℎ‘(𝐴 +ℎ 𝐵))↑2))) |
21 | 1 | adantr 479 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (normℎ‘𝐴) ∈ ℝ) |
22 | hvaddcl 30942 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ) | |
23 | normcl 31055 | . . . 4 ⊢ ((𝐴 +ℎ 𝐵) ∈ ℋ → (normℎ‘(𝐴 +ℎ 𝐵)) ∈ ℝ) | |
24 | 22, 23 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (normℎ‘(𝐴 +ℎ 𝐵)) ∈ ℝ) |
25 | normge0 31056 | . . . 4 ⊢ (𝐴 ∈ ℋ → 0 ≤ (normℎ‘𝐴)) | |
26 | 25 | adantr 479 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → 0 ≤ (normℎ‘𝐴)) |
27 | normge0 31056 | . . . 4 ⊢ ((𝐴 +ℎ 𝐵) ∈ ℋ → 0 ≤ (normℎ‘(𝐴 +ℎ 𝐵))) | |
28 | 22, 27 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → 0 ≤ (normℎ‘(𝐴 +ℎ 𝐵))) |
29 | 21, 24, 26, 28 | le2sqd 14269 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((normℎ‘𝐴) ≤ (normℎ‘(𝐴 +ℎ 𝐵)) ↔ ((normℎ‘𝐴)↑2) ≤ ((normℎ‘(𝐴 +ℎ 𝐵))↑2))) |
30 | 20, 29 | sylibrd 258 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 → (normℎ‘𝐴) ≤ (normℎ‘(𝐴 +ℎ 𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 class class class wbr 5145 ‘cfv 6546 (class class class)co 7416 ℝcr 11148 0cc0 11149 + caddc 11152 ≤ cle 11290 2c2 12313 ↑cexp 14075 ℋchba 30849 +ℎ cva 30850 ·ih csp 30852 normℎcno 30853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-hfvadd 30930 ax-hv0cl 30933 ax-hvmul0 30940 ax-hfi 31009 ax-his1 31012 ax-his2 31013 ax-his3 31014 ax-his4 31015 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-sup 9478 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-n0 12519 df-z 12605 df-uz 12869 df-rp 13023 df-seq 14016 df-exp 14076 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-hnorm 30898 |
This theorem is referenced by: pjnormi 31651 |
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