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Theorem iblcnlem1 25304
Description: Lemma for iblcnlem 25305. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
itgcnlem.r š‘… = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜šµ)), (ā„œā€˜šµ), 0)))
itgcnlem.s š‘† = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ -(ā„œā€˜šµ)), -(ā„œā€˜šµ), 0)))
itgcnlem.t š‘‡ = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„‘ā€˜šµ)), (ā„‘ā€˜šµ), 0)))
itgcnlem.u š‘ˆ = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ -(ā„‘ā€˜šµ)), -(ā„‘ā€˜šµ), 0)))
itgcnlem1.v ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ šµ āˆˆ ā„‚)
Assertion
Ref Expression
iblcnlem1 (šœ‘ ā†’ ((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ šæ1 ā†” ((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ MblFn āˆ§ (š‘… āˆˆ ā„ āˆ§ š‘† āˆˆ ā„) āˆ§ (š‘‡ āˆˆ ā„ āˆ§ š‘ˆ āˆˆ ā„))))
Distinct variable groups:   š‘„,š“   šœ‘,š‘„
Allowed substitution hints:   šµ(š‘„)   š‘…(š‘„)   š‘†(š‘„)   š‘‡(š‘„)   š‘ˆ(š‘„)
Proof of Theorem iblcnlem1
Dummy variable š‘˜ is distinct from all other variables.
StepHypRef Expression
1 eqidd 2733 . . 3 (šœ‘ ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0)))
2 eqidd 2733 . . 3 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜(šµ / (iā†‘š‘˜))) = (ā„œā€˜(šµ / (iā†‘š‘˜))))
3 itgcnlem1.v . . 3 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ šµ āˆˆ ā„‚)
41, 2, 3isibl2 25283 . 2 (šœ‘ ā†’ ((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ šæ1 ā†” ((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ MblFn āˆ§ āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„)))
5 c0ex 11207 . . . . . . . 8 0 āˆˆ V
6 1ex 11209 . . . . . . . 8 1 āˆˆ V
7 ax-icn 11168 . . . . . . . . . . 11 i āˆˆ ā„‚
8 exp0 14030 . . . . . . . . . . 11 (i āˆˆ ā„‚ ā†’ (iā†‘0) = 1)
97, 8ax-mp 5 . . . . . . . . . 10 (iā†‘0) = 1
109itgvallem 25301 . . . . . . . . 9 (š‘˜ = 0 ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / 1))), (ā„œā€˜(šµ / 1)), 0))))
1110eleq1d 2818 . . . . . . . 8 (š‘˜ = 0 ā†’ ((āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ ā†” (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / 1))), (ā„œā€˜(šµ / 1)), 0))) āˆˆ ā„))
12 exp1 14032 . . . . . . . . . . 11 (i āˆˆ ā„‚ ā†’ (iā†‘1) = i)
137, 12ax-mp 5 . . . . . . . . . 10 (iā†‘1) = i
1413itgvallem 25301 . . . . . . . . 9 (š‘˜ = 1 ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / i))), (ā„œā€˜(šµ / i)), 0))))
1514eleq1d 2818 . . . . . . . 8 (š‘˜ = 1 ā†’ ((āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ ā†” (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / i))), (ā„œā€˜(šµ / i)), 0))) āˆˆ ā„))
165, 6, 11, 15ralpr 4704 . . . . . . 7 (āˆ€š‘˜ āˆˆ {0, 1} (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ ā†” ((āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / 1))), (ā„œā€˜(šµ / 1)), 0))) āˆˆ ā„ āˆ§ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / i))), (ā„œā€˜(šµ / i)), 0))) āˆˆ ā„))
173div1d 11981 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (šµ / 1) = šµ)
1817fveq2d 6895 . . . . . . . . . . . . 13 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜(šµ / 1)) = (ā„œā€˜šµ))
1918ibllem 25281 . . . . . . . . . . . 12 (šœ‘ ā†’ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / 1))), (ā„œā€˜(šµ / 1)), 0) = if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜šµ)), (ā„œā€˜šµ), 0))
2019mpteq2dv 5250 . . . . . . . . . . 11 (šœ‘ ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / 1))), (ā„œā€˜(šµ / 1)), 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜šµ)), (ā„œā€˜šµ), 0)))
2120fveq2d 6895 . . . . . . . . . 10 (šœ‘ ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / 1))), (ā„œā€˜(šµ / 1)), 0))) = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜šµ)), (ā„œā€˜šµ), 0))))
22 itgcnlem.r . . . . . . . . . 10 š‘… = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜šµ)), (ā„œā€˜šµ), 0)))
2321, 22eqtr4di 2790 . . . . . . . . 9 (šœ‘ ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / 1))), (ā„œā€˜(šµ / 1)), 0))) = š‘…)
2423eleq1d 2818 . . . . . . . 8 (šœ‘ ā†’ ((āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / 1))), (ā„œā€˜(šµ / 1)), 0))) āˆˆ ā„ ā†” š‘… āˆˆ ā„))
25 itgcnlem.t . . . . . . . . . 10 š‘‡ = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„‘ā€˜šµ)), (ā„‘ā€˜šµ), 0)))
26 imval 15053 . . . . . . . . . . . . . 14 (šµ āˆˆ ā„‚ ā†’ (ā„‘ā€˜šµ) = (ā„œā€˜(šµ / i)))
273, 26syl 17 . . . . . . . . . . . . 13 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„‘ā€˜šµ) = (ā„œā€˜(šµ / i)))
2827ibllem 25281 . . . . . . . . . . . 12 (šœ‘ ā†’ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„‘ā€˜šµ)), (ā„‘ā€˜šµ), 0) = if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / i))), (ā„œā€˜(šµ / i)), 0))
2928mpteq2dv 5250 . . . . . . . . . . 11 (šœ‘ ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„‘ā€˜šµ)), (ā„‘ā€˜šµ), 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / i))), (ā„œā€˜(šµ / i)), 0)))
3029fveq2d 6895 . . . . . . . . . 10 (šœ‘ ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„‘ā€˜šµ)), (ā„‘ā€˜šµ), 0))) = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / i))), (ā„œā€˜(šµ / i)), 0))))
3125, 30eqtr2id 2785 . . . . . . . . 9 (šœ‘ ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / i))), (ā„œā€˜(šµ / i)), 0))) = š‘‡)
3231eleq1d 2818 . . . . . . . 8 (šœ‘ ā†’ ((āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / i))), (ā„œā€˜(šµ / i)), 0))) āˆˆ ā„ ā†” š‘‡ āˆˆ ā„))
3324, 32anbi12d 631 . . . . . . 7 (šœ‘ ā†’ (((āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / 1))), (ā„œā€˜(šµ / 1)), 0))) āˆˆ ā„ āˆ§ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / i))), (ā„œā€˜(šµ / i)), 0))) āˆˆ ā„) ā†” (š‘… āˆˆ ā„ āˆ§ š‘‡ āˆˆ ā„)))
3416, 33bitrid 282 . . . . . 6 (šœ‘ ā†’ (āˆ€š‘˜ āˆˆ {0, 1} (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ ā†” (š‘… āˆˆ ā„ āˆ§ š‘‡ āˆˆ ā„)))
35 2ex 12288 . . . . . . . 8 2 āˆˆ V
36 3ex 12293 . . . . . . . 8 3 āˆˆ V
37 i2 14165 . . . . . . . . . 10 (iā†‘2) = -1
3837itgvallem 25301 . . . . . . . . 9 (š‘˜ = 2 ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -1))), (ā„œā€˜(šµ / -1)), 0))))
3938eleq1d 2818 . . . . . . . 8 (š‘˜ = 2 ā†’ ((āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ ā†” (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -1))), (ā„œā€˜(šµ / -1)), 0))) āˆˆ ā„))
40 i3 14166 . . . . . . . . . 10 (iā†‘3) = -i
4140itgvallem 25301 . . . . . . . . 9 (š‘˜ = 3 ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -i))), (ā„œā€˜(šµ / -i)), 0))))
4241eleq1d 2818 . . . . . . . 8 (š‘˜ = 3 ā†’ ((āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ ā†” (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -i))), (ā„œā€˜(šµ / -i)), 0))) āˆˆ ā„))
4335, 36, 39, 42ralpr 4704 . . . . . . 7 (āˆ€š‘˜ āˆˆ {2, 3} (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ ā†” ((āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -1))), (ā„œā€˜(šµ / -1)), 0))) āˆˆ ā„ āˆ§ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -i))), (ā„œā€˜(šµ / -i)), 0))) āˆˆ ā„))
44 itgcnlem.s . . . . . . . . . 10 š‘† = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ -(ā„œā€˜šµ)), -(ā„œā€˜šµ), 0)))
453renegd 15155 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜-šµ) = -(ā„œā€˜šµ))
46 ax-1cn 11167 . . . . . . . . . . . . . . . . . . 19 1 āˆˆ ā„‚
4746negnegi 11529 . . . . . . . . . . . . . . . . . 18 --1 = 1
4847oveq2i 7419 . . . . . . . . . . . . . . . . 17 (-šµ / --1) = (-šµ / 1)
493negcld 11557 . . . . . . . . . . . . . . . . . 18 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ -šµ āˆˆ ā„‚)
5049div1d 11981 . . . . . . . . . . . . . . . . 17 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (-šµ / 1) = -šµ)
5148, 50eqtrid 2784 . . . . . . . . . . . . . . . 16 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (-šµ / --1) = -šµ)
5246negcli 11527 . . . . . . . . . . . . . . . . . 18 -1 āˆˆ ā„‚
53 neg1ne0 12327 . . . . . . . . . . . . . . . . . 18 -1 ā‰  0
54 div2neg 11936 . . . . . . . . . . . . . . . . . 18 ((šµ āˆˆ ā„‚ āˆ§ -1 āˆˆ ā„‚ āˆ§ -1 ā‰  0) ā†’ (-šµ / --1) = (šµ / -1))
5552, 53, 54mp3an23 1453 . . . . . . . . . . . . . . . . 17 (šµ āˆˆ ā„‚ ā†’ (-šµ / --1) = (šµ / -1))
563, 55syl 17 . . . . . . . . . . . . . . . 16 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (-šµ / --1) = (šµ / -1))
5751, 56eqtr3d 2774 . . . . . . . . . . . . . . 15 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ -šµ = (šµ / -1))
5857fveq2d 6895 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜-šµ) = (ā„œā€˜(šµ / -1)))
5945, 58eqtr3d 2774 . . . . . . . . . . . . 13 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ -(ā„œā€˜šµ) = (ā„œā€˜(šµ / -1)))
6059ibllem 25281 . . . . . . . . . . . 12 (šœ‘ ā†’ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ -(ā„œā€˜šµ)), -(ā„œā€˜šµ), 0) = if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -1))), (ā„œā€˜(šµ / -1)), 0))
6160mpteq2dv 5250 . . . . . . . . . . 11 (šœ‘ ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ -(ā„œā€˜šµ)), -(ā„œā€˜šµ), 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -1))), (ā„œā€˜(šµ / -1)), 0)))
6261fveq2d 6895 . . . . . . . . . 10 (šœ‘ ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ -(ā„œā€˜šµ)), -(ā„œā€˜šµ), 0))) = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -1))), (ā„œā€˜(šµ / -1)), 0))))
6344, 62eqtr2id 2785 . . . . . . . . 9 (šœ‘ ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -1))), (ā„œā€˜(šµ / -1)), 0))) = š‘†)
6463eleq1d 2818 . . . . . . . 8 (šœ‘ ā†’ ((āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -1))), (ā„œā€˜(šµ / -1)), 0))) āˆˆ ā„ ā†” š‘† āˆˆ ā„))
65 itgcnlem.u . . . . . . . . . 10 š‘ˆ = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ -(ā„‘ā€˜šµ)), -(ā„‘ā€˜šµ), 0)))
66 imval 15053 . . . . . . . . . . . . . . 15 (-šµ āˆˆ ā„‚ ā†’ (ā„‘ā€˜-šµ) = (ā„œā€˜(-šµ / i)))
6749, 66syl 17 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„‘ā€˜-šµ) = (ā„œā€˜(-šµ / i)))
683imnegd 15156 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„‘ā€˜-šµ) = -(ā„‘ā€˜šµ))
697negnegi 11529 . . . . . . . . . . . . . . . . . 18 --i = i
7069eqcomi 2741 . . . . . . . . . . . . . . . . 17 i = --i
7170oveq2i 7419 . . . . . . . . . . . . . . . 16 (-šµ / i) = (-šµ / --i)
727negcli 11527 . . . . . . . . . . . . . . . . . 18 -i āˆˆ ā„‚
73 ine0 11648 . . . . . . . . . . . . . . . . . . 19 i ā‰  0
747, 73negne0i 11534 . . . . . . . . . . . . . . . . . 18 -i ā‰  0
75 div2neg 11936 . . . . . . . . . . . . . . . . . 18 ((šµ āˆˆ ā„‚ āˆ§ -i āˆˆ ā„‚ āˆ§ -i ā‰  0) ā†’ (-šµ / --i) = (šµ / -i))
7672, 74, 75mp3an23 1453 . . . . . . . . . . . . . . . . 17 (šµ āˆˆ ā„‚ ā†’ (-šµ / --i) = (šµ / -i))
773, 76syl 17 . . . . . . . . . . . . . . . 16 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (-šµ / --i) = (šµ / -i))
7871, 77eqtrid 2784 . . . . . . . . . . . . . . 15 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (-šµ / i) = (šµ / -i))
7978fveq2d 6895 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜(-šµ / i)) = (ā„œā€˜(šµ / -i)))
8067, 68, 793eqtr3d 2780 . . . . . . . . . . . . 13 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ -(ā„‘ā€˜šµ) = (ā„œā€˜(šµ / -i)))
8180ibllem 25281 . . . . . . . . . . . 12 (šœ‘ ā†’ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ -(ā„‘ā€˜šµ)), -(ā„‘ā€˜šµ), 0) = if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -i))), (ā„œā€˜(šµ / -i)), 0))
8281mpteq2dv 5250 . . . . . . . . . . 11 (šœ‘ ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ -(ā„‘ā€˜šµ)), -(ā„‘ā€˜šµ), 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -i))), (ā„œā€˜(šµ / -i)), 0)))
8382fveq2d 6895 . . . . . . . . . 10 (šœ‘ ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ -(ā„‘ā€˜šµ)), -(ā„‘ā€˜šµ), 0))) = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -i))), (ā„œā€˜(šµ / -i)), 0))))
8465, 83eqtr2id 2785 . . . . . . . . 9 (šœ‘ ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -i))), (ā„œā€˜(šµ / -i)), 0))) = š‘ˆ)
8584eleq1d 2818 . . . . . . . 8 (šœ‘ ā†’ ((āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -i))), (ā„œā€˜(šµ / -i)), 0))) āˆˆ ā„ ā†” š‘ˆ āˆˆ ā„))
8664, 85anbi12d 631 . . . . . . 7 (šœ‘ ā†’ (((āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -1))), (ā„œā€˜(šµ / -1)), 0))) āˆˆ ā„ āˆ§ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / -i))), (ā„œā€˜(šµ / -i)), 0))) āˆˆ ā„) ā†” (š‘† āˆˆ ā„ āˆ§ š‘ˆ āˆˆ ā„)))
8743, 86bitrid 282 . . . . . 6 (šœ‘ ā†’ (āˆ€š‘˜ āˆˆ {2, 3} (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ ā†” (š‘† āˆˆ ā„ āˆ§ š‘ˆ āˆˆ ā„)))
8834, 87anbi12d 631 . . . . 5 (šœ‘ ā†’ ((āˆ€š‘˜ āˆˆ {0, 1} (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ āˆ§ āˆ€š‘˜ āˆˆ {2, 3} (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„) ā†” ((š‘… āˆˆ ā„ āˆ§ š‘‡ āˆˆ ā„) āˆ§ (š‘† āˆˆ ā„ āˆ§ š‘ˆ āˆˆ ā„))))
89 fz0to3un2pr 13602 . . . . . . 7 (0...3) = ({0, 1} āˆŖ {2, 3})
9089raleqi 3323 . . . . . 6 (āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ ā†” āˆ€š‘˜ āˆˆ ({0, 1} āˆŖ {2, 3})(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„)
91 ralunb 4191 . . . . . 6 (āˆ€š‘˜ āˆˆ ({0, 1} āˆŖ {2, 3})(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ ā†” (āˆ€š‘˜ āˆˆ {0, 1} (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ āˆ§ āˆ€š‘˜ āˆˆ {2, 3} (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„))
9290, 91bitri 274 . . . . 5 (āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ ā†” (āˆ€š‘˜ āˆˆ {0, 1} (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ āˆ§ āˆ€š‘˜ āˆˆ {2, 3} (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„))
93 an4 654 . . . . 5 (((š‘… āˆˆ ā„ āˆ§ š‘† āˆˆ ā„) āˆ§ (š‘‡ āˆˆ ā„ āˆ§ š‘ˆ āˆˆ ā„)) ā†” ((š‘… āˆˆ ā„ āˆ§ š‘‡ āˆˆ ā„) āˆ§ (š‘† āˆˆ ā„ āˆ§ š‘ˆ āˆˆ ā„)))
9488, 92, 933bitr4g 313 . . . 4 (šœ‘ ā†’ (āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ ā†” ((š‘… āˆˆ ā„ āˆ§ š‘† āˆˆ ā„) āˆ§ (š‘‡ āˆˆ ā„ āˆ§ š‘ˆ āˆˆ ā„))))
9594anbi2d 629 . . 3 (šœ‘ ā†’ (((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ MblFn āˆ§ āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„) ā†” ((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ MblFn āˆ§ ((š‘… āˆˆ ā„ āˆ§ š‘† āˆˆ ā„) āˆ§ (š‘‡ āˆˆ ā„ āˆ§ š‘ˆ āˆˆ ā„)))))
96 3anass 1095 . . 3 (((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ MblFn āˆ§ (š‘… āˆˆ ā„ āˆ§ š‘† āˆˆ ā„) āˆ§ (š‘‡ āˆˆ ā„ āˆ§ š‘ˆ āˆˆ ā„)) ā†” ((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ MblFn āˆ§ ((š‘… āˆˆ ā„ āˆ§ š‘† āˆˆ ā„) āˆ§ (š‘‡ āˆˆ ā„ āˆ§ š‘ˆ āˆˆ ā„))))
9795, 96bitr4di 288 . 2 (šœ‘ ā†’ (((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ MblFn āˆ§ āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„) ā†” ((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ MblFn āˆ§ (š‘… āˆˆ ā„ āˆ§ š‘† āˆˆ ā„) āˆ§ (š‘‡ āˆˆ ā„ āˆ§ š‘ˆ āˆˆ ā„))))
984, 97bitrd 278 1 (šœ‘ ā†’ ((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ šæ1 ā†” ((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ MblFn āˆ§ (š‘… āˆˆ ā„ āˆ§ š‘† āˆˆ ā„) āˆ§ (š‘‡ āˆˆ ā„ āˆ§ š‘ˆ āˆˆ ā„))))
Colors of variables: wff setvar class
Syntax hints:   ā†’ wi 4   ā†” wb 205   āˆ§ wa 396   āˆ§ w3a 1087   = wceq 1541   āˆˆ wcel 2106   ā‰  wne 2940  āˆ€wral 3061   āˆŖ cun 3946  ifcif 4528  {cpr 4630   class class class wbr 5148   ā†¦ cmpt 5231  ā€˜cfv 6543  (class class class)co 7408  ā„‚cc 11107  ā„cr 11108  0cc0 11109  1c1 11110  ici 11111   ā‰¤ cle 11248  -cneg 11444   / cdiv 11870  2c2 12266  3c3 12267  ...cfz 13483  ā†‘cexp 14026  ā„œcre 15043  ā„‘cim 15044  MblFncmbf 25130  āˆ«2citg2 25132  šæ1cibl 25133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-3 12275  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-seq 13966  df-exp 14027  df-cj 15045  df-re 15046  df-im 15047  df-ibl 25138
This theorem is referenced by:  iblcnlem  25305  iblcn  25315  bddiblnc  25358
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