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Mirrors > Home > MPE Home > Th. List > elimph | Structured version Visualization version GIF version |
Description: Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elimph.1 | β’ π = (BaseSetβπ) |
elimph.5 | β’ π = (0vecβπ) |
elimph.6 | β’ π β CPreHilOLD |
Ref | Expression |
---|---|
elimph | β’ if(π΄ β π, π΄, π) β π |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elimph.1 | . 2 β’ π = (BaseSetβπ) | |
2 | elimph.5 | . 2 β’ π = (0vecβπ) | |
3 | elimph.6 | . . 3 β’ π β CPreHilOLD | |
4 | 3 | phnvi 29807 | . 2 β’ π β NrmCVec |
5 | 1, 2, 4 | elimnv 29674 | 1 β’ if(π΄ β π, π΄, π) β π |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wcel 2107 ifcif 4490 βcfv 6500 BaseSetcba 29577 0veccn0v 29579 CPreHilOLDccphlo 29803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-1st 7925 df-2nd 7926 df-grpo 29484 df-gid 29485 df-ablo 29536 df-vc 29550 df-nv 29583 df-va 29586 df-ba 29587 df-sm 29588 df-0v 29589 df-nmcv 29591 df-ph 29804 |
This theorem is referenced by: ipdiri 29821 ipassi 29832 sii 29845 |
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