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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 11745 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
3 | hodseq.3 | . . . . . . 7 ⊢ 𝑇: ℋ⟶ ℋ | |
4 | homulcl 29530 | . . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (-1 ·op 𝑇): ℋ⟶ ℋ) | |
5 | 2, 3, 4 | mp2an 690 | . . . . . 6 ⊢ (-1 ·op 𝑇): ℋ⟶ ℋ |
6 | hosval 29511 | . . . . . 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 1447 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆‘𝑥) +ℎ ((-1 ·op 𝑇)‘𝑥))) |
8 | 1 | ffvelrni 6845 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
9 | 3 | ffvelrni 6845 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
10 | hvsubval 28787 | . . . . . . 7 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) = ((𝑆‘𝑥) +ℎ (-1 ·ℎ (𝑇‘𝑥)))) | |
11 | 8, 9, 10 | syl2anc 586 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) = ((𝑆‘𝑥) +ℎ (-1 ·ℎ (𝑇‘𝑥)))) |
12 | homval 29512 | . . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((-1 ·op 𝑇)‘𝑥) = (-1 ·ℎ (𝑇‘𝑥))) | |
13 | 2, 3, 12 | mp3an12 1447 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((-1 ·op 𝑇)‘𝑥) = (-1 ·ℎ (𝑇‘𝑥))) |
14 | 13 | oveq2d 7166 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) +ℎ ((-1 ·op 𝑇)‘𝑥)) = ((𝑆‘𝑥) +ℎ (-1 ·ℎ (𝑇‘𝑥)))) |
15 | 11, 14 | eqtr4d 2859 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) = ((𝑆‘𝑥) +ℎ ((-1 ·op 𝑇)‘𝑥))) |
16 | 7, 15 | eqtr4d 2859 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) |
17 | hodval 29513 | . . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) | |
18 | 1, 3, 17 | mp3an12 1447 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) |
19 | 16, 18 | eqtr4d 2859 | . . 3 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆 −op 𝑇)‘𝑥)) |
20 | 19 | rgen 3148 | . 2 ⊢ ∀𝑥 ∈ ℋ ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆 −op 𝑇)‘𝑥) |
21 | 1, 5 | hoaddcli 29539 | . . 3 ⊢ (𝑆 +op (-1 ·op 𝑇)): ℋ⟶ ℋ |
22 | 1, 3 | hosubcli 29540 | . . 3 ⊢ (𝑆 −op 𝑇): ℋ⟶ ℋ |
23 | 21, 22 | hoeqi 29532 | . 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 232 | 1 ⊢ (𝑆 +op (-1 ·op 𝑇)) = (𝑆 −op 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 1c1 10532 -cneg 10865 ℋchba 28690 +ℎ cva 28691 ·ℎ csm 28692 −ℎ cmv 28696 +op chos 28709 ·op chot 28710 −op chod 28711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-hilex 28770 ax-hfvadd 28771 ax-hfvmul 28776 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-sub 10866 df-neg 10867 df-hvsub 28742 df-hosum 29501 df-homul 29502 df-hodif 29503 |
This theorem is referenced by: honegsub 29570 hosubeq0i 29597 lnophdi 29773 bdophdi 29868 nmoptri2i 29870 |
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