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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 6852 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑅‘𝑥) ∈ ℋ) |
3 | hods.2 | . . . . . 6 ⊢ 𝑆: ℋ⟶ ℋ | |
4 | 3 | ffvelrni 6852 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
5 | hods.3 | . . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ | |
6 | 5 | ffvelrni 6852 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
7 | hvsubadd 28856 | . . . . 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 1367 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅‘𝑥) −ℎ (𝑆‘𝑥)) = (𝑇‘𝑥) ↔ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = (𝑅‘𝑥))) |
9 | hodval 29521 | . . . . . 6 ⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑅 −op 𝑆)‘𝑥) = ((𝑅‘𝑥) −ℎ (𝑆‘𝑥))) | |
10 | 1, 3, 9 | mp3an12 1447 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 −op 𝑆)‘𝑥) = ((𝑅‘𝑥) −ℎ (𝑆‘𝑥))) |
11 | 10 | eqeq1d 2825 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ ((𝑅‘𝑥) −ℎ (𝑆‘𝑥)) = (𝑇‘𝑥))) |
12 | hosval 29519 | . . . . . 6 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) | |
13 | 3, 5, 12 | mp3an12 1447 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) |
14 | 13 | eqeq1d 2825 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥) ↔ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = (𝑅‘𝑥))) |
15 | 8, 11, 14 | 3bitr4d 313 | . . 3 ⊢ (𝑥 ∈ ℋ → (((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ ((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥))) |
16 | 15 | ralbiia 3166 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ ∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥)) |
17 | 1, 3 | hosubcli 29548 | . . 3 ⊢ (𝑅 −op 𝑆): ℋ⟶ ℋ |
18 | 17, 5 | hoeqi 29540 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ (𝑅 −op 𝑆) = 𝑇) |
19 | 3, 5 | hoaddcli 29547 | . . 3 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ |
20 | 19, 1 | hoeqi 29540 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥) ↔ (𝑆 +op 𝑇) = 𝑅) |
21 | 16, 18, 20 | 3bitr3i 303 | 1 ⊢ ((𝑅 −op 𝑆) = 𝑇 ↔ (𝑆 +op 𝑇) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℋchba 28698 +ℎ cva 28699 −ℎ cmv 28704 +op chos 28717 −op chod 28719 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-hilex 28778 ax-hfvadd 28779 ax-hvcom 28780 ax-hvass 28781 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvdistr2 28788 ax-hvmul0 28789 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-neg 10875 df-hvsub 28750 df-hosum 29509 df-hodif 29511 |
This theorem is referenced by: hodidi 29566 hodseqi 29573 ho0subi 29574 hosd1i 29601 pjoci 29959 |
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