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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 28904 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ) | |
2 | 1 | resqcld 13614 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) ∈ ℝ) |
3 | 2 | recnd 10671 | . . . . . . . 8 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) ∈ ℂ) |
4 | 3 | addid1d 10842 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → (((normℎ‘𝐴)↑2) + 0) = ((normℎ‘𝐴)↑2)) |
5 | 4 | adantr 483 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((normℎ‘𝐴)↑2) + 0) = ((normℎ‘𝐴)↑2)) |
6 | normcl 28904 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℋ → (normℎ‘𝐵) ∈ ℝ) | |
7 | 6 | sqge0d 13615 | . . . . . . . 8 ⊢ (𝐵 ∈ ℋ → 0 ≤ ((normℎ‘𝐵)↑2)) |
8 | 7 | adantl 484 | . . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → 0 ≤ ((normℎ‘𝐵)↑2)) |
9 | 6 | resqcld 13614 | . . . . . . . 8 ⊢ (𝐵 ∈ ℋ → ((normℎ‘𝐵)↑2) ∈ ℝ) |
10 | 0re 10645 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
11 | leadd2 11111 | . . . . . . . . 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 1444 | . . . . . . . 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 598 | . . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (0 ≤ ((normℎ‘𝐵)↑2) ↔ (((normℎ‘𝐴)↑2) + 0) ≤ (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2)))) |
14 | 8, 13 | mpbid 234 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((normℎ‘𝐴)↑2) + 0) ≤ (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2))) |
15 | 5, 14 | eqbrtrrd 5092 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((normℎ‘𝐴)↑2) ≤ (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2))) |
16 | 15 | adantr 483 | . . . 4 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐴 ·ih 𝐵) = 0) → ((normℎ‘𝐴)↑2) ≤ (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2))) |
17 | normpyth 28924 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 → ((normℎ‘(𝐴 +ℎ 𝐵))↑2) = (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2)))) | |
18 | 17 | imp 409 | . . . 4 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐴 ·ih 𝐵) = 0) → ((normℎ‘(𝐴 +ℎ 𝐵))↑2) = (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2))) |
19 | 16, 18 | breqtrrd 5096 | . . 3 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐴 ·ih 𝐵) = 0) → ((normℎ‘𝐴)↑2) ≤ ((normℎ‘(𝐴 +ℎ 𝐵))↑2)) |
20 | 19 | ex 415 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 → ((normℎ‘𝐴)↑2) ≤ ((normℎ‘(𝐴 +ℎ 𝐵))↑2))) |
21 | 1 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (normℎ‘𝐴) ∈ ℝ) |
22 | hvaddcl 28791 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ) | |
23 | normcl 28904 | . . . 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 28905 | . . . 4 ⊢ (𝐴 ∈ ℋ → 0 ≤ (normℎ‘𝐴)) | |
26 | 25 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → 0 ≤ (normℎ‘𝐴)) |
27 | normge0 28905 | . . . 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 13623 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((normℎ‘𝐴) ≤ (normℎ‘(𝐴 +ℎ 𝐵)) ↔ ((normℎ‘𝐴)↑2) ≤ ((normℎ‘(𝐴 +ℎ 𝐵))↑2))) |
30 | 20, 29 | sylibrd 261 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 + caddc 10542 ≤ cle 10678 2c2 11695 ↑cexp 13432 ℋchba 28698 +ℎ cva 28699 ·ih csp 28701 normℎcno 28702 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-hfvadd 28779 ax-hv0cl 28782 ax-hvmul0 28789 ax-hfi 28858 ax-his1 28861 ax-his2 28862 ax-his3 28863 ax-his4 28864 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-hnorm 28747 |
This theorem is referenced by: pjnormi 29500 |
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