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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 6673 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑅‘𝑥) ∈ ℋ) |
3 | hods.2 | . . . . . 6 ⊢ 𝑆: ℋ⟶ ℋ | |
4 | 3 | ffvelrni 6673 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
5 | hods.3 | . . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ | |
6 | 5 | ffvelrni 6673 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
7 | hvsubadd 28645 | . . . . 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 1351 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅‘𝑥) −ℎ (𝑆‘𝑥)) = (𝑇‘𝑥) ↔ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = (𝑅‘𝑥))) |
9 | hodval 29312 | . . . . . 6 ⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑅 −op 𝑆)‘𝑥) = ((𝑅‘𝑥) −ℎ (𝑆‘𝑥))) | |
10 | 1, 3, 9 | mp3an12 1430 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 −op 𝑆)‘𝑥) = ((𝑅‘𝑥) −ℎ (𝑆‘𝑥))) |
11 | 10 | eqeq1d 2774 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ ((𝑅‘𝑥) −ℎ (𝑆‘𝑥)) = (𝑇‘𝑥))) |
12 | hosval 29310 | . . . . . 6 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) | |
13 | 3, 5, 12 | mp3an12 1430 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) |
14 | 13 | eqeq1d 2774 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥) ↔ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = (𝑅‘𝑥))) |
15 | 8, 11, 14 | 3bitr4d 303 | . . 3 ⊢ (𝑥 ∈ ℋ → (((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ ((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥))) |
16 | 15 | ralbiia 3108 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ ∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥)) |
17 | 1, 3 | hosubcli 29339 | . . 3 ⊢ (𝑅 −op 𝑆): ℋ⟶ ℋ |
18 | 17, 5 | hoeqi 29331 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑅 −op 𝑆)‘𝑥) = (𝑇‘𝑥) ↔ (𝑅 −op 𝑆) = 𝑇) |
19 | 3, 5 | hoaddcli 29338 | . . 3 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ |
20 | 19, 1 | hoeqi 29331 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = (𝑅‘𝑥) ↔ (𝑆 +op 𝑇) = 𝑅) |
21 | 16, 18, 20 | 3bitr3i 293 | 1 ⊢ ((𝑅 −op 𝑆) = 𝑇 ↔ (𝑆 +op 𝑇) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1507 ∈ wcel 2050 ∀wral 3082 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 ℋchba 28487 +ℎ cva 28488 −ℎ cmv 28493 +op chos 28506 −op chod 28508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-hilex 28567 ax-hfvadd 28568 ax-hvcom 28569 ax-hvass 28570 ax-hv0cl 28571 ax-hvaddid 28572 ax-hfvmul 28573 ax-hvmulid 28574 ax-hvdistr2 28577 ax-hvmul0 28578 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-po 5322 df-so 5323 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-ltxr 10477 df-sub 10670 df-neg 10671 df-hvsub 28539 df-hosum 29300 df-hodif 29302 |
This theorem is referenced by: hodidi 29357 hodseqi 29364 ho0subi 29365 hosd1i 29392 pjoci 29750 |
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