Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sibfof Structured version   Visualization version   GIF version

Theorem sibfof 30723
Description: Applying function operations on simple functions results in simple functions with regard to the destination space, provided the operation fulfills a simple condition. (Contributed by Thierry Arnoux, 12-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)
sibfmbl.1 (𝜑𝐹 ∈ dom (𝑊sitg𝑀))
sibfof.c 𝐶 = (Base‘𝐾)
sibfof.0 (𝜑𝑊 ∈ TopSp)
sibfof.1 (𝜑+ :(𝐵 × 𝐵)⟶𝐶)
sibfof.2 (𝜑𝐺 ∈ dom (𝑊sitg𝑀))
sibfof.3 (𝜑𝐾 ∈ TopSp)
sibfof.4 (𝜑𝐽 ∈ Fre)
sibfof.5 (𝜑 → ( 0 + 0 ) = (0g𝐾))
Assertion
Ref Expression
sibfof (𝜑 → (𝐹𝑓 + 𝐺) ∈ dom (𝐾sitg𝑀))

Proof of Theorem sibfof
Dummy variables 𝑥 𝑦 𝑧 𝑝 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sibfof.1 . . . . . . . 8 (𝜑+ :(𝐵 × 𝐵)⟶𝐶)
2 sibfof.0 . . . . . . . . . . 11 (𝜑𝑊 ∈ TopSp)
3 sitgval.b . . . . . . . . . . . 12 𝐵 = (Base‘𝑊)
4 sitgval.j . . . . . . . . . . . 12 𝐽 = (TopOpen‘𝑊)
53, 4tpsuni 20950 . . . . . . . . . . 11 (𝑊 ∈ TopSp → 𝐵 = 𝐽)
62, 5syl 17 . . . . . . . . . 10 (𝜑𝐵 = 𝐽)
76sqxpeqd 5342 . . . . . . . . 9 (𝜑 → (𝐵 × 𝐵) = ( 𝐽 × 𝐽))
87feq2d 6238 . . . . . . . 8 (𝜑 → ( + :(𝐵 × 𝐵)⟶𝐶+ :( 𝐽 × 𝐽)⟶𝐶))
91, 8mpbid 223 . . . . . . 7 (𝜑+ :( 𝐽 × 𝐽)⟶𝐶)
109fovrnda 7031 . . . . . 6 ((𝜑 ∧ (𝑧 𝐽𝑥 𝐽)) → (𝑧 + 𝑥) ∈ 𝐶)
11 sitgval.s . . . . . . 7 𝑆 = (sigaGen‘𝐽)
12 sitgval.0 . . . . . . 7 0 = (0g𝑊)
13 sitgval.x . . . . . . 7 · = ( ·𝑠𝑊)
14 sitgval.h . . . . . . 7 𝐻 = (ℝHom‘(Scalar‘𝑊))
15 sitgval.1 . . . . . . 7 (𝜑𝑊𝑉)
16 sitgval.2 . . . . . . 7 (𝜑𝑀 ran measures)
17 sibfmbl.1 . . . . . . 7 (𝜑𝐹 ∈ dom (𝑊sitg𝑀))
183, 4, 11, 12, 13, 14, 15, 16, 17sibff 30719 . . . . . 6 (𝜑𝐹: dom 𝑀 𝐽)
19 sibfof.2 . . . . . . 7 (𝜑𝐺 ∈ dom (𝑊sitg𝑀))
203, 4, 11, 12, 13, 14, 15, 16, 19sibff 30719 . . . . . 6 (𝜑𝐺: dom 𝑀 𝐽)
21 dmexg 7323 . . . . . . 7 (𝑀 ran measures → dom 𝑀 ∈ V)
22 uniexg 7181 . . . . . . 7 (dom 𝑀 ∈ V → dom 𝑀 ∈ V)
2316, 21, 223syl 18 . . . . . 6 (𝜑 dom 𝑀 ∈ V)
24 inidm 4019 . . . . . 6 ( dom 𝑀 dom 𝑀) = dom 𝑀
2510, 18, 20, 23, 23, 24off 7138 . . . . 5 (𝜑 → (𝐹𝑓 + 𝐺): dom 𝑀𝐶)
26 sibfof.3 . . . . . . . 8 (𝜑𝐾 ∈ TopSp)
27 sibfof.c . . . . . . . . 9 𝐶 = (Base‘𝐾)
28 eqid 2806 . . . . . . . . 9 (TopOpen‘𝐾) = (TopOpen‘𝐾)
2927, 28tpsuni 20950 . . . . . . . 8 (𝐾 ∈ TopSp → 𝐶 = (TopOpen‘𝐾))
3026, 29syl 17 . . . . . . 7 (𝜑𝐶 = (TopOpen‘𝐾))
31 fvex 6417 . . . . . . . 8 (TopOpen‘𝐾) ∈ V
32 unisg 30527 . . . . . . . 8 ((TopOpen‘𝐾) ∈ V → (sigaGen‘(TopOpen‘𝐾)) = (TopOpen‘𝐾))
3331, 32ax-mp 5 . . . . . . 7 (sigaGen‘(TopOpen‘𝐾)) = (TopOpen‘𝐾)
3430, 33syl6eqr 2858 . . . . . 6 (𝜑𝐶 = (sigaGen‘(TopOpen‘𝐾)))
3534feq3d 6239 . . . . 5 (𝜑 → ((𝐹𝑓 + 𝐺): dom 𝑀𝐶 ↔ (𝐹𝑓 + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾))))
3625, 35mpbid 223 . . . 4 (𝜑 → (𝐹𝑓 + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾)))
3731a1i 11 . . . . . . 7 (𝜑 → (TopOpen‘𝐾) ∈ V)
3837sgsiga 30526 . . . . . 6 (𝜑 → (sigaGen‘(TopOpen‘𝐾)) ∈ ran sigAlgebra)
39 uniexg 7181 . . . . . 6 ((sigaGen‘(TopOpen‘𝐾)) ∈ ran sigAlgebra → (sigaGen‘(TopOpen‘𝐾)) ∈ V)
4038, 39syl 17 . . . . 5 (𝜑 (sigaGen‘(TopOpen‘𝐾)) ∈ V)
4140, 23elmapd 8102 . . . 4 (𝜑 → ((𝐹𝑓 + 𝐺) ∈ ( (sigaGen‘(TopOpen‘𝐾)) ↑𝑚 dom 𝑀) ↔ (𝐹𝑓 + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾))))
4236, 41mpbird 248 . . 3 (𝜑 → (𝐹𝑓 + 𝐺) ∈ ( (sigaGen‘(TopOpen‘𝐾)) ↑𝑚 dom 𝑀))
43 inundif 4242 . . . . . . 7 ((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ∪ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) = 𝑏
4443imaeq2i 5674 . . . . . 6 ((𝐹𝑓 + 𝐺) “ ((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ∪ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))) = ((𝐹𝑓 + 𝐺) “ 𝑏)
45 ffun 6255 . . . . . . . 8 ((𝐹𝑓 + 𝐺): dom 𝑀𝐶 → Fun (𝐹𝑓 + 𝐺))
46 unpreima 6559 . . . . . . . 8 (Fun (𝐹𝑓 + 𝐺) → ((𝐹𝑓 + 𝐺) “ ((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ∪ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))) = (((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∪ ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))))
4725, 45, 463syl 18 . . . . . . 7 (𝜑 → ((𝐹𝑓 + 𝐺) “ ((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ∪ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))) = (((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∪ ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))))
4847adantr 468 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ ((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ∪ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))) = (((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∪ ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))))
4944, 48syl5eqr 2854 . . . . 5 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ 𝑏) = (((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∪ ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))))
50 dmmeas 30585 . . . . . . . 8 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
5116, 50syl 17 . . . . . . 7 (𝜑 → dom 𝑀 ran sigAlgebra)
5251adantr 468 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → dom 𝑀 ran sigAlgebra)
53 imaiun 6723 . . . . . . . 8 ((𝐹𝑓 + 𝐺) “ 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)){𝑧}) = 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧})
54 iunid 4767 . . . . . . . . 9 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)){𝑧} = (𝑏 ∩ ran (𝐹𝑓 + 𝐺))
5554imaeq2i 5674 . . . . . . . 8 ((𝐹𝑓 + 𝐺) “ 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)){𝑧}) = ((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)))
5653, 55eqtr3i 2830 . . . . . . 7 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) = ((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)))
57 inss2 4030 . . . . . . . . . 10 (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ⊆ ran (𝐹𝑓 + 𝐺)
586feq3d 6239 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹: dom 𝑀𝐵𝐹: dom 𝑀 𝐽))
5918, 58mpbird 248 . . . . . . . . . . . . . 14 (𝜑𝐹: dom 𝑀𝐵)
606feq3d 6239 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺: dom 𝑀𝐵𝐺: dom 𝑀 𝐽))
6120, 60mpbird 248 . . . . . . . . . . . . . 14 (𝜑𝐺: dom 𝑀𝐵)
621ffnd 6253 . . . . . . . . . . . . . 14 (𝜑+ Fn (𝐵 × 𝐵))
6359, 61, 23, 62ofpreima2 29789 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝑓 + 𝐺) “ {𝑧}) = 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
6463adantr 468 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → ((𝐹𝑓 + 𝐺) “ {𝑧}) = 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
6551adantr 468 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → dom 𝑀 ran sigAlgebra)
6651ad2antrr 708 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → dom 𝑀 ran sigAlgebra)
67 simpll 774 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
68 inss1 4029 . . . . . . . . . . . . . . . . . 18 (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ ( + “ {𝑧})
69 cnvimass 5695 . . . . . . . . . . . . . . . . . . . 20 ( + “ {𝑧}) ⊆ dom +
7069, 1fssdm 6268 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ( + “ {𝑧}) ⊆ (𝐵 × 𝐵))
7170adantr 468 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → ( + “ {𝑧}) ⊆ (𝐵 × 𝐵))
7268, 71syl5ss 3809 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐵))
7372sselda 3798 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐵))
7451adantr 468 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → dom 𝑀 ran sigAlgebra)
75 sibfof.4 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ Fre)
7675sgsiga 30526 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
7711, 76syl5eqel 2889 . . . . . . . . . . . . . . . . . 18 (𝜑𝑆 ran sigAlgebra)
7877adantr 468 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝑆 ran sigAlgebra)
793, 4, 11, 12, 13, 14, 15, 16, 17sibfmbl 30718 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
8079adantr 468 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
814tpstop 20951 . . . . . . . . . . . . . . . . . . . . 21 (𝑊 ∈ TopSp → 𝐽 ∈ Top)
82 cldssbrsiga 30571 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
832, 81, 823syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
8483adantr 468 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
8575adantr 468 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐽 ∈ Fre)
86 xp1st 7426 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ (𝐵 × 𝐵) → (1st𝑝) ∈ 𝐵)
8786adantl 469 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (1st𝑝) ∈ 𝐵)
886adantr 468 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐵 = 𝐽)
8987, 88eleqtrd 2887 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (1st𝑝) ∈ 𝐽)
90 eqid 2806 . . . . . . . . . . . . . . . . . . . . 21 𝐽 = 𝐽
9190t1sncld 21340 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ Fre ∧ (1st𝑝) ∈ 𝐽) → {(1st𝑝)} ∈ (Clsd‘𝐽))
9285, 89, 91syl2anc 575 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ (Clsd‘𝐽))
9384, 92sseldd 3799 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ (sigaGen‘𝐽))
9493, 11syl6eleqr 2896 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ 𝑆)
9574, 78, 80, 94mbfmcnvima 30640 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀)
9667, 73, 95syl2anc 575 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀)
973, 4, 11, 12, 13, 14, 15, 16, 19sibfmbl 30718 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 ∈ (dom 𝑀MblFnM𝑆))
9897adantr 468 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
99 xp2nd 7427 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ (𝐵 × 𝐵) → (2nd𝑝) ∈ 𝐵)
10099adantl 469 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (2nd𝑝) ∈ 𝐵)
101100, 88eleqtrd 2887 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (2nd𝑝) ∈ 𝐽)
10290t1sncld 21340 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ Fre ∧ (2nd𝑝) ∈ 𝐽) → {(2nd𝑝)} ∈ (Clsd‘𝐽))
10385, 101, 102syl2anc 575 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ (Clsd‘𝐽))
10484, 103sseldd 3799 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ (sigaGen‘𝐽))
105104, 11syl6eleqr 2896 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ 𝑆)
10674, 78, 98, 105mbfmcnvima 30640 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀)
10767, 73, 106syl2anc 575 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀)
108 inelsiga 30519 . . . . . . . . . . . . . . 15 ((dom 𝑀 ran sigAlgebra ∧ (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀 ∧ (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
10966, 96, 107, 108syl3anc 1483 . . . . . . . . . . . . . 14 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
110109ralrimiva 3154 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → ∀𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
1113, 4, 11, 12, 13, 14, 15, 16, 17sibfrn 30720 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝐹 ∈ Fin)
1123, 4, 11, 12, 13, 14, 15, 16, 19sibfrn 30720 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝐺 ∈ Fin)
113 xpfi 8466 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
114111, 112, 113syl2anc 575 . . . . . . . . . . . . . . . 16 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
115 inss2 4030 . . . . . . . . . . . . . . . 16 (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)
116 ssdomg 8234 . . . . . . . . . . . . . . . 16 ((ran 𝐹 × ran 𝐺) ∈ Fin → ((( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺)))
117114, 115, 116mpisyl 21 . . . . . . . . . . . . . . 15 (𝜑 → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺))
118 isfinite 8792 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 × ran 𝐺) ∈ Fin ↔ (ran 𝐹 × ran 𝐺) ≺ ω)
119118biimpi 207 . . . . . . . . . . . . . . . 16 ((ran 𝐹 × ran 𝐺) ∈ Fin → (ran 𝐹 × ran 𝐺) ≺ ω)
120 sdomdom 8216 . . . . . . . . . . . . . . . 16 ((ran 𝐹 × ran 𝐺) ≺ ω → (ran 𝐹 × ran 𝐺) ≼ ω)
121114, 119, 1203syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (ran 𝐹 × ran 𝐺) ≼ ω)
122 domtr 8241 . . . . . . . . . . . . . . 15 (((( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺) ∧ (ran 𝐹 × ran 𝐺) ≼ ω) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
123117, 121, 122syl2anc 575 . . . . . . . . . . . . . 14 (𝜑 → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
124123adantr 468 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
125 nfcv 2948 . . . . . . . . . . . . . 14 𝑝(( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))
126125sigaclcuni 30502 . . . . . . . . . . . . 13 ((dom 𝑀 ran sigAlgebra ∧ ∀𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀 ∧ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω) → 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
12765, 110, 124, 126syl3anc 1483 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
12864, 127eqeltrd 2885 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → ((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
129128ralrimiva 3154 . . . . . . . . . 10 (𝜑 → ∀𝑧 ∈ ran (𝐹𝑓 + 𝐺)((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
130 ssralv 3863 . . . . . . . . . 10 ((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ⊆ ran (𝐹𝑓 + 𝐺) → (∀𝑧 ∈ ran (𝐹𝑓 + 𝐺)((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀 → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀))
13157, 129, 130mpsyl 68 . . . . . . . . 9 (𝜑 → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
132131adantr 468 . . . . . . . 8 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
1331ffund 6256 . . . . . . . . . . . . 13 (𝜑 → Fun + )
134 imafi 8494 . . . . . . . . . . . . 13 ((Fun + ∧ (ran 𝐹 × ran 𝐺) ∈ Fin) → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
135133, 114, 134syl2anc 575 . . . . . . . . . . . 12 (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
13618, 20, 9, 23ofrn2 29765 . . . . . . . . . . . 12 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
137 ssfi 8415 . . . . . . . . . . . 12 ((( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin ∧ ran (𝐹𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺))) → ran (𝐹𝑓 + 𝐺) ∈ Fin)
138135, 136, 137syl2anc 575 . . . . . . . . . . 11 (𝜑 → ran (𝐹𝑓 + 𝐺) ∈ Fin)
139 ssdomg 8234 . . . . . . . . . . 11 (ran (𝐹𝑓 + 𝐺) ∈ Fin → ((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ⊆ ran (𝐹𝑓 + 𝐺) → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ran (𝐹𝑓 + 𝐺)))
140138, 57, 139mpisyl 21 . . . . . . . . . 10 (𝜑 → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ran (𝐹𝑓 + 𝐺))
141 isfinite 8792 . . . . . . . . . . . 12 (ran (𝐹𝑓 + 𝐺) ∈ Fin ↔ ran (𝐹𝑓 + 𝐺) ≺ ω)
142138, 141sylib 209 . . . . . . . . . . 11 (𝜑 → ran (𝐹𝑓 + 𝐺) ≺ ω)
143 sdomdom 8216 . . . . . . . . . . 11 (ran (𝐹𝑓 + 𝐺) ≺ ω → ran (𝐹𝑓 + 𝐺) ≼ ω)
144142, 143syl 17 . . . . . . . . . 10 (𝜑 → ran (𝐹𝑓 + 𝐺) ≼ ω)
145 domtr 8241 . . . . . . . . . 10 (((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ran (𝐹𝑓 + 𝐺) ∧ ran (𝐹𝑓 + 𝐺) ≼ ω) → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ω)
146140, 144, 145syl2anc 575 . . . . . . . . 9 (𝜑 → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ω)
147146adantr 468 . . . . . . . 8 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ω)
148 nfcv 2948 . . . . . . . . 9 𝑧(𝑏 ∩ ran (𝐹𝑓 + 𝐺))
149148sigaclcuni 30502 . . . . . . . 8 ((dom 𝑀 ran sigAlgebra ∧ ∀𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀 ∧ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ω) → 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
15052, 132, 147, 149syl3anc 1483 . . . . . . 7 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
15156, 150syl5eqelr 2890 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∈ dom 𝑀)
152 difpreima 6561 . . . . . . . . . 10 (Fun (𝐹𝑓 + 𝐺) → ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) = (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺))))
15325, 45, 1523syl 18 . . . . . . . . 9 (𝜑 → ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) = (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺))))
154 cnvimarndm 5696 . . . . . . . . . . 11 ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺)) = dom (𝐹𝑓 + 𝐺)
155154difeq2i 3924 . . . . . . . . . 10 (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺))) = (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ dom (𝐹𝑓 + 𝐺))
156 cnvimass 5695 . . . . . . . . . . 11 ((𝐹𝑓 + 𝐺) “ 𝑏) ⊆ dom (𝐹𝑓 + 𝐺)
157 ssdif0 4143 . . . . . . . . . . 11 (((𝐹𝑓 + 𝐺) “ 𝑏) ⊆ dom (𝐹𝑓 + 𝐺) ↔ (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ dom (𝐹𝑓 + 𝐺)) = ∅)
158156, 157mpbi 221 . . . . . . . . . 10 (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ dom (𝐹𝑓 + 𝐺)) = ∅
159155, 158eqtri 2828 . . . . . . . . 9 (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺))) = ∅
160153, 159syl6eq 2856 . . . . . . . 8 (𝜑 → ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) = ∅)
161 0elsiga 30498 . . . . . . . . 9 (dom 𝑀 ran sigAlgebra → ∅ ∈ dom 𝑀)
16216, 50, 1613syl 18 . . . . . . . 8 (𝜑 → ∅ ∈ dom 𝑀)
163160, 162eqeltrd 2885 . . . . . . 7 (𝜑 → ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) ∈ dom 𝑀)
164163adantr 468 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) ∈ dom 𝑀)
165 unelsiga 30518 . . . . . 6 ((dom 𝑀 ran sigAlgebra ∧ ((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∈ dom 𝑀 ∧ ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) ∈ dom 𝑀) → (((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∪ ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))) ∈ dom 𝑀)
16652, 151, 164, 165syl3anc 1483 . . . . 5 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∪ ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))) ∈ dom 𝑀)
16749, 166eqeltrd 2885 . . . 4 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ 𝑏) ∈ dom 𝑀)
168167ralrimiva 3154 . . 3 (𝜑 → ∀𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))((𝐹𝑓 + 𝐺) “ 𝑏) ∈ dom 𝑀)
16951, 38ismbfm 30635 . . 3 (𝜑 → ((𝐹𝑓 + 𝐺) ∈ (dom 𝑀MblFnM(sigaGen‘(TopOpen‘𝐾))) ↔ ((𝐹𝑓 + 𝐺) ∈ ( (sigaGen‘(TopOpen‘𝐾)) ↑𝑚 dom 𝑀) ∧ ∀𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))((𝐹𝑓 + 𝐺) “ 𝑏) ∈ dom 𝑀)))
17042, 168, 169mpbir2and 695 . 2 (𝜑 → (𝐹𝑓 + 𝐺) ∈ (dom 𝑀MblFnM(sigaGen‘(TopOpen‘𝐾))))
17163adantr 468 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → ((𝐹𝑓 + 𝐺) “ {𝑧}) = 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
172171fveq2d 6408 . . . . . 6 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) = (𝑀 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
173 measbasedom 30586 . . . . . . . . 9 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
17416, 173sylib 209 . . . . . . . 8 (𝜑𝑀 ∈ (measures‘dom 𝑀))
175174adantr 468 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → 𝑀 ∈ (measures‘dom 𝑀))
176 eldifi 3931 . . . . . . . 8 (𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)}) → 𝑧 ∈ ran (𝐹𝑓 + 𝐺))
177176, 110sylan2 582 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → ∀𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
178123adantr 468 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
179 sneq 4380 . . . . . . . . . . 11 (𝑥 = (1st𝑝) → {𝑥} = {(1st𝑝)})
180179imaeq2d 5676 . . . . . . . . . 10 (𝑥 = (1st𝑝) → (𝐹 “ {𝑥}) = (𝐹 “ {(1st𝑝)}))
181 sneq 4380 . . . . . . . . . . 11 (𝑦 = (2nd𝑝) → {𝑦} = {(2nd𝑝)})
182181imaeq2d 5676 . . . . . . . . . 10 (𝑦 = (2nd𝑝) → (𝐺 “ {𝑦}) = (𝐺 “ {(2nd𝑝)}))
18318ffund 6256 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
184 sndisj 4836 . . . . . . . . . . 11 Disj 𝑥 ∈ ran 𝐹{𝑥}
185 disjpreima 29718 . . . . . . . . . . 11 ((Fun 𝐹Disj 𝑥 ∈ ran 𝐹{𝑥}) → Disj 𝑥 ∈ ran 𝐹(𝐹 “ {𝑥}))
186183, 184, 185sylancl 576 . . . . . . . . . 10 (𝜑Disj 𝑥 ∈ ran 𝐹(𝐹 “ {𝑥}))
18720ffund 6256 . . . . . . . . . . 11 (𝜑 → Fun 𝐺)
188 sndisj 4836 . . . . . . . . . . 11 Disj 𝑦 ∈ ran 𝐺{𝑦}
189 disjpreima 29718 . . . . . . . . . . 11 ((Fun 𝐺Disj 𝑦 ∈ ran 𝐺{𝑦}) → Disj 𝑦 ∈ ran 𝐺(𝐺 “ {𝑦}))
190187, 188, 189sylancl 576 . . . . . . . . . 10 (𝜑Disj 𝑦 ∈ ran 𝐺(𝐺 “ {𝑦}))
191180, 182, 186, 190disjxpin 29722 . . . . . . . . 9 (𝜑Disj 𝑝 ∈ (ran 𝐹 × ran 𝐺)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
192 disjss1 4818 . . . . . . . . 9 ((( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺) → (Disj 𝑝 ∈ (ran 𝐹 × ran 𝐺)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) → Disj 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
193115, 191, 192mpsyl 68 . . . . . . . 8 (𝜑Disj 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
194193adantr 468 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → Disj 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
195 measvuni 30598 . . . . . . 7 ((𝑀 ∈ (measures‘dom 𝑀) ∧ ∀𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀 ∧ ((( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω ∧ Disj 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))) → (𝑀 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) = Σ*𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
196175, 177, 178, 194, 195syl112anc 1486 . . . . . 6 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) = Σ*𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
197 ssfi 8415 . . . . . . . . 9 (((ran 𝐹 × ran 𝐺) ∈ Fin ∧ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
198114, 115, 197sylancl 576 . . . . . . . 8 (𝜑 → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
199198adantr 468 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
200 simpll 774 . . . . . . . 8 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
201 simpr 473 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)))
202115, 201sseldi 3796 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (ran 𝐹 × ran 𝐺))
203 xp1st 7426 . . . . . . . . 9 (𝑝 ∈ (ran 𝐹 × ran 𝐺) → (1st𝑝) ∈ ran 𝐹)
204202, 203syl 17 . . . . . . . 8 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (1st𝑝) ∈ ran 𝐹)
205 xp2nd 7427 . . . . . . . . 9 (𝑝 ∈ (ran 𝐹 × ran 𝐺) → (2nd𝑝) ∈ ran 𝐺)
206202, 205syl 17 . . . . . . . 8 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (2nd𝑝) ∈ ran 𝐺)
207 oveq12 6879 . . . . . . . . . . . . . . . 16 ((𝑥 = 0𝑦 = 0 ) → (𝑥 + 𝑦) = ( 0 + 0 ))
208 sibfof.5 . . . . . . . . . . . . . . . 16 (𝜑 → ( 0 + 0 ) = (0g𝐾))
209207, 208sylan9eqr 2862 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 = 0𝑦 = 0 )) → (𝑥 + 𝑦) = (0g𝐾))
210209ex 399 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥 = 0𝑦 = 0 ) → (𝑥 + 𝑦) = (0g𝐾)))
211210necon3ad 2991 . . . . . . . . . . . . 13 (𝜑 → ((𝑥 + 𝑦) ≠ (0g𝐾) → ¬ (𝑥 = 0𝑦 = 0 )))
212 neorian 3072 . . . . . . . . . . . . 13 ((𝑥0𝑦0 ) ↔ ¬ (𝑥 = 0𝑦 = 0 ))
213211, 212syl6ibr 243 . . . . . . . . . . . 12 (𝜑 → ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
214213adantr 468 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
215214ralrimivva 3159 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
216200, 215syl 17 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
21768a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ ( + “ {𝑧}))
218217sselda 3798 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ( + “ {𝑧}))
219 fniniseg 6556 . . . . . . . . . . . . 13 ( + Fn (𝐵 × 𝐵) → (𝑝 ∈ ( + “ {𝑧}) ↔ (𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧)))
220200, 62, 2193syl 18 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑝 ∈ ( + “ {𝑧}) ↔ (𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧)))
221218, 220mpbid 223 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧))
222 simpr 473 . . . . . . . . . . . 12 ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧) → ( +𝑝) = 𝑧)
223 1st2nd2 7433 . . . . . . . . . . . . . . 15 (𝑝 ∈ (𝐵 × 𝐵) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
224223fveq2d 6408 . . . . . . . . . . . . . 14 (𝑝 ∈ (𝐵 × 𝐵) → ( +𝑝) = ( + ‘⟨(1st𝑝), (2nd𝑝)⟩))
225 df-ov 6873 . . . . . . . . . . . . . 14 ((1st𝑝) + (2nd𝑝)) = ( + ‘⟨(1st𝑝), (2nd𝑝)⟩)
226224, 225syl6eqr 2858 . . . . . . . . . . . . 13 (𝑝 ∈ (𝐵 × 𝐵) → ( +𝑝) = ((1st𝑝) + (2nd𝑝)))
227226adantr 468 . . . . . . . . . . . 12 ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧) → ( +𝑝) = ((1st𝑝) + (2nd𝑝)))
228222, 227eqtr3d 2842 . . . . . . . . . . 11 ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧) → 𝑧 = ((1st𝑝) + (2nd𝑝)))
229221, 228syl 17 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 = ((1st𝑝) + (2nd𝑝)))
230 simplr 776 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)}))
231230eldifbd 3782 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ¬ 𝑧 ∈ {(0g𝐾)})
232 velsn 4386 . . . . . . . . . . . 12 (𝑧 ∈ {(0g𝐾)} ↔ 𝑧 = (0g𝐾))
233232necon3bbii 3025 . . . . . . . . . . 11 𝑧 ∈ {(0g𝐾)} ↔ 𝑧 ≠ (0g𝐾))
234231, 233sylib 209 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ≠ (0g𝐾))
235229, 234eqnetrrd 3046 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾))
236176, 73sylanl2 663 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐵))
237236, 86syl 17 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (1st𝑝) ∈ 𝐵)
238236, 99syl 17 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (2nd𝑝) ∈ 𝐵)
239 oveq1 6877 . . . . . . . . . . . . 13 (𝑥 = (1st𝑝) → (𝑥 + 𝑦) = ((1st𝑝) + 𝑦))
240239neeq1d 3037 . . . . . . . . . . . 12 (𝑥 = (1st𝑝) → ((𝑥 + 𝑦) ≠ (0g𝐾) ↔ ((1st𝑝) + 𝑦) ≠ (0g𝐾)))
241 neeq1 3040 . . . . . . . . . . . . 13 (𝑥 = (1st𝑝) → (𝑥0 ↔ (1st𝑝) ≠ 0 ))
242241orbi1d 931 . . . . . . . . . . . 12 (𝑥 = (1st𝑝) → ((𝑥0𝑦0 ) ↔ ((1st𝑝) ≠ 0𝑦0 )))
243240, 242imbi12d 335 . . . . . . . . . . 11 (𝑥 = (1st𝑝) → (((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )) ↔ (((1st𝑝) + 𝑦) ≠ (0g𝐾) → ((1st𝑝) ≠ 0𝑦0 ))))
244 oveq2 6878 . . . . . . . . . . . . 13 (𝑦 = (2nd𝑝) → ((1st𝑝) + 𝑦) = ((1st𝑝) + (2nd𝑝)))
245244neeq1d 3037 . . . . . . . . . . . 12 (𝑦 = (2nd𝑝) → (((1st𝑝) + 𝑦) ≠ (0g𝐾) ↔ ((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾)))
246 neeq1 3040 . . . . . . . . . . . . 13 (𝑦 = (2nd𝑝) → (𝑦0 ↔ (2nd𝑝) ≠ 0 ))
247246orbi2d 930 . . . . . . . . . . . 12 (𝑦 = (2nd𝑝) → (((1st𝑝) ≠ 0𝑦0 ) ↔ ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 )))
248245, 247imbi12d 335 . . . . . . . . . . 11 (𝑦 = (2nd𝑝) → ((((1st𝑝) + 𝑦) ≠ (0g𝐾) → ((1st𝑝) ≠ 0𝑦0 )) ↔ (((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))))
249243, 248rspc2v 3515 . . . . . . . . . 10 (((1st𝑝) ∈ 𝐵 ∧ (2nd𝑝) ∈ 𝐵) → (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )) → (((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))))
250237, 238, 249syl2anc 575 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )) → (((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))))
251216, 235, 250mp2d 49 . . . . . . . 8 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))
2523, 4, 11, 12, 13, 14, 15, 16, 17, 19, 2, 75sibfinima 30722 . . . . . . . 8 (((𝜑 ∧ (1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) ∧ ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 )) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ (0[,)+∞))
253200, 204, 206, 251, 252syl31anc 1485 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ (0[,)+∞))
254199, 253esumpfinval 30458 . . . . . 6 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → Σ*𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) = Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
255172, 196, 2543eqtrd 2844 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) = Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
256 rge0ssre 12496 . . . . . . 7 (0[,)+∞) ⊆ ℝ
257256, 253sseldi 3796 . . . . . 6 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ ℝ)
258199, 257fsumrecl 14684 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ ℝ)
259255, 258eqeltrd 2885 . . . 4 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ ℝ)
260175adantr 468 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑀 ∈ (measures‘dom 𝑀))
261176, 109sylanl2 663 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
262 measge0 30591 . . . . . . 7 ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀) → 0 ≤ (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
263260, 261, 262syl2anc 575 . . . . . 6 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 0 ≤ (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
264199, 257, 263fsumge0 14745 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → 0 ≤ Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
265264, 255breqtrrd 4872 . . . 4 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → 0 ≤ (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})))
266 elrege0 12494 . . . 4 ((𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞) ↔ ((𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ ℝ ∧ 0 ≤ (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧}))))
267259, 265, 266sylanbrc 574 . . 3 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
268267ralrimiva 3154 . 2 (𝜑 → ∀𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})(𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
269 eqid 2806 . . 3 (sigaGen‘(TopOpen‘𝐾)) = (sigaGen‘(TopOpen‘𝐾))
270 eqid 2806 . . 3 (0g𝐾) = (0g𝐾)
271 eqid 2806 . . 3 ( ·𝑠𝐾) = ( ·𝑠𝐾)
272 eqid 2806 . . 3 (ℝHom‘(Scalar‘𝐾)) = (ℝHom‘(Scalar‘𝐾))
27327, 28, 269, 270, 271, 272, 26, 16issibf 30716 . 2 (𝜑 → ((𝐹𝑓 + 𝐺) ∈ dom (𝐾sitg𝑀) ↔ ((𝐹𝑓 + 𝐺) ∈ (dom 𝑀MblFnM(sigaGen‘(TopOpen‘𝐾))) ∧ ran (𝐹𝑓 + 𝐺) ∈ Fin ∧ ∀𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})(𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞))))
274170, 138, 268, 273mpbir3and 1435 1 (𝜑 → (𝐹𝑓 + 𝐺) ∈ dom (𝐾sitg𝑀))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865   = wceq 1637  wcel 2156  wne 2978  wral 3096  Vcvv 3391  cdif 3766  cun 3767  cin 3768  wss 3769  c0 4116  {csn 4370  cop 4376   cuni 4630   ciun 4712  Disj wdisj 4812   class class class wbr 4844   × cxp 5309  ccnv 5310  dom cdm 5311  ran crn 5312  cima 5314  Fun wfun 6091   Fn wfn 6092  wf 6093  cfv 6097  (class class class)co 6870  𝑓 cof 7121  ωcom 7291  1st c1st 7392  2nd c2nd 7393  𝑚 cmap 8088  cdom 8186  csdm 8187  Fincfn 8188  cr 10216  0cc0 10217  +∞cpnf 10352  cle 10356  [,)cico 12391  Σcsu 14635  Basecbs 16064  Scalarcsca 16152   ·𝑠 cvsca 16153  TopOpenctopn 16283  0gc0g 16301  Topctop 20907  TopSpctps 20946  Clsdccld 21030  Frect1 21321  ℝHomcrrh 30358  Σ*cesum 30410  sigAlgebracsiga 30491  sigaGencsigagen 30522  measurescmeas 30579  MblFnMcmbfm 30633  sitgcsitg 30712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-inf2 8781  ax-ac2 9566  ax-cnex 10273  ax-resscn 10274  ax-1cn 10275  ax-icn 10276  ax-addcl 10277  ax-addrcl 10278  ax-mulcl 10279  ax-mulrcl 10280  ax-mulcom 10281  ax-addass 10282  ax-mulass 10283  ax-distr 10284  ax-i2m1 10285  ax-1ne0 10286  ax-1rid 10287  ax-rnegex 10288  ax-rrecex 10289  ax-cnre 10290  ax-pre-lttri 10291  ax-pre-lttrn 10292  ax-pre-ltadd 10293  ax-pre-mulgt0 10294  ax-pre-sup 10295  ax-addf 10296  ax-mulf 10297
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-disj 4813  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-isom 6106  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-of 7123  df-om 7292  df-1st 7394  df-2nd 7395  df-supp 7526  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-1o 7792  df-2o 7793  df-oadd 7796  df-er 7975  df-map 8090  df-pm 8091  df-ixp 8142  df-en 8189  df-dom 8190  df-sdom 8191  df-fin 8192  df-fsupp 8511  df-fi 8552  df-sup 8583  df-inf 8584  df-oi 8650  df-card 9044  df-acn 9047  df-ac 9218  df-cda 9271  df-pnf 10357  df-mnf 10358  df-xr 10359  df-ltxr 10360  df-le 10361  df-sub 10549  df-neg 10550  df-div 10966  df-nn 11302  df-2 11360  df-3 11361  df-4 11362  df-5 11363  df-6 11364  df-7 11365  df-8 11366  df-9 11367  df-n0 11556  df-z 11640  df-dec 11756  df-uz 11901  df-q 12004  df-rp 12043  df-xneg 12158  df-xadd 12159  df-xmul 12160  df-ioo 12393  df-ioc 12394  df-ico 12395  df-icc 12396  df-fz 12546  df-fzo 12686  df-fl 12813  df-mod 12889  df-seq 13021  df-exp 13080  df-fac 13277  df-bc 13306  df-hash 13334  df-shft 14026  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-limsup 14421  df-clim 14438  df-rlim 14439  df-sum 14636  df-ef 15014  df-sin 15016  df-cos 15017  df-pi 15019  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-starv 16164  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16171  df-unif 16172  df-hom 16173  df-cco 16174  df-rest 16284  df-topn 16285  df-0g 16303  df-gsum 16304  df-topgen 16305  df-pt 16306  df-prds 16309  df-ordt 16362  df-xrs 16363  df-qtop 16368  df-imas 16369  df-xps 16371  df-mre 16447  df-mrc 16448  df-acs 16450  df-ps 17401  df-tsr 17402  df-plusf 17442  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-submnd 17537  df-grp 17626  df-minusg 17627  df-sbg 17628  df-mulg 17742  df-subg 17789  df-cntz 17947  df-cmn 18392  df-abl 18393  df-mgp 18688  df-ur 18700  df-ring 18747  df-cring 18748  df-subrg 18978  df-abv 19017  df-lmod 19065  df-scaf 19066  df-sra 19377  df-rgmod 19378  df-psmet 19942  df-xmet 19943  df-met 19944  df-bl 19945  df-mopn 19946  df-fbas 19947  df-fg 19948  df-cnfld 19951  df-top 20908  df-topon 20925  df-topsp 20947  df-bases 20960  df-cld 21033  df-ntr 21034  df-cls 21035  df-nei 21112  df-lp 21150  df-perf 21151  df-cn 21241  df-cnp 21242  df-t1 21328  df-haus 21329  df-tx 21575  df-hmeo 21768  df-fil 21859  df-fm 21951  df-flim 21952  df-flf 21953  df-tmd 22085  df-tgp 22086  df-tsms 22139  df-trg 22172  df-xms 22334  df-ms 22335  df-tms 22336  df-nm 22596  df-ngp 22597  df-nrg 22599  df-nlm 22600  df-ii 22889  df-cncf 22890  df-limc 23840  df-dv 23841  df-log 24513  df-esum 30411  df-siga 30492  df-sigagen 30523  df-meas 30580  df-mbfm 30634  df-sitg 30713
This theorem is referenced by:  sitmcl  30734
  Copyright terms: Public domain W3C validator