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Mirrors > Home > HSE Home > Th. List > honegsubi | Structured version Visualization version GIF version |
Description: Relationship between Hilbert operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hodseq.2 | ⊢ 𝑆: ℋ⟶ ℋ |
hodseq.3 | ⊢ 𝑇: ℋ⟶ ℋ |
Ref | Expression |
---|---|
honegsubi | ⊢ (𝑆 +op (-1 ·op 𝑇)) = (𝑆 −op 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hodseq.2 | . . . . . 6 ⊢ 𝑆: ℋ⟶ ℋ | |
2 | neg1cn 11909 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
3 | hodseq.3 | . . . . . . 7 ⊢ 𝑇: ℋ⟶ ℋ | |
4 | homulcl 29794 | . . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (-1 ·op 𝑇): ℋ⟶ ℋ) | |
5 | 2, 3, 4 | mp2an 692 | . . . . . 6 ⊢ (-1 ·op 𝑇): ℋ⟶ ℋ |
6 | hosval 29775 | . . . . . 6 ⊢ ((𝑆: ℋ⟶ ℋ ∧ (-1 ·op 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆‘𝑥) +ℎ ((-1 ·op 𝑇)‘𝑥))) | |
7 | 1, 5, 6 | mp3an12 1453 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆‘𝑥) +ℎ ((-1 ·op 𝑇)‘𝑥))) |
8 | 1 | ffvelrni 6881 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
9 | 3 | ffvelrni 6881 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
10 | hvsubval 29051 | . . . . . . 7 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) = ((𝑆‘𝑥) +ℎ (-1 ·ℎ (𝑇‘𝑥)))) | |
11 | 8, 9, 10 | syl2anc 587 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) = ((𝑆‘𝑥) +ℎ (-1 ·ℎ (𝑇‘𝑥)))) |
12 | homval 29776 | . . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((-1 ·op 𝑇)‘𝑥) = (-1 ·ℎ (𝑇‘𝑥))) | |
13 | 2, 3, 12 | mp3an12 1453 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((-1 ·op 𝑇)‘𝑥) = (-1 ·ℎ (𝑇‘𝑥))) |
14 | 13 | oveq2d 7207 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) +ℎ ((-1 ·op 𝑇)‘𝑥)) = ((𝑆‘𝑥) +ℎ (-1 ·ℎ (𝑇‘𝑥)))) |
15 | 11, 14 | eqtr4d 2774 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) = ((𝑆‘𝑥) +ℎ ((-1 ·op 𝑇)‘𝑥))) |
16 | 7, 15 | eqtr4d 2774 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) |
17 | hodval 29777 | . . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) | |
18 | 1, 3, 17 | mp3an12 1453 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) |
19 | 16, 18 | eqtr4d 2774 | . . 3 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆 −op 𝑇)‘𝑥)) |
20 | 19 | rgen 3061 | . 2 ⊢ ∀𝑥 ∈ ℋ ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆 −op 𝑇)‘𝑥) |
21 | 1, 5 | hoaddcli 29803 | . . 3 ⊢ (𝑆 +op (-1 ·op 𝑇)): ℋ⟶ ℋ |
22 | 1, 3 | hosubcli 29804 | . . 3 ⊢ (𝑆 −op 𝑇): ℋ⟶ ℋ |
23 | 21, 22 | hoeqi 29796 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆 −op 𝑇)‘𝑥) ↔ (𝑆 +op (-1 ·op 𝑇)) = (𝑆 −op 𝑇)) |
24 | 20, 23 | mpbi 233 | 1 ⊢ (𝑆 +op (-1 ·op 𝑇)) = (𝑆 −op 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 ∀wral 3051 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 1c1 10695 -cneg 11028 ℋchba 28954 +ℎ cva 28955 ·ℎ csm 28956 −ℎ cmv 28960 +op chos 28973 ·op chot 28974 −op chod 28975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-hilex 29034 ax-hfvadd 29035 ax-hfvmul 29040 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 df-sub 11029 df-neg 11030 df-hvsub 29006 df-hosum 29765 df-homul 29766 df-hodif 29767 |
This theorem is referenced by: honegsub 29834 hosubeq0i 29861 lnophdi 30037 bdophdi 30132 nmoptri2i 30134 |
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