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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 15208 | . . . . . . . 8 ⊢ (∗‘𝑆) ∈ ℂ |
| 4 | normlem1.2 | . . . . . . . . 9 ⊢ 𝐹 ∈ ℋ | |
| 5 | normlem1.3 | . . . . . . . . 9 ⊢ 𝐺 ∈ ℋ | |
| 6 | 4, 5 | hicli 31100 | . . . . . . . 8 ⊢ (𝐹 ·ih 𝐺) ∈ ℂ |
| 7 | 3, 6 | mulcli 11268 | . . . . . . 7 ⊢ ((∗‘𝑆) · (𝐹 ·ih 𝐺)) ∈ ℂ |
| 8 | 5, 4 | hicli 31100 | . . . . . . . 8 ⊢ (𝐺 ·ih 𝐹) ∈ ℂ |
| 9 | 2, 8 | mulcli 11268 | . . . . . . 7 ⊢ (𝑆 · (𝐺 ·ih 𝐹)) ∈ ℂ |
| 10 | 7, 9 | cjaddi 15227 | . . . . . 6 ⊢ (∗‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) = ((∗‘((∗‘𝑆) · (𝐹 ·ih 𝐺))) + (∗‘(𝑆 · (𝐺 ·ih 𝐹)))) |
| 11 | 2 | cjcji 15210 | . . . . . . . . . 10 ⊢ (∗‘(∗‘𝑆)) = 𝑆 |
| 12 | 11 | eqcomi 2746 | . . . . . . . . 9 ⊢ 𝑆 = (∗‘(∗‘𝑆)) |
| 13 | 5, 4 | his1i 31119 | . . . . . . . . 9 ⊢ (𝐺 ·ih 𝐹) = (∗‘(𝐹 ·ih 𝐺)) |
| 14 | 12, 13 | oveq12i 7443 | . . . . . . . 8 ⊢ (𝑆 · (𝐺 ·ih 𝐹)) = ((∗‘(∗‘𝑆)) · (∗‘(𝐹 ·ih 𝐺))) |
| 15 | 3, 6 | cjmuli 15228 | . . . . . . . 8 ⊢ (∗‘((∗‘𝑆) · (𝐹 ·ih 𝐺))) = ((∗‘(∗‘𝑆)) · (∗‘(𝐹 ·ih 𝐺))) |
| 16 | 14, 15 | eqtr4i 2768 | . . . . . . 7 ⊢ (𝑆 · (𝐺 ·ih 𝐹)) = (∗‘((∗‘𝑆) · (𝐹 ·ih 𝐺))) |
| 17 | 4, 5 | his1i 31119 | . . . . . . . . 9 ⊢ (𝐹 ·ih 𝐺) = (∗‘(𝐺 ·ih 𝐹)) |
| 18 | 17 | oveq2i 7442 | . . . . . . . 8 ⊢ ((∗‘𝑆) · (𝐹 ·ih 𝐺)) = ((∗‘𝑆) · (∗‘(𝐺 ·ih 𝐹))) |
| 19 | 2, 8 | cjmuli 15228 | . . . . . . . 8 ⊢ (∗‘(𝑆 · (𝐺 ·ih 𝐹))) = ((∗‘𝑆) · (∗‘(𝐺 ·ih 𝐹))) |
| 20 | 18, 19 | eqtr4i 2768 | . . . . . . 7 ⊢ ((∗‘𝑆) · (𝐹 ·ih 𝐺)) = (∗‘(𝑆 · (𝐺 ·ih 𝐹))) |
| 21 | 16, 20 | oveq12i 7443 | . . . . . 6 ⊢ ((𝑆 · (𝐺 ·ih 𝐹)) + ((∗‘𝑆) · (𝐹 ·ih 𝐺))) = ((∗‘((∗‘𝑆) · (𝐹 ·ih 𝐺))) + (∗‘(𝑆 · (𝐺 ·ih 𝐹)))) |
| 22 | 10, 21 | eqtr4i 2768 | . . . . 5 ⊢ (∗‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) = ((𝑆 · (𝐺 ·ih 𝐹)) + ((∗‘𝑆) · (𝐹 ·ih 𝐺))) |
| 23 | 7, 9 | addcomi 11452 | . . . . 5 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) = ((𝑆 · (𝐺 ·ih 𝐹)) + ((∗‘𝑆) · (𝐹 ·ih 𝐺))) |
| 24 | 22, 23 | eqtr4i 2768 | . . . 4 ⊢ (∗‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) = (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) |
| 25 | 7, 9 | addcli 11267 | . . . . 5 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℂ |
| 26 | 25 | cjrebi 15213 | . . . 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 11570 | . 2 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ |
| 29 | 1, 28 | eqeltri 2837 | 1 ⊢ 𝐵 ∈ ℝ |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 + caddc 11158 · cmul 11160 -cneg 11493 ∗ccj 15135 ℋchba 30938 ·ih csp 30941 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-hfi 31098 ax-his1 31101 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-2 12329 df-cj 15138 df-re 15139 df-im 15140 |
| This theorem is referenced by: normlem3 31131 normlem6 31134 normlem7 31135 norm-ii-i 31156 |
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