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Mirrors > Home > HSE Home > Th. List > normlem2 | 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, 27-Jul-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
normlem1.1 | ⊢ 𝑆 ∈ ℂ |
normlem1.2 | ⊢ 𝐹 ∈ ℋ |
normlem1.3 | ⊢ 𝐺 ∈ ℋ |
normlem2.4 | ⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) |
Ref | Expression |
---|---|
normlem2 | ⊢ 𝐵 ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | normlem2.4 | . 2 ⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) | |
2 | normlem1.1 | . . . . . . . . 9 ⊢ 𝑆 ∈ ℂ | |
3 | 2 | cjcli 15204 | . . . . . . . 8 ⊢ (∗‘𝑆) ∈ ℂ |
4 | normlem1.2 | . . . . . . . . 9 ⊢ 𝐹 ∈ ℋ | |
5 | normlem1.3 | . . . . . . . . 9 ⊢ 𝐺 ∈ ℋ | |
6 | 4, 5 | hicli 31109 | . . . . . . . 8 ⊢ (𝐹 ·ih 𝐺) ∈ ℂ |
7 | 3, 6 | mulcli 11265 | . . . . . . 7 ⊢ ((∗‘𝑆) · (𝐹 ·ih 𝐺)) ∈ ℂ |
8 | 5, 4 | hicli 31109 | . . . . . . . 8 ⊢ (𝐺 ·ih 𝐹) ∈ ℂ |
9 | 2, 8 | mulcli 11265 | . . . . . . 7 ⊢ (𝑆 · (𝐺 ·ih 𝐹)) ∈ ℂ |
10 | 7, 9 | cjaddi 15223 | . . . . . 6 ⊢ (∗‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) = ((∗‘((∗‘𝑆) · (𝐹 ·ih 𝐺))) + (∗‘(𝑆 · (𝐺 ·ih 𝐹)))) |
11 | 2 | cjcji 15206 | . . . . . . . . . 10 ⊢ (∗‘(∗‘𝑆)) = 𝑆 |
12 | 11 | eqcomi 2743 | . . . . . . . . 9 ⊢ 𝑆 = (∗‘(∗‘𝑆)) |
13 | 5, 4 | his1i 31128 | . . . . . . . . 9 ⊢ (𝐺 ·ih 𝐹) = (∗‘(𝐹 ·ih 𝐺)) |
14 | 12, 13 | oveq12i 7442 | . . . . . . . 8 ⊢ (𝑆 · (𝐺 ·ih 𝐹)) = ((∗‘(∗‘𝑆)) · (∗‘(𝐹 ·ih 𝐺))) |
15 | 3, 6 | cjmuli 15224 | . . . . . . . 8 ⊢ (∗‘((∗‘𝑆) · (𝐹 ·ih 𝐺))) = ((∗‘(∗‘𝑆)) · (∗‘(𝐹 ·ih 𝐺))) |
16 | 14, 15 | eqtr4i 2765 | . . . . . . 7 ⊢ (𝑆 · (𝐺 ·ih 𝐹)) = (∗‘((∗‘𝑆) · (𝐹 ·ih 𝐺))) |
17 | 4, 5 | his1i 31128 | . . . . . . . . 9 ⊢ (𝐹 ·ih 𝐺) = (∗‘(𝐺 ·ih 𝐹)) |
18 | 17 | oveq2i 7441 | . . . . . . . 8 ⊢ ((∗‘𝑆) · (𝐹 ·ih 𝐺)) = ((∗‘𝑆) · (∗‘(𝐺 ·ih 𝐹))) |
19 | 2, 8 | cjmuli 15224 | . . . . . . . 8 ⊢ (∗‘(𝑆 · (𝐺 ·ih 𝐹))) = ((∗‘𝑆) · (∗‘(𝐺 ·ih 𝐹))) |
20 | 18, 19 | eqtr4i 2765 | . . . . . . 7 ⊢ ((∗‘𝑆) · (𝐹 ·ih 𝐺)) = (∗‘(𝑆 · (𝐺 ·ih 𝐹))) |
21 | 16, 20 | oveq12i 7442 | . . . . . 6 ⊢ ((𝑆 · (𝐺 ·ih 𝐹)) + ((∗‘𝑆) · (𝐹 ·ih 𝐺))) = ((∗‘((∗‘𝑆) · (𝐹 ·ih 𝐺))) + (∗‘(𝑆 · (𝐺 ·ih 𝐹)))) |
22 | 10, 21 | eqtr4i 2765 | . . . . 5 ⊢ (∗‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) = ((𝑆 · (𝐺 ·ih 𝐹)) + ((∗‘𝑆) · (𝐹 ·ih 𝐺))) |
23 | 7, 9 | addcomi 11449 | . . . . 5 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) = ((𝑆 · (𝐺 ·ih 𝐹)) + ((∗‘𝑆) · (𝐹 ·ih 𝐺))) |
24 | 22, 23 | eqtr4i 2765 | . . . 4 ⊢ (∗‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) = (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) |
25 | 7, 9 | addcli 11264 | . . . . 5 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℂ |
26 | 25 | cjrebi 15209 | . . . 4 ⊢ ((((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ ↔ (∗‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) = (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
27 | 24, 26 | mpbir 231 | . . 3 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ |
28 | 27 | renegcli 11567 | . 2 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ |
29 | 1, 28 | eqeltri 2834 | 1 ⊢ 𝐵 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 + caddc 11155 · cmul 11157 -cneg 11490 ∗ccj 15131 ℋchba 30947 ·ih csp 30950 |
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-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-hfi 31107 ax-his1 31110 |
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-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-id 5582 df-po 5596 df-so 5597 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-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-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 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-2 12326 df-cj 15134 df-re 15135 df-im 15136 |
This theorem is referenced by: normlem3 31140 normlem6 31143 normlem7 31144 norm-ii-i 31165 |
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