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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 14531 | . . . . . . . 8 ⊢ (∗‘𝑆) ∈ ℂ |
4 | normlem1.2 | . . . . . . . . 9 ⊢ 𝐹 ∈ ℋ | |
5 | normlem1.3 | . . . . . . . . 9 ⊢ 𝐺 ∈ ℋ | |
6 | 4, 5 | hicli 28861 | . . . . . . . 8 ⊢ (𝐹 ·ih 𝐺) ∈ ℂ |
7 | 3, 6 | mulcli 10651 | . . . . . . 7 ⊢ ((∗‘𝑆) · (𝐹 ·ih 𝐺)) ∈ ℂ |
8 | 5, 4 | hicli 28861 | . . . . . . . 8 ⊢ (𝐺 ·ih 𝐹) ∈ ℂ |
9 | 2, 8 | mulcli 10651 | . . . . . . 7 ⊢ (𝑆 · (𝐺 ·ih 𝐹)) ∈ ℂ |
10 | 7, 9 | cjaddi 14550 | . . . . . 6 ⊢ (∗‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) = ((∗‘((∗‘𝑆) · (𝐹 ·ih 𝐺))) + (∗‘(𝑆 · (𝐺 ·ih 𝐹)))) |
11 | 2 | cjcji 14533 | . . . . . . . . . 10 ⊢ (∗‘(∗‘𝑆)) = 𝑆 |
12 | 11 | eqcomi 2833 | . . . . . . . . 9 ⊢ 𝑆 = (∗‘(∗‘𝑆)) |
13 | 5, 4 | his1i 28880 | . . . . . . . . 9 ⊢ (𝐺 ·ih 𝐹) = (∗‘(𝐹 ·ih 𝐺)) |
14 | 12, 13 | oveq12i 7171 | . . . . . . . 8 ⊢ (𝑆 · (𝐺 ·ih 𝐹)) = ((∗‘(∗‘𝑆)) · (∗‘(𝐹 ·ih 𝐺))) |
15 | 3, 6 | cjmuli 14551 | . . . . . . . 8 ⊢ (∗‘((∗‘𝑆) · (𝐹 ·ih 𝐺))) = ((∗‘(∗‘𝑆)) · (∗‘(𝐹 ·ih 𝐺))) |
16 | 14, 15 | eqtr4i 2850 | . . . . . . 7 ⊢ (𝑆 · (𝐺 ·ih 𝐹)) = (∗‘((∗‘𝑆) · (𝐹 ·ih 𝐺))) |
17 | 4, 5 | his1i 28880 | . . . . . . . . 9 ⊢ (𝐹 ·ih 𝐺) = (∗‘(𝐺 ·ih 𝐹)) |
18 | 17 | oveq2i 7170 | . . . . . . . 8 ⊢ ((∗‘𝑆) · (𝐹 ·ih 𝐺)) = ((∗‘𝑆) · (∗‘(𝐺 ·ih 𝐹))) |
19 | 2, 8 | cjmuli 14551 | . . . . . . . 8 ⊢ (∗‘(𝑆 · (𝐺 ·ih 𝐹))) = ((∗‘𝑆) · (∗‘(𝐺 ·ih 𝐹))) |
20 | 18, 19 | eqtr4i 2850 | . . . . . . 7 ⊢ ((∗‘𝑆) · (𝐹 ·ih 𝐺)) = (∗‘(𝑆 · (𝐺 ·ih 𝐹))) |
21 | 16, 20 | oveq12i 7171 | . . . . . 6 ⊢ ((𝑆 · (𝐺 ·ih 𝐹)) + ((∗‘𝑆) · (𝐹 ·ih 𝐺))) = ((∗‘((∗‘𝑆) · (𝐹 ·ih 𝐺))) + (∗‘(𝑆 · (𝐺 ·ih 𝐹)))) |
22 | 10, 21 | eqtr4i 2850 | . . . . 5 ⊢ (∗‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) = ((𝑆 · (𝐺 ·ih 𝐹)) + ((∗‘𝑆) · (𝐹 ·ih 𝐺))) |
23 | 7, 9 | addcomi 10834 | . . . . 5 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) = ((𝑆 · (𝐺 ·ih 𝐹)) + ((∗‘𝑆) · (𝐹 ·ih 𝐺))) |
24 | 22, 23 | eqtr4i 2850 | . . . 4 ⊢ (∗‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) = (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) |
25 | 7, 9 | addcli 10650 | . . . . 5 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℂ |
26 | 25 | cjrebi 14536 | . . . 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 233 | . . 3 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ |
28 | 27 | renegcli 10950 | . 2 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ |
29 | 1, 28 | eqeltri 2912 | 1 ⊢ 𝐵 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 + caddc 10543 · cmul 10545 -cneg 10874 ∗ccj 14458 ℋchba 28699 ·ih csp 28702 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-hfi 28859 ax-his1 28862 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-2 11703 df-cj 14461 df-re 14462 df-im 14463 |
This theorem is referenced by: normlem3 28892 normlem6 28895 normlem7 28896 norm-ii-i 28917 |
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