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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 11275 | . . . 4 ⊢ 𝑅 ∈ ℂ |
| 4 | 1, 3 | mulcli 11268 | . . 3 ⊢ (𝑆 · 𝑅) ∈ ℂ |
| 5 | normlem1.2 | . . 3 ⊢ 𝐹 ∈ ℋ | |
| 6 | normlem1.3 | . . 3 ⊢ 𝐺 ∈ ℋ | |
| 7 | 4, 5, 6 | normlem0 31128 | . 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 15228 | . . . . . . . 8 ⊢ (∗‘(𝑆 · 𝑅)) = ((∗‘𝑆) · (∗‘𝑅)) |
| 9 | 3 | cjrebi 15213 | . . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ ↔ (∗‘𝑅) = 𝑅) |
| 10 | 2, 9 | mpbi 230 | . . . . . . . . 9 ⊢ (∗‘𝑅) = 𝑅 |
| 11 | 10 | oveq2i 7442 | . . . . . . . 8 ⊢ ((∗‘𝑆) · (∗‘𝑅)) = ((∗‘𝑆) · 𝑅) |
| 12 | 8, 11 | eqtri 2765 | . . . . . . 7 ⊢ (∗‘(𝑆 · 𝑅)) = ((∗‘𝑆) · 𝑅) |
| 13 | 12 | negeqi 11501 | . . . . . 6 ⊢ -(∗‘(𝑆 · 𝑅)) = -((∗‘𝑆) · 𝑅) |
| 14 | 1 | cjcli 15208 | . . . . . . 7 ⊢ (∗‘𝑆) ∈ ℂ |
| 15 | 14, 3 | mulneg2i 11710 | . . . . . 6 ⊢ ((∗‘𝑆) · -𝑅) = -((∗‘𝑆) · 𝑅) |
| 16 | 13, 15 | eqtr4i 2768 | . . . . 5 ⊢ -(∗‘(𝑆 · 𝑅)) = ((∗‘𝑆) · -𝑅) |
| 17 | 16 | oveq1i 7441 | . . . 4 ⊢ (-(∗‘(𝑆 · 𝑅)) · (𝐹 ·ih 𝐺)) = (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺)) |
| 18 | 17 | oveq2i 7442 | . . 3 ⊢ ((𝐹 ·ih 𝐹) + (-(∗‘(𝑆 · 𝑅)) · (𝐹 ·ih 𝐺))) = ((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) |
| 19 | 1, 3 | mulneg2i 11710 | . . . . . 6 ⊢ (𝑆 · -𝑅) = -(𝑆 · 𝑅) |
| 20 | 19 | eqcomi 2746 | . . . . 5 ⊢ -(𝑆 · 𝑅) = (𝑆 · -𝑅) |
| 21 | 20 | oveq1i 7441 | . . . 4 ⊢ (-(𝑆 · 𝑅) · (𝐺 ·ih 𝐹)) = ((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) |
| 22 | 8 | oveq2i 7442 | . . . . . . 7 ⊢ ((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) = ((𝑆 · 𝑅) · ((∗‘𝑆) · (∗‘𝑅))) |
| 23 | 3 | cjcli 15208 | . . . . . . . . 9 ⊢ (∗‘𝑅) ∈ ℂ |
| 24 | 1, 3, 14, 23 | mul4i 11458 | . . . . . . . 8 ⊢ ((𝑆 · 𝑅) · ((∗‘𝑆) · (∗‘𝑅))) = ((𝑆 · (∗‘𝑆)) · (𝑅 · (∗‘𝑅))) |
| 25 | normlem1.5 | . . . . . . . . . . . 12 ⊢ (abs‘𝑆) = 1 | |
| 26 | 25 | oveq1i 7441 | . . . . . . . . . . 11 ⊢ ((abs‘𝑆)↑2) = (1↑2) |
| 27 | 1 | absvalsqi 15432 | . . . . . . . . . . 11 ⊢ ((abs‘𝑆)↑2) = (𝑆 · (∗‘𝑆)) |
| 28 | sq1 14234 | . . . . . . . . . . 11 ⊢ (1↑2) = 1 | |
| 29 | 26, 27, 28 | 3eqtr3i 2773 | . . . . . . . . . 10 ⊢ (𝑆 · (∗‘𝑆)) = 1 |
| 30 | 10 | oveq2i 7442 | . . . . . . . . . 10 ⊢ (𝑅 · (∗‘𝑅)) = (𝑅 · 𝑅) |
| 31 | 29, 30 | oveq12i 7443 | . . . . . . . . 9 ⊢ ((𝑆 · (∗‘𝑆)) · (𝑅 · (∗‘𝑅))) = (1 · (𝑅 · 𝑅)) |
| 32 | 3, 3 | mulcli 11268 | . . . . . . . . . 10 ⊢ (𝑅 · 𝑅) ∈ ℂ |
| 33 | 32 | mullidi 11266 | . . . . . . . . 9 ⊢ (1 · (𝑅 · 𝑅)) = (𝑅 · 𝑅) |
| 34 | 31, 33 | eqtri 2765 | . . . . . . . 8 ⊢ ((𝑆 · (∗‘𝑆)) · (𝑅 · (∗‘𝑅))) = (𝑅 · 𝑅) |
| 35 | 24, 34 | eqtri 2765 | . . . . . . 7 ⊢ ((𝑆 · 𝑅) · ((∗‘𝑆) · (∗‘𝑅))) = (𝑅 · 𝑅) |
| 36 | 22, 35 | eqtri 2765 | . . . . . 6 ⊢ ((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) = (𝑅 · 𝑅) |
| 37 | 3 | sqvali 14219 | . . . . . 6 ⊢ (𝑅↑2) = (𝑅 · 𝑅) |
| 38 | 36, 37 | eqtr4i 2768 | . . . . 5 ⊢ ((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) = (𝑅↑2) |
| 39 | 38 | oveq1i 7441 | . . . 4 ⊢ (((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) · (𝐺 ·ih 𝐺)) = ((𝑅↑2) · (𝐺 ·ih 𝐺)) |
| 40 | 21, 39 | oveq12i 7443 | . . 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 7443 | . 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 2765 | 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 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 1c1 11156 + caddc 11158 · cmul 11160 -cneg 11493 2c2 12321 ↑cexp 14102 ∗ccj 15135 abscabs 15273 ℋchba 30938 ·ℎ csm 30940 ·ih csp 30941 −ℎ cmv 30944 |
| 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-cnex 11211 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-pre-sup 11233 ax-hfvadd 31019 ax-hfvmul 31024 ax-hvmulass 31026 ax-hfi 31098 ax-his1 31101 ax-his2 31102 ax-his3 31103 |
| 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-pss 3971 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-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 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-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 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-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 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-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-hvsub 30990 |
| This theorem is referenced by: normlem4 31132 |
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