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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 12333 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
3 | hodseq.3 | . . . . . . 7 ⊢ 𝑇: ℋ⟶ ℋ | |
4 | homulcl 31446 | . . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (-1 ·op 𝑇): ℋ⟶ ℋ) | |
5 | 2, 3, 4 | mp2an 689 | . . . . . 6 ⊢ (-1 ·op 𝑇): ℋ⟶ ℋ |
6 | hosval 31427 | . . . . . 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 1450 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆‘𝑥) +ℎ ((-1 ·op 𝑇)‘𝑥))) |
8 | 1 | ffvelcdmi 7085 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
9 | 3 | ffvelcdmi 7085 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
10 | hvsubval 30703 | . . . . . . 7 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) = ((𝑆‘𝑥) +ℎ (-1 ·ℎ (𝑇‘𝑥)))) | |
11 | 8, 9, 10 | syl2anc 583 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) = ((𝑆‘𝑥) +ℎ (-1 ·ℎ (𝑇‘𝑥)))) |
12 | homval 31428 | . . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((-1 ·op 𝑇)‘𝑥) = (-1 ·ℎ (𝑇‘𝑥))) | |
13 | 2, 3, 12 | mp3an12 1450 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((-1 ·op 𝑇)‘𝑥) = (-1 ·ℎ (𝑇‘𝑥))) |
14 | 13 | oveq2d 7428 | . . . . . 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 31429 | . . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) | |
18 | 1, 3, 17 | mp3an12 1450 | . . . 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 3062 | . 2 ⊢ ∀𝑥 ∈ ℋ ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆 −op 𝑇)‘𝑥) |
21 | 1, 5 | hoaddcli 31455 | . . 3 ⊢ (𝑆 +op (-1 ·op 𝑇)): ℋ⟶ ℋ |
22 | 1, 3 | hosubcli 31456 | . . 3 ⊢ (𝑆 −op 𝑇): ℋ⟶ ℋ |
23 | 21, 22 | hoeqi 31448 | . 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 229 | 1 ⊢ (𝑆 +op (-1 ·op 𝑇)) = (𝑆 −op 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 ∀wral 3060 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 1c1 11117 -cneg 11452 ℋchba 30606 +ℎ cva 30607 ·ℎ csm 30608 −ℎ cmv 30612 +op chos 30625 ·op chot 30626 −op chod 30627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-hilex 30686 ax-hfvadd 30687 ax-hfvmul 30692 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-ltxr 11260 df-sub 11453 df-neg 11454 df-hvsub 30658 df-hosum 31417 df-homul 31418 df-hodif 31419 |
This theorem is referenced by: honegsub 31486 hosubeq0i 31513 lnophdi 31689 bdophdi 31784 nmoptri2i 31786 |
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