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Mirrors > Home > HSE Home > Th. List > hodsi | Structured version Visualization version GIF version |
Description: Relationship between Hilbert space operator difference and sum. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hods.1 | ⊢ 𝑅: ℋ⟶ ℋ |
hods.2 | ⊢ 𝑆: ℋ⟶ ℋ |
hods.3 | ⊢ 𝑇: ℋ⟶ ℋ |
Ref | Expression |
---|---|
hodsi | ⊢ ((𝑅 −op 𝑆) = 𝑇 ↔ (𝑆 +op 𝑇) = 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hods.1 | . . . . . 6 ⊢ 𝑅: ℋ⟶ ℋ | |
2 | 1 | ffvelcdmi 7093 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑅‘𝑥) ∈ ℋ) |
3 | hods.2 | . . . . . 6 ⊢ 𝑆: ℋ⟶ ℋ | |
4 | 3 | ffvelcdmi 7093 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
5 | hods.3 | . . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ | |
6 | 5 | ffvelcdmi 7093 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
7 | hvsubadd 30886 | . . . . 5 ⊢ (((𝑅‘𝑥) ∈ ℋ ∧ (𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → (((𝑅‘𝑥) −ℎ (𝑆‘𝑥)) = (𝑇‘𝑥) ↔ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = (𝑅‘𝑥))) | |
8 | 2, 4, 6, 7 | syl3anc 1369 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅‘𝑥) −ℎ (𝑆‘𝑥)) = (𝑇‘𝑥) ↔ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = (𝑅‘𝑥))) |
9 | hodval 31551 | . . . . . 6 ⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑅 −op 𝑆)‘𝑥) = ((𝑅‘𝑥) −ℎ (𝑆‘𝑥))) | |
10 | 1, 3, 9 | mp3an12 1448 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 −op 𝑆)‘𝑥) = ((𝑅‘𝑥) −ℎ (𝑆‘𝑥))) |
11 | 10 | eqeq1d 2730 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ ((𝑅‘𝑥) −ℎ (𝑆‘𝑥)) = (𝑇‘𝑥))) |
12 | hosval 31549 | . . . . . 6 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) | |
13 | 3, 5, 12 | mp3an12 1448 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) |
14 | 13 | eqeq1d 2730 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥) ↔ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = (𝑅‘𝑥))) |
15 | 8, 11, 14 | 3bitr4d 311 | . . 3 ⊢ (𝑥 ∈ ℋ → (((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ ((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥))) |
16 | 15 | ralbiia 3088 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ ∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥)) |
17 | 1, 3 | hosubcli 31578 | . . 3 ⊢ (𝑅 −op 𝑆): ℋ⟶ ℋ |
18 | 17, 5 | hoeqi 31570 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ (𝑅 −op 𝑆) = 𝑇) |
19 | 3, 5 | hoaddcli 31577 | . . 3 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ |
20 | 19, 1 | hoeqi 31570 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥) ↔ (𝑆 +op 𝑇) = 𝑅) |
21 | 16, 18, 20 | 3bitr3i 301 | 1 ⊢ ((𝑅 −op 𝑆) = 𝑇 ↔ (𝑆 +op 𝑇) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ℋchba 30728 +ℎ cva 30729 −ℎ cmv 30734 +op chos 30747 −op chod 30749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-hilex 30808 ax-hfvadd 30809 ax-hvcom 30810 ax-hvass 30811 ax-hv0cl 30812 ax-hvaddid 30813 ax-hfvmul 30814 ax-hvmulid 30815 ax-hvdistr2 30818 ax-hvmul0 30819 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-ltxr 11283 df-sub 11476 df-neg 11477 df-hvsub 30780 df-hosum 31539 df-hodif 31541 |
This theorem is referenced by: hodidi 31596 hodseqi 31603 ho0subi 31604 hosd1i 31631 pjoci 31989 |
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