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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 11001 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
3 | hodseq.3 | . . . . . . 7 ⊢ 𝑇: ℋ⟶ ℋ | |
4 | homulcl 28002 | . . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (-1 ·op 𝑇): ℋ⟶ ℋ) | |
5 | 2, 3, 4 | mp2an 704 | . . . . . 6 ⊢ (-1 ·op 𝑇): ℋ⟶ ℋ |
6 | hosval 27983 | . . . . . 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 1406 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆‘𝑥) +ℎ ((-1 ·op 𝑇)‘𝑥))) |
8 | 1 | ffvelrni 6266 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
9 | 3 | ffvelrni 6266 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
10 | hvsubval 27257 | . . . . . . 7 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) = ((𝑆‘𝑥) +ℎ (-1 ·ℎ (𝑇‘𝑥)))) | |
11 | 8, 9, 10 | syl2anc 691 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) = ((𝑆‘𝑥) +ℎ (-1 ·ℎ (𝑇‘𝑥)))) |
12 | homval 27984 | . . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((-1 ·op 𝑇)‘𝑥) = (-1 ·ℎ (𝑇‘𝑥))) | |
13 | 2, 3, 12 | mp3an12 1406 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((-1 ·op 𝑇)‘𝑥) = (-1 ·ℎ (𝑇‘𝑥))) |
14 | 13 | oveq2d 6565 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) +ℎ ((-1 ·op 𝑇)‘𝑥)) = ((𝑆‘𝑥) +ℎ (-1 ·ℎ (𝑇‘𝑥)))) |
15 | 11, 14 | eqtr4d 2647 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) = ((𝑆‘𝑥) +ℎ ((-1 ·op 𝑇)‘𝑥))) |
16 | 7, 15 | eqtr4d 2647 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) |
17 | hodval 27985 | . . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) | |
18 | 1, 3, 17 | mp3an12 1406 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) |
19 | 16, 18 | eqtr4d 2647 | . . 3 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆 −op 𝑇)‘𝑥)) |
20 | 19 | rgen 2906 | . 2 ⊢ ∀𝑥 ∈ ℋ ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆 −op 𝑇)‘𝑥) |
21 | 1, 5 | hoaddcli 28011 | . . 3 ⊢ (𝑆 +op (-1 ·op 𝑇)): ℋ⟶ ℋ |
22 | 1, 3 | hosubcli 28012 | . . 3 ⊢ (𝑆 −op 𝑇): ℋ⟶ ℋ |
23 | 21, 22 | hoeqi 28004 | . 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 219 | 1 ⊢ (𝑆 +op (-1 ·op 𝑇)) = (𝑆 −op 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 1c1 9816 -cneg 10146 ℋchil 27160 +ℎ cva 27161 ·ℎ csm 27162 −ℎ cmv 27166 +op chos 27179 ·op chot 27180 −op chod 27181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-hilex 27240 ax-hfvadd 27241 ax-hfvmul 27246 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-neg 10148 df-hvsub 27212 df-hosum 27973 df-homul 27974 df-hodif 27975 |
This theorem is referenced by: honegsub 28042 hosubeq0i 28069 lnophdi 28245 bdophdi 28340 nmoptri2i 28342 |
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