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Theorem sibf0 29723
Description: The constant zero function is a simple function. (Contributed by Thierry Arnoux, 4-Mar-2018.)
Hypotheses
Ref Expression
sitgval.b 𝐵 = (Base‘𝑊)
sitgval.j 𝐽 = (TopOpen‘𝑊)
sitgval.s 𝑆 = (sigaGen‘𝐽)
sitgval.0 0 = (0g𝑊)
sitgval.x · = ( ·𝑠𝑊)
sitgval.h 𝐻 = (ℝHom‘(Scalar‘𝑊))
sitgval.1 (𝜑𝑊𝑉)
sitgval.2 (𝜑𝑀 ran measures)
sibf0.1 (𝜑𝑊 ∈ TopSp)
sibf0.2 (𝜑𝑊 ∈ Mnd)
Assertion
Ref Expression
sibf0 (𝜑 → ( dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀))

Proof of Theorem sibf0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sitgval.2 . . . 4 (𝜑𝑀 ran measures)
2 dmmeas 29591 . . . 4 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
31, 2syl 17 . . 3 (𝜑 → dom 𝑀 ran sigAlgebra)
4 sitgval.s . . . 4 𝑆 = (sigaGen‘𝐽)
5 sitgval.j . . . . . . 7 𝐽 = (TopOpen‘𝑊)
6 fvex 6113 . . . . . . 7 (TopOpen‘𝑊) ∈ V
75, 6eqeltri 2684 . . . . . 6 𝐽 ∈ V
87a1i 11 . . . . 5 (𝜑𝐽 ∈ V)
98sgsiga 29532 . . . 4 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
104, 9syl5eqel 2692 . . 3 (𝜑𝑆 ran sigAlgebra)
11 fconstmpt 5085 . . . 4 ( dom 𝑀 × { 0 }) = (𝑥 dom 𝑀0 )
1211a1i 11 . . 3 (𝜑 → ( dom 𝑀 × { 0 }) = (𝑥 dom 𝑀0 ))
13 sibf0.2 . . . . 5 (𝜑𝑊 ∈ Mnd)
14 sitgval.b . . . . . 6 𝐵 = (Base‘𝑊)
15 sitgval.0 . . . . . 6 0 = (0g𝑊)
1614, 15mndidcl 17131 . . . . 5 (𝑊 ∈ Mnd → 0𝐵)
1713, 16syl 17 . . . 4 (𝜑0𝐵)
18 sibf0.1 . . . . . 6 (𝜑𝑊 ∈ TopSp)
1914, 5tpsuni 20553 . . . . . 6 (𝑊 ∈ TopSp → 𝐵 = 𝐽)
2018, 19syl 17 . . . . 5 (𝜑𝐵 = 𝐽)
214unieqi 4381 . . . . . 6 𝑆 = (sigaGen‘𝐽)
22 unisg 29533 . . . . . . 7 (𝐽 ∈ V → (sigaGen‘𝐽) = 𝐽)
237, 22mp1i 13 . . . . . 6 (𝜑 (sigaGen‘𝐽) = 𝐽)
2421, 23syl5eq 2656 . . . . 5 (𝜑 𝑆 = 𝐽)
2520, 24eqtr4d 2647 . . . 4 (𝜑𝐵 = 𝑆)
2617, 25eleqtrd 2690 . . 3 (𝜑0 𝑆)
273, 10, 12, 26mbfmcst 29648 . 2 (𝜑 → ( dom 𝑀 × { 0 }) ∈ (dom 𝑀MblFnM𝑆))
28 xpeq1 5052 . . . . . . . 8 ( dom 𝑀 = ∅ → ( dom 𝑀 × { 0 }) = (∅ × { 0 }))
29 0xp 5122 . . . . . . . 8 (∅ × { 0 }) = ∅
3028, 29syl6eq 2660 . . . . . . 7 ( dom 𝑀 = ∅ → ( dom 𝑀 × { 0 }) = ∅)
3130rneqd 5274 . . . . . 6 ( dom 𝑀 = ∅ → ran ( dom 𝑀 × { 0 }) = ran ∅)
32 rn0 5298 . . . . . 6 ran ∅ = ∅
3331, 32syl6eq 2660 . . . . 5 ( dom 𝑀 = ∅ → ran ( dom 𝑀 × { 0 }) = ∅)
34 0fin 8073 . . . . 5 ∅ ∈ Fin
3533, 34syl6eqel 2696 . . . 4 ( dom 𝑀 = ∅ → ran ( dom 𝑀 × { 0 }) ∈ Fin)
36 rnxp 5483 . . . . 5 ( dom 𝑀 ≠ ∅ → ran ( dom 𝑀 × { 0 }) = { 0 })
37 snfi 7923 . . . . 5 { 0 } ∈ Fin
3836, 37syl6eqel 2696 . . . 4 ( dom 𝑀 ≠ ∅ → ran ( dom 𝑀 × { 0 }) ∈ Fin)
3935, 38pm2.61ine 2865 . . 3 ran ( dom 𝑀 × { 0 }) ∈ Fin
4039a1i 11 . 2 (𝜑 → ran ( dom 𝑀 × { 0 }) ∈ Fin)
41 noel 3878 . . . . . 6 ¬ 𝑥 ∈ ∅
4233difeq1d 3689 . . . . . . . . 9 ( dom 𝑀 = ∅ → (ran ( dom 𝑀 × { 0 }) ∖ { 0 }) = (∅ ∖ { 0 }))
43 0dif 3929 . . . . . . . . 9 (∅ ∖ { 0 }) = ∅
4442, 43syl6eq 2660 . . . . . . . 8 ( dom 𝑀 = ∅ → (ran ( dom 𝑀 × { 0 }) ∖ { 0 }) = ∅)
4536difeq1d 3689 . . . . . . . . 9 ( dom 𝑀 ≠ ∅ → (ran ( dom 𝑀 × { 0 }) ∖ { 0 }) = ({ 0 } ∖ { 0 }))
46 difid 3902 . . . . . . . . 9 ({ 0 } ∖ { 0 }) = ∅
4745, 46syl6eq 2660 . . . . . . . 8 ( dom 𝑀 ≠ ∅ → (ran ( dom 𝑀 × { 0 }) ∖ { 0 }) = ∅)
4844, 47pm2.61ine 2865 . . . . . . 7 (ran ( dom 𝑀 × { 0 }) ∖ { 0 }) = ∅
4948eleq2i 2680 . . . . . 6 (𝑥 ∈ (ran ( dom 𝑀 × { 0 }) ∖ { 0 }) ↔ 𝑥 ∈ ∅)
5041, 49mtbir 312 . . . . 5 ¬ 𝑥 ∈ (ran ( dom 𝑀 × { 0 }) ∖ { 0 })
5150pm2.21i 115 . . . 4 (𝑥 ∈ (ran ( dom 𝑀 × { 0 }) ∖ { 0 }) → (𝑀‘(( dom 𝑀 × { 0 }) “ {𝑥})) ∈ (0[,)+∞))
5251adantl 481 . . 3 ((𝜑𝑥 ∈ (ran ( dom 𝑀 × { 0 }) ∖ { 0 })) → (𝑀‘(( dom 𝑀 × { 0 }) “ {𝑥})) ∈ (0[,)+∞))
5352ralrimiva 2949 . 2 (𝜑 → ∀𝑥 ∈ (ran ( dom 𝑀 × { 0 }) ∖ { 0 })(𝑀‘(( dom 𝑀 × { 0 }) “ {𝑥})) ∈ (0[,)+∞))
54 sitgval.x . . 3 · = ( ·𝑠𝑊)
55 sitgval.h . . 3 𝐻 = (ℝHom‘(Scalar‘𝑊))
56 sitgval.1 . . 3 (𝜑𝑊𝑉)
5714, 5, 4, 15, 54, 55, 56, 1issibf 29722 . 2 (𝜑 → (( dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀) ↔ (( dom 𝑀 × { 0 }) ∈ (dom 𝑀MblFnM𝑆) ∧ ran ( dom 𝑀 × { 0 }) ∈ Fin ∧ ∀𝑥 ∈ (ran ( dom 𝑀 × { 0 }) ∖ { 0 })(𝑀‘(( dom 𝑀 × { 0 }) “ {𝑥})) ∈ (0[,)+∞))))
5827, 40, 53, 57mpbir3and 1238 1 (𝜑 → ( dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  wne 2780  wral 2896  Vcvv 3173  cdif 3537  c0 3874  {csn 4125   cuni 4372  cmpt 4643   × cxp 5036  ccnv 5037  dom cdm 5038  ran crn 5039  cima 5041  cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  +∞cpnf 9950  [,)cico 12048  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  TopOpenctopn 15905  0gc0g 15923  Mndcmnd 17117  TopSpctps 20519  ℝHomcrrh 29365  sigAlgebracsiga 29497  sigaGencsigagen 29528  measurescmeas 29585  MblFnMcmbfm 29639  sitgcsitg 29718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1o 7447  df-map 7746  df-en 7842  df-fin 7845  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-top 20521  df-topon 20523  df-topsp 20524  df-esum 29417  df-siga 29498  df-sigagen 29529  df-meas 29586  df-mbfm 29640  df-sitg 29719
This theorem is referenced by:  sitg0  29735
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