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Mirrors > Home > HSE Home > Th. List > normlem1 | Structured version Visualization version GIF version |
Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 22-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
normlem1.1 | ⊢ 𝑆 ∈ ℂ |
normlem1.2 | ⊢ 𝐹 ∈ ℋ |
normlem1.3 | ⊢ 𝐺 ∈ ℋ |
normlem1.4 | ⊢ 𝑅 ∈ ℝ |
normlem1.5 | ⊢ (abs‘𝑆) = 1 |
Ref | Expression |
---|---|
normlem1 | ⊢ ((𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ·ih (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺))) = (((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) + (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | normlem1.1 | . . . 4 ⊢ 𝑆 ∈ ℂ | |
2 | normlem1.4 | . . . . 5 ⊢ 𝑅 ∈ ℝ | |
3 | 2 | recni 11272 | . . . 4 ⊢ 𝑅 ∈ ℂ |
4 | 1, 3 | mulcli 11265 | . . 3 ⊢ (𝑆 · 𝑅) ∈ ℂ |
5 | normlem1.2 | . . 3 ⊢ 𝐹 ∈ ℋ | |
6 | normlem1.3 | . . 3 ⊢ 𝐺 ∈ ℋ | |
7 | 4, 5, 6 | normlem0 31137 | . 2 ⊢ ((𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ·ih (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺))) = (((𝐹 ·ih 𝐹) + (-(∗‘(𝑆 · 𝑅)) · (𝐹 ·ih 𝐺))) + ((-(𝑆 · 𝑅) · (𝐺 ·ih 𝐹)) + (((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) · (𝐺 ·ih 𝐺)))) |
8 | 1, 3 | cjmuli 15224 | . . . . . . . 8 ⊢ (∗‘(𝑆 · 𝑅)) = ((∗‘𝑆) · (∗‘𝑅)) |
9 | 3 | cjrebi 15209 | . . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ ↔ (∗‘𝑅) = 𝑅) |
10 | 2, 9 | mpbi 230 | . . . . . . . . 9 ⊢ (∗‘𝑅) = 𝑅 |
11 | 10 | oveq2i 7441 | . . . . . . . 8 ⊢ ((∗‘𝑆) · (∗‘𝑅)) = ((∗‘𝑆) · 𝑅) |
12 | 8, 11 | eqtri 2762 | . . . . . . 7 ⊢ (∗‘(𝑆 · 𝑅)) = ((∗‘𝑆) · 𝑅) |
13 | 12 | negeqi 11498 | . . . . . 6 ⊢ -(∗‘(𝑆 · 𝑅)) = -((∗‘𝑆) · 𝑅) |
14 | 1 | cjcli 15204 | . . . . . . 7 ⊢ (∗‘𝑆) ∈ ℂ |
15 | 14, 3 | mulneg2i 11707 | . . . . . 6 ⊢ ((∗‘𝑆) · -𝑅) = -((∗‘𝑆) · 𝑅) |
16 | 13, 15 | eqtr4i 2765 | . . . . 5 ⊢ -(∗‘(𝑆 · 𝑅)) = ((∗‘𝑆) · -𝑅) |
17 | 16 | oveq1i 7440 | . . . 4 ⊢ (-(∗‘(𝑆 · 𝑅)) · (𝐹 ·ih 𝐺)) = (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺)) |
18 | 17 | oveq2i 7441 | . . 3 ⊢ ((𝐹 ·ih 𝐹) + (-(∗‘(𝑆 · 𝑅)) · (𝐹 ·ih 𝐺))) = ((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) |
19 | 1, 3 | mulneg2i 11707 | . . . . . 6 ⊢ (𝑆 · -𝑅) = -(𝑆 · 𝑅) |
20 | 19 | eqcomi 2743 | . . . . 5 ⊢ -(𝑆 · 𝑅) = (𝑆 · -𝑅) |
21 | 20 | oveq1i 7440 | . . . 4 ⊢ (-(𝑆 · 𝑅) · (𝐺 ·ih 𝐹)) = ((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) |
22 | 8 | oveq2i 7441 | . . . . . . 7 ⊢ ((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) = ((𝑆 · 𝑅) · ((∗‘𝑆) · (∗‘𝑅))) |
23 | 3 | cjcli 15204 | . . . . . . . . 9 ⊢ (∗‘𝑅) ∈ ℂ |
24 | 1, 3, 14, 23 | mul4i 11455 | . . . . . . . 8 ⊢ ((𝑆 · 𝑅) · ((∗‘𝑆) · (∗‘𝑅))) = ((𝑆 · (∗‘𝑆)) · (𝑅 · (∗‘𝑅))) |
25 | normlem1.5 | . . . . . . . . . . . 12 ⊢ (abs‘𝑆) = 1 | |
26 | 25 | oveq1i 7440 | . . . . . . . . . . 11 ⊢ ((abs‘𝑆)↑2) = (1↑2) |
27 | 1 | absvalsqi 15428 | . . . . . . . . . . 11 ⊢ ((abs‘𝑆)↑2) = (𝑆 · (∗‘𝑆)) |
28 | sq1 14230 | . . . . . . . . . . 11 ⊢ (1↑2) = 1 | |
29 | 26, 27, 28 | 3eqtr3i 2770 | . . . . . . . . . 10 ⊢ (𝑆 · (∗‘𝑆)) = 1 |
30 | 10 | oveq2i 7441 | . . . . . . . . . 10 ⊢ (𝑅 · (∗‘𝑅)) = (𝑅 · 𝑅) |
31 | 29, 30 | oveq12i 7442 | . . . . . . . . 9 ⊢ ((𝑆 · (∗‘𝑆)) · (𝑅 · (∗‘𝑅))) = (1 · (𝑅 · 𝑅)) |
32 | 3, 3 | mulcli 11265 | . . . . . . . . . 10 ⊢ (𝑅 · 𝑅) ∈ ℂ |
33 | 32 | mullidi 11263 | . . . . . . . . 9 ⊢ (1 · (𝑅 · 𝑅)) = (𝑅 · 𝑅) |
34 | 31, 33 | eqtri 2762 | . . . . . . . 8 ⊢ ((𝑆 · (∗‘𝑆)) · (𝑅 · (∗‘𝑅))) = (𝑅 · 𝑅) |
35 | 24, 34 | eqtri 2762 | . . . . . . 7 ⊢ ((𝑆 · 𝑅) · ((∗‘𝑆) · (∗‘𝑅))) = (𝑅 · 𝑅) |
36 | 22, 35 | eqtri 2762 | . . . . . 6 ⊢ ((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) = (𝑅 · 𝑅) |
37 | 3 | sqvali 14215 | . . . . . 6 ⊢ (𝑅↑2) = (𝑅 · 𝑅) |
38 | 36, 37 | eqtr4i 2765 | . . . . 5 ⊢ ((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) = (𝑅↑2) |
39 | 38 | oveq1i 7440 | . . . 4 ⊢ (((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) · (𝐺 ·ih 𝐺)) = ((𝑅↑2) · (𝐺 ·ih 𝐺)) |
40 | 21, 39 | oveq12i 7442 | . . 3 ⊢ ((-(𝑆 · 𝑅) · (𝐺 ·ih 𝐹)) + (((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) · (𝐺 ·ih 𝐺))) = (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺))) |
41 | 18, 40 | oveq12i 7442 | . 2 ⊢ (((𝐹 ·ih 𝐹) + (-(∗‘(𝑆 · 𝑅)) · (𝐹 ·ih 𝐺))) + ((-(𝑆 · 𝑅) · (𝐺 ·ih 𝐹)) + (((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) · (𝐺 ·ih 𝐺)))) = (((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) + (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺)))) |
42 | 7, 41 | eqtri 2762 | 1 ⊢ ((𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ·ih (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺))) = (((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) + (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺)))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 1c1 11153 + caddc 11155 · cmul 11157 -cneg 11490 2c2 12318 ↑cexp 14098 ∗ccj 15131 abscabs 15269 ℋchba 30947 ·ℎ csm 30949 ·ih csp 30950 −ℎ cmv 30953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-hfvadd 31028 ax-hfvmul 31033 ax-hvmulass 31035 ax-hfi 31107 ax-his1 31110 ax-his2 31111 ax-his3 31112 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-hvsub 30999 |
This theorem is referenced by: normlem4 31141 |
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