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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 | ffvelrni 6842 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑅‘𝑥) ∈ ℋ) |
3 | hods.2 | . . . . . 6 ⊢ 𝑆: ℋ⟶ ℋ | |
4 | 3 | ffvelrni 6842 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
5 | hods.3 | . . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ | |
6 | 5 | ffvelrni 6842 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
7 | hvsubadd 28781 | . . . . 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 1363 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅‘𝑥) −ℎ (𝑆‘𝑥)) = (𝑇‘𝑥) ↔ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = (𝑅‘𝑥))) |
9 | hodval 29446 | . . . . . 6 ⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑅 −op 𝑆)‘𝑥) = ((𝑅‘𝑥) −ℎ (𝑆‘𝑥))) | |
10 | 1, 3, 9 | mp3an12 1442 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 −op 𝑆)‘𝑥) = ((𝑅‘𝑥) −ℎ (𝑆‘𝑥))) |
11 | 10 | eqeq1d 2820 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ ((𝑅‘𝑥) −ℎ (𝑆‘𝑥)) = (𝑇‘𝑥))) |
12 | hosval 29444 | . . . . . 6 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) | |
13 | 3, 5, 12 | mp3an12 1442 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) |
14 | 13 | eqeq1d 2820 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥) ↔ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = (𝑅‘𝑥))) |
15 | 8, 11, 14 | 3bitr4d 312 | . . 3 ⊢ (𝑥 ∈ ℋ → (((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ ((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥))) |
16 | 15 | ralbiia 3161 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ ∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥)) |
17 | 1, 3 | hosubcli 29473 | . . 3 ⊢ (𝑅 −op 𝑆): ℋ⟶ ℋ |
18 | 17, 5 | hoeqi 29465 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ (𝑅 −op 𝑆) = 𝑇) |
19 | 3, 5 | hoaddcli 29472 | . . 3 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ |
20 | 19, 1 | hoeqi 29465 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥) ↔ (𝑆 +op 𝑇) = 𝑅) |
21 | 16, 18, 20 | 3bitr3i 302 | 1 ⊢ ((𝑅 −op 𝑆) = 𝑇 ↔ (𝑆 +op 𝑇) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℋchba 28623 +ℎ cva 28624 −ℎ cmv 28629 +op chos 28642 −op chod 28644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-hilex 28703 ax-hfvadd 28704 ax-hvcom 28705 ax-hvass 28706 ax-hv0cl 28707 ax-hvaddid 28708 ax-hfvmul 28709 ax-hvmulid 28710 ax-hvdistr2 28713 ax-hvmul0 28714 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-sub 10860 df-neg 10861 df-hvsub 28675 df-hosum 29434 df-hodif 29436 |
This theorem is referenced by: hodidi 29491 hodseqi 29498 ho0subi 29499 hosd1i 29526 pjoci 29884 |
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