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Mirrors > Home > HSE Home > Th. List > hoaddcli | Structured version Visualization version GIF version |
Description: Mapping of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hoeq.1 | ⊢ 𝑆: ℋ⟶ ℋ |
hoeq.2 | ⊢ 𝑇: ℋ⟶ ℋ |
Ref | Expression |
---|---|
hoaddcli | ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoeq.1 | . 2 ⊢ 𝑆: ℋ⟶ ℋ | |
2 | hoeq.2 | . 2 ⊢ 𝑇: ℋ⟶ ℋ | |
3 | hoaddcl 31781 | . 2 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇): ℋ⟶ ℋ) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ⟶wf 6568 (class class class)co 7445 ℋchba 30942 +op chos 30961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-hilex 31022 ax-hfvadd 31023 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-ov 7448 df-oprab 7449 df-mpo 7450 df-map 8882 df-hosum 31753 |
This theorem is referenced by: hoaddfni 31793 hoaddcomi 31795 hodsi 31798 hoaddassi 31799 hocadddiri 31802 hoaddridi 31809 ho0subi 31818 honegsubi 31819 hosd1i 31845 lnophsi 32024 nmoptrii 32117 bdophsi 32119 nmoptri2i 32122 pjsdii 32178 pjscji 32193 pjtoi 32202 |
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