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Mirrors > Home > HSE Home > Th. List > dmadjrn | Structured version Visualization version GIF version |
Description: The adjoint of an operator belongs to the adjoint function's domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dmadjrn | ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ dom adjℎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | adj1o 29674 | . . 3 ⊢ adjℎ:dom adjℎ–1-1-onto→dom adjℎ | |
2 | f1of 6618 | . . 3 ⊢ (adjℎ:dom adjℎ–1-1-onto→dom adjℎ → adjℎ:dom adjℎ⟶dom adjℎ) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ adjℎ:dom adjℎ⟶dom adjℎ |
4 | 3 | ffvelrni 6853 | 1 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ dom adjℎ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 dom cdm 5558 ⟶wf 6354 –1-1-onto→wf1o 6357 ‘cfv 6358 adjℎcado 28735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-hfvadd 28780 ax-hvcom 28781 ax-hvass 28782 ax-hv0cl 28783 ax-hvaddid 28784 ax-hfvmul 28785 ax-hvmulid 28786 ax-hvdistr2 28789 ax-hvmul0 28790 ax-hfi 28859 ax-his1 28862 ax-his2 28863 ax-his3 28864 ax-his4 28865 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-2 11703 df-cj 14461 df-re 14462 df-im 14463 df-hvsub 28751 df-adjh 29629 |
This theorem is referenced by: dmadjrnb 29686 adjcl 29712 adjadj 29716 adjlnop 29866 adjsslnop 29867 adjmul 29872 adjadd 29873 |
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