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Definition df-bj-oppc 36116
Description: Define the negation (operation giving the opposite) on the set of extended complex numbers and the complex projective line (Riemann sphere). (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
df-bj-oppc -ā„‚Ģ… = (š‘„ āˆˆ (ā„‚Ģ… āˆŖ ā„‚Ģ‚) ā†¦ if(š‘„ = āˆž, āˆž, if(š‘„ āˆˆ ā„‚, (ā„©š‘¦ āˆˆ ā„‚ (š‘„ +ā„‚Ģ… š‘¦) = 0), (+āˆžeiĻ„ā€˜(š‘„ +ā„‚Ģ… āŸØ1/2, 0RāŸ©)))))
Distinct variable group:   š‘„,š‘¦

Detailed syntax breakdown of Definition df-bj-oppc
StepHypRef Expression
1 coppcc 36115 . 2 class -ā„‚Ģ…
2 vx . . 3 setvar š‘„
3 cccbar 36091 . . . 4 class ā„‚Ģ…
4 ccchat 36108 . . . 4 class ā„‚Ģ‚
53, 4cun 3946 . . 3 class (ā„‚Ģ… āˆŖ ā„‚Ģ‚)
62cv 1540 . . . . 5 class š‘„
7 cinfty 36106 . . . . 5 class āˆž
86, 7wceq 1541 . . . 4 wff š‘„ = āˆž
9 cc 11107 . . . . . 6 class ā„‚
106, 9wcel 2106 . . . . 5 wff š‘„ āˆˆ ā„‚
11 vy . . . . . . . . 9 setvar š‘¦
1211cv 1540 . . . . . . . 8 class š‘¦
13 caddcc 36113 . . . . . . . 8 class +ā„‚Ģ…
146, 12, 13co 7408 . . . . . . 7 class (š‘„ +ā„‚Ģ… š‘¦)
15 cc0 11109 . . . . . . 7 class 0
1614, 15wceq 1541 . . . . . 6 wff (š‘„ +ā„‚Ģ… š‘¦) = 0
1716, 11, 9crio 7363 . . . . 5 class (ā„©š‘¦ āˆˆ ā„‚ (š‘„ +ā„‚Ģ… š‘¦) = 0)
18 chalf 36079 . . . . . . . 8 class 1/2
19 c0r 10860 . . . . . . . 8 class 0R
2018, 19cop 4634 . . . . . . 7 class āŸØ1/2, 0RāŸ©
216, 20, 13co 7408 . . . . . 6 class (š‘„ +ā„‚Ģ… āŸØ1/2, 0RāŸ©)
22 cinftyexpitau 36074 . . . . . 6 class +āˆžeiĻ„
2321, 22cfv 6543 . . . . 5 class (+āˆžeiĻ„ā€˜(š‘„ +ā„‚Ģ… āŸØ1/2, 0RāŸ©))
2410, 17, 23cif 4528 . . . 4 class if(š‘„ āˆˆ ā„‚, (ā„©š‘¦ āˆˆ ā„‚ (š‘„ +ā„‚Ģ… š‘¦) = 0), (+āˆžeiĻ„ā€˜(š‘„ +ā„‚Ģ… āŸØ1/2, 0RāŸ©)))
258, 7, 24cif 4528 . . 3 class if(š‘„ = āˆž, āˆž, if(š‘„ āˆˆ ā„‚, (ā„©š‘¦ āˆˆ ā„‚ (š‘„ +ā„‚Ģ… š‘¦) = 0), (+āˆžeiĻ„ā€˜(š‘„ +ā„‚Ģ… āŸØ1/2, 0RāŸ©))))
262, 5, 25cmpt 5231 . 2 class (š‘„ āˆˆ (ā„‚Ģ… āˆŖ ā„‚Ģ‚) ā†¦ if(š‘„ = āˆž, āˆž, if(š‘„ āˆˆ ā„‚, (ā„©š‘¦ āˆˆ ā„‚ (š‘„ +ā„‚Ģ… š‘¦) = 0), (+āˆžeiĻ„ā€˜(š‘„ +ā„‚Ģ… āŸØ1/2, 0RāŸ©)))))
271, 26wceq 1541 1 wff -ā„‚Ģ… = (š‘„ āˆˆ (ā„‚Ģ… āˆŖ ā„‚Ģ‚) ā†¦ if(š‘„ = āˆž, āˆž, if(š‘„ āˆˆ ā„‚, (ā„©š‘¦ āˆˆ ā„‚ (š‘„ +ā„‚Ģ… š‘¦) = 0), (+āˆžeiĻ„ā€˜(š‘„ +ā„‚Ģ… āŸØ1/2, 0RāŸ©)))))
Colors of variables: wff setvar class
This definition is referenced by: (None)
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