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Theorem iblcnlem1 25175
Description: Lemma for iblcnlem 25176. (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 2734 . . 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 2734 . . 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 25154 . 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 11157 . . . . . . . 8 0 āˆˆ V
6 1ex 11159 . . . . . . . 8 1 āˆˆ V
7 ax-icn 11118 . . . . . . . . . . 11 i āˆˆ ā„‚
8 exp0 13980 . . . . . . . . . . 11 (i āˆˆ ā„‚ ā†’ (iā†‘0) = 1)
97, 8ax-mp 5 . . . . . . . . . 10 (iā†‘0) = 1
109itgvallem 25172 . . . . . . . . 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 2819 . . . . . . . 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 13982 . . . . . . . . . . 11 (i āˆˆ ā„‚ ā†’ (iā†‘1) = i)
137, 12ax-mp 5 . . . . . . . . . 10 (iā†‘1) = i
1413itgvallem 25172 . . . . . . . . 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 2819 . . . . . . . 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 4665 . . . . . . 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 11931 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (šµ / 1) = šµ)
1817fveq2d 6850 . . . . . . . . . . . . 13 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜(šµ / 1)) = (ā„œā€˜šµ))
1918ibllem 25152 . . . . . . . . . . . 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 5211 . . . . . . . . . . 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 6850 . . . . . . . . . 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 2791 . . . . . . . . 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 2819 . . . . . . . 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 15001 . . . . . . . . . . . . . 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 25152 . . . . . . . . . . . 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 5211 . . . . . . . . . . 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 6850 . . . . . . . . . 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 2786 . . . . . . . . 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 2819 . . . . . . . 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 632 . . . . . . 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 12238 . . . . . . . 8 2 āˆˆ V
36 3ex 12243 . . . . . . . 8 3 āˆˆ V
37 i2 14115 . . . . . . . . . 10 (iā†‘2) = -1
3837itgvallem 25172 . . . . . . . . 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 2819 . . . . . . . 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 14116 . . . . . . . . . 10 (iā†‘3) = -i
4140itgvallem 25172 . . . . . . . . 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 2819 . . . . . . . 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 4665 . . . . . . 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 15103 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜-šµ) = -(ā„œā€˜šµ))
46 ax-1cn 11117 . . . . . . . . . . . . . . . . . . 19 1 āˆˆ ā„‚
4746negnegi 11479 . . . . . . . . . . . . . . . . . 18 --1 = 1
4847oveq2i 7372 . . . . . . . . . . . . . . . . 17 (-šµ / --1) = (-šµ / 1)
493negcld 11507 . . . . . . . . . . . . . . . . . 18 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ -šµ āˆˆ ā„‚)
5049div1d 11931 . . . . . . . . . . . . . . . . 17 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (-šµ / 1) = -šµ)
5148, 50eqtrid 2785 . . . . . . . . . . . . . . . 16 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (-šµ / --1) = -šµ)
5246negcli 11477 . . . . . . . . . . . . . . . . . 18 -1 āˆˆ ā„‚
53 neg1ne0 12277 . . . . . . . . . . . . . . . . . 18 -1 ā‰  0
54 div2neg 11886 . . . . . . . . . . . . . . . . . 18 ((šµ āˆˆ ā„‚ āˆ§ -1 āˆˆ ā„‚ āˆ§ -1 ā‰  0) ā†’ (-šµ / --1) = (šµ / -1))
5552, 53, 54mp3an23 1454 . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . 15 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ -šµ = (šµ / -1))
5857fveq2d 6850 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜-šµ) = (ā„œā€˜(šµ / -1)))
5945, 58eqtr3d 2775 . . . . . . . . . . . . 13 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ -(ā„œā€˜šµ) = (ā„œā€˜(šµ / -1)))
6059ibllem 25152 . . . . . . . . . . . 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 5211 . . . . . . . . . . 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 6850 . . . . . . . . . 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 2786 . . . . . . . . 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 2819 . . . . . . . 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 15001 . . . . . . . . . . . . . . 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 15104 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„‘ā€˜-šµ) = -(ā„‘ā€˜šµ))
697negnegi 11479 . . . . . . . . . . . . . . . . . 18 --i = i
7069eqcomi 2742 . . . . . . . . . . . . . . . . 17 i = --i
7170oveq2i 7372 . . . . . . . . . . . . . . . 16 (-šµ / i) = (-šµ / --i)
727negcli 11477 . . . . . . . . . . . . . . . . . 18 -i āˆˆ ā„‚
73 ine0 11598 . . . . . . . . . . . . . . . . . . 19 i ā‰  0
747, 73negne0i 11484 . . . . . . . . . . . . . . . . . 18 -i ā‰  0
75 div2neg 11886 . . . . . . . . . . . . . . . . . 18 ((šµ āˆˆ ā„‚ āˆ§ -i āˆˆ ā„‚ āˆ§ -i ā‰  0) ā†’ (-šµ / --i) = (šµ / -i))
7672, 74, 75mp3an23 1454 . . . . . . . . . . . . . . . . 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 2785 . . . . . . . . . . . . . . 15 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (-šµ / i) = (šµ / -i))
7978fveq2d 6850 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜(-šµ / i)) = (ā„œā€˜(šµ / -i)))
8067, 68, 793eqtr3d 2781 . . . . . . . . . . . . 13 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ -(ā„‘ā€˜šµ) = (ā„œā€˜(šµ / -i)))
8180ibllem 25152 . . . . . . . . . . . 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 5211 . . . . . . . . . . 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 6850 . . . . . . . . . 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 2786 . . . . . . . . 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 2819 . . . . . . . 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 632 . . . . . . 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 632 . . . . 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 13552 . . . . . . 7 (0...3) = ({0, 1} āˆŖ {2, 3})
9089raleqi 3310 . . . . . 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 4155 . . . . . 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 655 . . . . 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 630 . . 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 1096 . . 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 397   āˆ§ w3a 1088   = wceq 1542   āˆˆ wcel 2107   ā‰  wne 2940  āˆ€wral 3061   āˆŖ cun 3912  ifcif 4490  {cpr 4592   class class class wbr 5109   ā†¦ cmpt 5192  ā€˜cfv 6500  (class class class)co 7361  ā„‚cc 11057  ā„cr 11058  0cc0 11059  1c1 11060  ici 11061   ā‰¤ cle 11198  -cneg 11394   / cdiv 11820  2c2 12216  3c3 12217  ...cfz 13433  ā†‘cexp 13976  ā„œcre 14991  ā„‘cim 14992  MblFncmbf 25001  āˆ«2citg2 25003  šæ1cibl 25004
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-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  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-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  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-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  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-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  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-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-seq 13916  df-exp 13977  df-cj 14993  df-re 14994  df-im 14995  df-ibl 25009
This theorem is referenced by:  iblcnlem  25176  iblcn  25186  bddiblnc  25229
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