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Theorem iblcnlem1 25672
Description: Lemma for iblcnlem 25673. (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 2727 . . 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 2727 . . 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 25651 . 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 11212 . . . . . . . 8 0 āˆˆ V
6 1ex 11214 . . . . . . . 8 1 āˆˆ V
7 ax-icn 11171 . . . . . . . . . . 11 i āˆˆ ā„‚
8 exp0 14036 . . . . . . . . . . 11 (i āˆˆ ā„‚ ā†’ (iā†‘0) = 1)
97, 8ax-mp 5 . . . . . . . . . 10 (iā†‘0) = 1
109itgvallem 25669 . . . . . . . . 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 2812 . . . . . . . 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 14038 . . . . . . . . . . 11 (i āˆˆ ā„‚ ā†’ (iā†‘1) = i)
137, 12ax-mp 5 . . . . . . . . . 10 (iā†‘1) = i
1413itgvallem 25669 . . . . . . . . 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 2812 . . . . . . . 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 4699 . . . . . . 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 11986 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (šµ / 1) = šµ)
1817fveq2d 6889 . . . . . . . . . . . . 13 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜(šµ / 1)) = (ā„œā€˜šµ))
1918ibllem 25649 . . . . . . . . . . . 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 5243 . . . . . . . . . . 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 6889 . . . . . . . . . 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 2784 . . . . . . . . 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 2812 . . . . . . . 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 15060 . . . . . . . . . . . . . 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 25649 . . . . . . . . . . . 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 5243 . . . . . . . . . . 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 6889 . . . . . . . . . 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 2779 . . . . . . . . 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 2812 . . . . . . . 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 630 . . . . . . 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 283 . . . . . 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 12293 . . . . . . . 8 2 āˆˆ V
36 3ex 12298 . . . . . . . 8 3 āˆˆ V
37 i2 14171 . . . . . . . . . 10 (iā†‘2) = -1
3837itgvallem 25669 . . . . . . . . 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 2812 . . . . . . . 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 14172 . . . . . . . . . 10 (iā†‘3) = -i
4140itgvallem 25669 . . . . . . . . 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 2812 . . . . . . . 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 4699 . . . . . . 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 15162 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜-šµ) = -(ā„œā€˜šµ))
46 ax-1cn 11170 . . . . . . . . . . . . . . . . . . 19 1 āˆˆ ā„‚
4746negnegi 11534 . . . . . . . . . . . . . . . . . 18 --1 = 1
4847oveq2i 7416 . . . . . . . . . . . . . . . . 17 (-šµ / --1) = (-šµ / 1)
493negcld 11562 . . . . . . . . . . . . . . . . . 18 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ -šµ āˆˆ ā„‚)
5049div1d 11986 . . . . . . . . . . . . . . . . 17 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (-šµ / 1) = -šµ)
5148, 50eqtrid 2778 . . . . . . . . . . . . . . . 16 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (-šµ / --1) = -šµ)
5246negcli 11532 . . . . . . . . . . . . . . . . . 18 -1 āˆˆ ā„‚
53 neg1ne0 12332 . . . . . . . . . . . . . . . . . 18 -1 ā‰  0
54 div2neg 11941 . . . . . . . . . . . . . . . . . 18 ((šµ āˆˆ ā„‚ āˆ§ -1 āˆˆ ā„‚ āˆ§ -1 ā‰  0) ā†’ (-šµ / --1) = (šµ / -1))
5552, 53, 54mp3an23 1449 . . . . . . . . . . . . . . . . 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 2768 . . . . . . . . . . . . . . 15 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ -šµ = (šµ / -1))
5857fveq2d 6889 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜-šµ) = (ā„œā€˜(šµ / -1)))
5945, 58eqtr3d 2768 . . . . . . . . . . . . 13 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ -(ā„œā€˜šµ) = (ā„œā€˜(šµ / -1)))
6059ibllem 25649 . . . . . . . . . . . 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 5243 . . . . . . . . . . 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 6889 . . . . . . . . . 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 2779 . . . . . . . . 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 2812 . . . . . . . 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 15060 . . . . . . . . . . . . . . 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 15163 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„‘ā€˜-šµ) = -(ā„‘ā€˜šµ))
697negnegi 11534 . . . . . . . . . . . . . . . . . 18 --i = i
7069eqcomi 2735 . . . . . . . . . . . . . . . . 17 i = --i
7170oveq2i 7416 . . . . . . . . . . . . . . . 16 (-šµ / i) = (-šµ / --i)
727negcli 11532 . . . . . . . . . . . . . . . . . 18 -i āˆˆ ā„‚
73 ine0 11653 . . . . . . . . . . . . . . . . . . 19 i ā‰  0
747, 73negne0i 11539 . . . . . . . . . . . . . . . . . 18 -i ā‰  0
75 div2neg 11941 . . . . . . . . . . . . . . . . . 18 ((šµ āˆˆ ā„‚ āˆ§ -i āˆˆ ā„‚ āˆ§ -i ā‰  0) ā†’ (-šµ / --i) = (šµ / -i))
7672, 74, 75mp3an23 1449 . . . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . . 15 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (-šµ / i) = (šµ / -i))
7978fveq2d 6889 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜(-šµ / i)) = (ā„œā€˜(šµ / -i)))
8067, 68, 793eqtr3d 2774 . . . . . . . . . . . . 13 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ -(ā„‘ā€˜šµ) = (ā„œā€˜(šµ / -i)))
8180ibllem 25649 . . . . . . . . . . . 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 5243 . . . . . . . . . . 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 6889 . . . . . . . . . 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 2779 . . . . . . . . 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 2812 . . . . . . . 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 630 . . . . . . 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 283 . . . . . 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 630 . . . . 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 13609 . . . . . . 7 (0...3) = ({0, 1} āˆŖ {2, 3})
9089raleqi 3317 . . . . . 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 4186 . . . . . 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 275 . . . . 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 653 . . . . 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 314 . . . 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 628 . . 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 1092 . . 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 289 . 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 279 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 395   āˆ§ w3a 1084   = wceq 1533   āˆˆ wcel 2098   ā‰  wne 2934  āˆ€wral 3055   āˆŖ cun 3941  ifcif 4523  {cpr 4625   class class class wbr 5141   ā†¦ cmpt 5224  ā€˜cfv 6537  (class class class)co 7405  ā„‚cc 11110  ā„cr 11111  0cc0 11112  1c1 11113  ici 11114   ā‰¤ cle 11253  -cneg 11449   / cdiv 11875  2c2 12271  3c3 12272  ...cfz 13490  ā†‘cexp 14032  ā„œcre 15050  ā„‘cim 15051  MblFncmbf 25498  āˆ«2citg2 25500  šæ1cibl 25501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-seq 13973  df-exp 14033  df-cj 15052  df-re 15053  df-im 15054  df-ibl 25506
This theorem is referenced by:  iblcnlem  25673  iblcn  25683  bddiblnc  25726
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