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

Theorem sibfof 31672
 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 21539 . . . . . . . . . . 11 (𝑊 ∈ TopSp → 𝐵 = 𝐽)
62, 5syl 17 . . . . . . . . . 10 (𝜑𝐵 = 𝐽)
76sqxpeqd 5564 . . . . . . . . 9 (𝜑 → (𝐵 × 𝐵) = ( 𝐽 × 𝐽))
87feq2d 6480 . . . . . . . 8 (𝜑 → ( + :(𝐵 × 𝐵)⟶𝐶+ :( 𝐽 × 𝐽)⟶𝐶))
91, 8mpbid 235 . . . . . . 7 (𝜑+ :( 𝐽 × 𝐽)⟶𝐶)
109fovrnda 7304 . . . . . 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 31668 . . . . . 6 (𝜑𝐹: dom 𝑀 𝐽)
19 sibfof.2 . . . . . . 7 (𝜑𝐺 ∈ dom (𝑊sitg𝑀))
203, 4, 11, 12, 13, 14, 15, 16, 19sibff 31668 . . . . . 6 (𝜑𝐺: dom 𝑀 𝐽)
21 dmexg 7599 . . . . . . 7 (𝑀 ran measures → dom 𝑀 ∈ V)
22 uniexg 7451 . . . . . . 7 (dom 𝑀 ∈ V → dom 𝑀 ∈ V)
2316, 21, 223syl 18 . . . . . 6 (𝜑 dom 𝑀 ∈ V)
24 inidm 4169 . . . . . 6 ( dom 𝑀 dom 𝑀) = dom 𝑀
2510, 18, 20, 23, 23, 24off 7409 . . . . 5 (𝜑 → (𝐹f + 𝐺): dom 𝑀𝐶)
26 sibfof.3 . . . . . . . 8 (𝜑𝐾 ∈ TopSp)
27 sibfof.c . . . . . . . . 9 𝐶 = (Base‘𝐾)
28 eqid 2822 . . . . . . . . 9 (TopOpen‘𝐾) = (TopOpen‘𝐾)
2927, 28tpsuni 21539 . . . . . . . 8 (𝐾 ∈ TopSp → 𝐶 = (TopOpen‘𝐾))
3026, 29syl 17 . . . . . . 7 (𝜑𝐶 = (TopOpen‘𝐾))
31 fvex 6665 . . . . . . . 8 (TopOpen‘𝐾) ∈ V
32 unisg 31476 . . . . . . . 8 ((TopOpen‘𝐾) ∈ V → (sigaGen‘(TopOpen‘𝐾)) = (TopOpen‘𝐾))
3331, 32ax-mp 5 . . . . . . 7 (sigaGen‘(TopOpen‘𝐾)) = (TopOpen‘𝐾)
3430, 33eqtr4di 2875 . . . . . 6 (𝜑𝐶 = (sigaGen‘(TopOpen‘𝐾)))
3534feq3d 6481 . . . . 5 (𝜑 → ((𝐹f + 𝐺): dom 𝑀𝐶 ↔ (𝐹f + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾))))
3625, 35mpbid 235 . . . 4 (𝜑 → (𝐹f + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾)))
3731a1i 11 . . . . . . 7 (𝜑 → (TopOpen‘𝐾) ∈ V)
3837sgsiga 31475 . . . . . 6 (𝜑 → (sigaGen‘(TopOpen‘𝐾)) ∈ ran sigAlgebra)
3938uniexd 7453 . . . . 5 (𝜑 (sigaGen‘(TopOpen‘𝐾)) ∈ V)
4039, 23elmapd 8407 . . . 4 (𝜑 → ((𝐹f + 𝐺) ∈ ( (sigaGen‘(TopOpen‘𝐾)) ↑m dom 𝑀) ↔ (𝐹f + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾))))
4136, 40mpbird 260 . . 3 (𝜑 → (𝐹f + 𝐺) ∈ ( (sigaGen‘(TopOpen‘𝐾)) ↑m dom 𝑀))
42 inundif 4399 . . . . . . 7 ((𝑏 ∩ ran (𝐹f + 𝐺)) ∪ (𝑏 ∖ ran (𝐹f + 𝐺))) = 𝑏
4342imaeq2i 5905 . . . . . 6 ((𝐹f + 𝐺) “ ((𝑏 ∩ ran (𝐹f + 𝐺)) ∪ (𝑏 ∖ ran (𝐹f + 𝐺)))) = ((𝐹f + 𝐺) “ 𝑏)
44 ffun 6497 . . . . . . . 8 ((𝐹f + 𝐺): dom 𝑀𝐶 → Fun (𝐹f + 𝐺))
45 unpreima 6815 . . . . . . . 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)))))
4625, 44, 453syl 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)))))
4746adantr 484 . . . . . 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)))))
4843, 47syl5eqr 2871 . . . . 5 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹f + 𝐺) “ 𝑏) = (((𝐹f + 𝐺) “ (𝑏 ∩ ran (𝐹f + 𝐺))) ∪ ((𝐹f + 𝐺) “ (𝑏 ∖ ran (𝐹f + 𝐺)))))
49 dmmeas 31534 . . . . . . . 8 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
5016, 49syl 17 . . . . . . 7 (𝜑 → dom 𝑀 ran sigAlgebra)
5150adantr 484 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → dom 𝑀 ran sigAlgebra)
52 imaiun 6987 . . . . . . . 8 ((𝐹f + 𝐺) “ 𝑧 ∈ (𝑏 ∩ ran (𝐹f + 𝐺)){𝑧}) = 𝑧 ∈ (𝑏 ∩ ran (𝐹f + 𝐺))((𝐹f + 𝐺) “ {𝑧})
53 iunid 4959 . . . . . . . . 9 𝑧 ∈ (𝑏 ∩ ran (𝐹f + 𝐺)){𝑧} = (𝑏 ∩ ran (𝐹f + 𝐺))
5453imaeq2i 5905 . . . . . . . 8 ((𝐹f + 𝐺) “ 𝑧 ∈ (𝑏 ∩ ran (𝐹f + 𝐺)){𝑧}) = ((𝐹f + 𝐺) “ (𝑏 ∩ ran (𝐹f + 𝐺)))
5552, 54eqtr3i 2847 . . . . . . 7 𝑧 ∈ (𝑏 ∩ ran (𝐹f + 𝐺))((𝐹f + 𝐺) “ {𝑧}) = ((𝐹f + 𝐺) “ (𝑏 ∩ ran (𝐹f + 𝐺)))
56 inss2 4180 . . . . . . . . . 10 (𝑏 ∩ ran (𝐹f + 𝐺)) ⊆ ran (𝐹f + 𝐺)
576feq3d 6481 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹: dom 𝑀𝐵𝐹: dom 𝑀 𝐽))
5818, 57mpbird 260 . . . . . . . . . . . . . 14 (𝜑𝐹: dom 𝑀𝐵)
596feq3d 6481 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺: dom 𝑀𝐵𝐺: dom 𝑀 𝐽))
6020, 59mpbird 260 . . . . . . . . . . . . . 14 (𝜑𝐺: dom 𝑀𝐵)
611ffnd 6495 . . . . . . . . . . . . . 14 (𝜑+ Fn (𝐵 × 𝐵))
6258, 60, 23, 61ofpreima2 30419 . . . . . . . . . . . . 13 (𝜑 → ((𝐹f + 𝐺) “ {𝑧}) = 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
6362adantr 484 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) → ((𝐹f + 𝐺) “ {𝑧}) = 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
6450adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) → dom 𝑀 ran sigAlgebra)
6550ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → dom 𝑀 ran sigAlgebra)
66 simpll 766 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
67 inss1 4179 . . . . . . . . . . . . . . . . . 18 (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ ( + “ {𝑧})
68 cnvimass 5927 . . . . . . . . . . . . . . . . . . . 20 ( + “ {𝑧}) ⊆ dom +
6968, 1fssdm 6511 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ( + “ {𝑧}) ⊆ (𝐵 × 𝐵))
7069adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) → ( + “ {𝑧}) ⊆ (𝐵 × 𝐵))
7167, 70sstrid 3953 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐵))
7271sselda 3942 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐵))
7350adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → dom 𝑀 ran sigAlgebra)
74 sibfof.4 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ Fre)
7574sgsiga 31475 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
7611, 75eqeltrid 2918 . . . . . . . . . . . . . . . . . 18 (𝜑𝑆 ran sigAlgebra)
7776adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝑆 ran sigAlgebra)
783, 4, 11, 12, 13, 14, 15, 16, 17sibfmbl 31667 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
7978adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
804tpstop 21540 . . . . . . . . . . . . . . . . . . . . 21 (𝑊 ∈ TopSp → 𝐽 ∈ Top)
81 cldssbrsiga 31520 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
822, 80, 813syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
8382adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
8474adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐽 ∈ Fre)
85 xp1st 7707 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ (𝐵 × 𝐵) → (1st𝑝) ∈ 𝐵)
8685adantl 485 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (1st𝑝) ∈ 𝐵)
876adantr 484 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐵 = 𝐽)
8886, 87eleqtrd 2916 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (1st𝑝) ∈ 𝐽)
89 eqid 2822 . . . . . . . . . . . . . . . . . . . . 21 𝐽 = 𝐽
9089t1sncld 21929 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ Fre ∧ (1st𝑝) ∈ 𝐽) → {(1st𝑝)} ∈ (Clsd‘𝐽))
9184, 88, 90syl2anc 587 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ (Clsd‘𝐽))
9283, 91sseldd 3943 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ (sigaGen‘𝐽))
9392, 11eleqtrrdi 2925 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ 𝑆)
9473, 77, 79, 93mbfmcnvima 31589 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀)
9566, 72, 94syl2anc 587 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀)
963, 4, 11, 12, 13, 14, 15, 16, 19sibfmbl 31667 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 ∈ (dom 𝑀MblFnM𝑆))
9796adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
98 xp2nd 7708 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ (𝐵 × 𝐵) → (2nd𝑝) ∈ 𝐵)
9998adantl 485 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (2nd𝑝) ∈ 𝐵)
10099, 87eleqtrd 2916 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (2nd𝑝) ∈ 𝐽)
10189t1sncld 21929 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ Fre ∧ (2nd𝑝) ∈ 𝐽) → {(2nd𝑝)} ∈ (Clsd‘𝐽))
10284, 100, 101syl2anc 587 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ (Clsd‘𝐽))
10383, 102sseldd 3943 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ (sigaGen‘𝐽))
104103, 11eleqtrrdi 2925 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ 𝑆)
10573, 77, 97, 104mbfmcnvima 31589 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀)
10666, 72, 105syl2anc 587 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀)
107 inelsiga 31468 . . . . . . . . . . . . . . 15 ((dom 𝑀 ran sigAlgebra ∧ (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀 ∧ (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
10865, 95, 106, 107syl3anc 1368 . . . . . . . . . . . . . 14 (((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
109108ralrimiva 3174 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) → ∀𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
1103, 4, 11, 12, 13, 14, 15, 16, 17sibfrn 31669 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝐹 ∈ Fin)
1113, 4, 11, 12, 13, 14, 15, 16, 19sibfrn 31669 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝐺 ∈ Fin)
112 xpfi 8777 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
113110, 111, 112syl2anc 587 . . . . . . . . . . . . . . . 16 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
114 inss2 4180 . . . . . . . . . . . . . . . 16 (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)
115 ssdomg 8542 . . . . . . . . . . . . . . . 16 ((ran 𝐹 × ran 𝐺) ∈ Fin → ((( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺)))
116113, 114, 115mpisyl 21 . . . . . . . . . . . . . . 15 (𝜑 → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺))
117 isfinite 9103 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 × ran 𝐺) ∈ Fin ↔ (ran 𝐹 × ran 𝐺) ≺ ω)
118117biimpi 219 . . . . . . . . . . . . . . . 16 ((ran 𝐹 × ran 𝐺) ∈ Fin → (ran 𝐹 × ran 𝐺) ≺ ω)
119 sdomdom 8524 . . . . . . . . . . . . . . . 16 ((ran 𝐹 × ran 𝐺) ≺ ω → (ran 𝐹 × ran 𝐺) ≼ ω)
120113, 118, 1193syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (ran 𝐹 × ran 𝐺) ≼ ω)
121 domtr 8549 . . . . . . . . . . . . . . 15 (((( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺) ∧ (ran 𝐹 × ran 𝐺) ≼ ω) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
122116, 120, 121syl2anc 587 . . . . . . . . . . . . . 14 (𝜑 → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
123122adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
124 nfcv 2979 . . . . . . . . . . . . . 14 𝑝(( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))
125124sigaclcuni 31451 . . . . . . . . . . . . 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)
12664, 109, 123, 125syl3anc 1368 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) → 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
12763, 126eqeltrd 2914 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) → ((𝐹f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
128127ralrimiva 3174 . . . . . . . . . 10 (𝜑 → ∀𝑧 ∈ ran (𝐹f + 𝐺)((𝐹f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
129 ssralv 4008 . . . . . . . . . 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))
13056, 128, 129mpsyl 68 . . . . . . . . 9 (𝜑 → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹f + 𝐺))((𝐹f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
131130adantr 484 . . . . . . . 8 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹f + 𝐺))((𝐹f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
1321ffund 6498 . . . . . . . . . . . . 13 (𝜑 → Fun + )
133 imafi 8805 . . . . . . . . . . . . 13 ((Fun + ∧ (ran 𝐹 × ran 𝐺) ∈ Fin) → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
134132, 113, 133syl2anc 587 . . . . . . . . . . . 12 (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
13518, 20, 9, 23ofrn2 30395 . . . . . . . . . . . 12 (𝜑 → ran (𝐹f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
136 ssfi 8726 . . . . . . . . . . . 12 ((( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin ∧ ran (𝐹f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺))) → ran (𝐹f + 𝐺) ∈ Fin)
137134, 135, 136syl2anc 587 . . . . . . . . . . 11 (𝜑 → ran (𝐹f + 𝐺) ∈ Fin)
138 ssdomg 8542 . . . . . . . . . . 11 (ran (𝐹f + 𝐺) ∈ Fin → ((𝑏 ∩ ran (𝐹f + 𝐺)) ⊆ ran (𝐹f + 𝐺) → (𝑏 ∩ ran (𝐹f + 𝐺)) ≼ ran (𝐹f + 𝐺)))
139137, 56, 138mpisyl 21 . . . . . . . . . 10 (𝜑 → (𝑏 ∩ ran (𝐹f + 𝐺)) ≼ ran (𝐹f + 𝐺))
140 isfinite 9103 . . . . . . . . . . . 12 (ran (𝐹f + 𝐺) ∈ Fin ↔ ran (𝐹f + 𝐺) ≺ ω)
141137, 140sylib 221 . . . . . . . . . . 11 (𝜑 → ran (𝐹f + 𝐺) ≺ ω)
142 sdomdom 8524 . . . . . . . . . . 11 (ran (𝐹f + 𝐺) ≺ ω → ran (𝐹f + 𝐺) ≼ ω)
143141, 142syl 17 . . . . . . . . . 10 (𝜑 → ran (𝐹f + 𝐺) ≼ ω)
144 domtr 8549 . . . . . . . . . 10 (((𝑏 ∩ ran (𝐹f + 𝐺)) ≼ ran (𝐹f + 𝐺) ∧ ran (𝐹f + 𝐺) ≼ ω) → (𝑏 ∩ ran (𝐹f + 𝐺)) ≼ ω)
145139, 143, 144syl2anc 587 . . . . . . . . 9 (𝜑 → (𝑏 ∩ ran (𝐹f + 𝐺)) ≼ ω)
146145adantr 484 . . . . . . . 8 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (𝑏 ∩ ran (𝐹f + 𝐺)) ≼ ω)
147 nfcv 2979 . . . . . . . . 9 𝑧(𝑏 ∩ ran (𝐹f + 𝐺))
148147sigaclcuni 31451 . . . . . . . 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)
14951, 131, 146, 148syl3anc 1368 . . . . . . 7 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → 𝑧 ∈ (𝑏 ∩ ran (𝐹f + 𝐺))((𝐹f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
15055, 149eqeltrrid 2919 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹f + 𝐺) “ (𝑏 ∩ ran (𝐹f + 𝐺))) ∈ dom 𝑀)
151 difpreima 6817 . . . . . . . . . 10 (Fun (𝐹f + 𝐺) → ((𝐹f + 𝐺) “ (𝑏 ∖ ran (𝐹f + 𝐺))) = (((𝐹f + 𝐺) “ 𝑏) ∖ ((𝐹f + 𝐺) “ ran (𝐹f + 𝐺))))
15225, 44, 1513syl 18 . . . . . . . . 9 (𝜑 → ((𝐹f + 𝐺) “ (𝑏 ∖ ran (𝐹f + 𝐺))) = (((𝐹f + 𝐺) “ 𝑏) ∖ ((𝐹f + 𝐺) “ ran (𝐹f + 𝐺))))
153 cnvimarndm 5928 . . . . . . . . . . 11 ((𝐹f + 𝐺) “ ran (𝐹f + 𝐺)) = dom (𝐹f + 𝐺)
154153difeq2i 4071 . . . . . . . . . 10 (((𝐹f + 𝐺) “ 𝑏) ∖ ((𝐹f + 𝐺) “ ran (𝐹f + 𝐺))) = (((𝐹f + 𝐺) “ 𝑏) ∖ dom (𝐹f + 𝐺))
155 cnvimass 5927 . . . . . . . . . . 11 ((𝐹f + 𝐺) “ 𝑏) ⊆ dom (𝐹f + 𝐺)
156 ssdif0 4295 . . . . . . . . . . 11 (((𝐹f + 𝐺) “ 𝑏) ⊆ dom (𝐹f + 𝐺) ↔ (((𝐹f + 𝐺) “ 𝑏) ∖ dom (𝐹f + 𝐺)) = ∅)
157155, 156mpbi 233 . . . . . . . . . 10 (((𝐹f + 𝐺) “ 𝑏) ∖ dom (𝐹f + 𝐺)) = ∅
158154, 157eqtri 2845 . . . . . . . . 9 (((𝐹f + 𝐺) “ 𝑏) ∖ ((𝐹f + 𝐺) “ ran (𝐹f + 𝐺))) = ∅
159152, 158syl6eq 2873 . . . . . . . 8 (𝜑 → ((𝐹f + 𝐺) “ (𝑏 ∖ ran (𝐹f + 𝐺))) = ∅)
160 0elsiga 31447 . . . . . . . . 9 (dom 𝑀 ran sigAlgebra → ∅ ∈ dom 𝑀)
16116, 49, 1603syl 18 . . . . . . . 8 (𝜑 → ∅ ∈ dom 𝑀)
162159, 161eqeltrd 2914 . . . . . . 7 (𝜑 → ((𝐹f + 𝐺) “ (𝑏 ∖ ran (𝐹f + 𝐺))) ∈ dom 𝑀)
163162adantr 484 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹f + 𝐺) “ (𝑏 ∖ ran (𝐹f + 𝐺))) ∈ dom 𝑀)
164 unelsiga 31467 . . . . . 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)
16551, 150, 163, 164syl3anc 1368 . . . . 5 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (((𝐹f + 𝐺) “ (𝑏 ∩ ran (𝐹f + 𝐺))) ∪ ((𝐹f + 𝐺) “ (𝑏 ∖ ran (𝐹f + 𝐺)))) ∈ dom 𝑀)
16648, 165eqeltrd 2914 . . . 4 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹f + 𝐺) “ 𝑏) ∈ dom 𝑀)
167166ralrimiva 3174 . . 3 (𝜑 → ∀𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))((𝐹f + 𝐺) “ 𝑏) ∈ dom 𝑀)
16850, 38ismbfm 31584 . . 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)))
16941, 167, 168mpbir2and 712 . 2 (𝜑 → (𝐹f + 𝐺) ∈ (dom 𝑀MblFnM(sigaGen‘(TopOpen‘𝐾))))
17062adantr 484 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → ((𝐹f + 𝐺) “ {𝑧}) = 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
171170fveq2d 6656 . . . . . 6 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹f + 𝐺) “ {𝑧})) = (𝑀 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
172 measbasedom 31535 . . . . . . . . 9 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
17316, 172sylib 221 . . . . . . . 8 (𝜑𝑀 ∈ (measures‘dom 𝑀))
174173adantr 484 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → 𝑀 ∈ (measures‘dom 𝑀))
175 eldifi 4078 . . . . . . . 8 (𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)}) → 𝑧 ∈ ran (𝐹f + 𝐺))
176175, 109sylan2 595 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → ∀𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
177122adantr 484 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
178 sneq 4549 . . . . . . . . . . 11 (𝑥 = (1st𝑝) → {𝑥} = {(1st𝑝)})
179178imaeq2d 5907 . . . . . . . . . 10 (𝑥 = (1st𝑝) → (𝐹 “ {𝑥}) = (𝐹 “ {(1st𝑝)}))
180 sneq 4549 . . . . . . . . . . 11 (𝑦 = (2nd𝑝) → {𝑦} = {(2nd𝑝)})
181180imaeq2d 5907 . . . . . . . . . 10 (𝑦 = (2nd𝑝) → (𝐺 “ {𝑦}) = (𝐺 “ {(2nd𝑝)}))
18218ffund 6498 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
183 sndisj 5033 . . . . . . . . . . 11 Disj 𝑥 ∈ ran 𝐹{𝑥}
184 disjpreima 30342 . . . . . . . . . . 11 ((Fun 𝐹Disj 𝑥 ∈ ran 𝐹{𝑥}) → Disj 𝑥 ∈ ran 𝐹(𝐹 “ {𝑥}))
185182, 183, 184sylancl 589 . . . . . . . . . 10 (𝜑Disj 𝑥 ∈ ran 𝐹(𝐹 “ {𝑥}))
18620ffund 6498 . . . . . . . . . . 11 (𝜑 → Fun 𝐺)
187 sndisj 5033 . . . . . . . . . . 11 Disj 𝑦 ∈ ran 𝐺{𝑦}
188 disjpreima 30342 . . . . . . . . . . 11 ((Fun 𝐺Disj 𝑦 ∈ ran 𝐺{𝑦}) → Disj 𝑦 ∈ ran 𝐺(𝐺 “ {𝑦}))
189186, 187, 188sylancl 589 . . . . . . . . . 10 (𝜑Disj 𝑦 ∈ ran 𝐺(𝐺 “ {𝑦}))
190179, 181, 185, 189disjxpin 30346 . . . . . . . . 9 (𝜑Disj 𝑝 ∈ (ran 𝐹 × ran 𝐺)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
191 disjss1 5013 . . . . . . . . 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)}))))
192114, 190, 191mpsyl 68 . . . . . . . 8 (𝜑Disj 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
193192adantr 484 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → Disj 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
194 measvuni 31547 . . . . . . 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)}))))
195174, 176, 177, 193, 194syl112anc 1371 . . . . . 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)}))))
196 ssfi 8726 . . . . . . . . 9 (((ran 𝐹 × ran 𝐺) ∈ Fin ∧ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
197113, 114, 196sylancl 589 . . . . . . . 8 (𝜑 → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
198197adantr 484 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
199 simpll 766 . . . . . . . 8 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
200 simpr 488 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)))
201114, 200sseldi 3940 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (ran 𝐹 × ran 𝐺))
202 xp1st 7707 . . . . . . . . 9 (𝑝 ∈ (ran 𝐹 × ran 𝐺) → (1st𝑝) ∈ ran 𝐹)
203201, 202syl 17 . . . . . . . 8 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (1st𝑝) ∈ ran 𝐹)
204 xp2nd 7708 . . . . . . . . 9 (𝑝 ∈ (ran 𝐹 × ran 𝐺) → (2nd𝑝) ∈ ran 𝐺)
205201, 204syl 17 . . . . . . . 8 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (2nd𝑝) ∈ ran 𝐺)
206 oveq12 7149 . . . . . . . . . . . . . . . 16 ((𝑥 = 0𝑦 = 0 ) → (𝑥 + 𝑦) = ( 0 + 0 ))
207 sibfof.5 . . . . . . . . . . . . . . . 16 (𝜑 → ( 0 + 0 ) = (0g𝐾))
208206, 207sylan9eqr 2879 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 = 0𝑦 = 0 )) → (𝑥 + 𝑦) = (0g𝐾))
209208ex 416 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥 = 0𝑦 = 0 ) → (𝑥 + 𝑦) = (0g𝐾)))
210209necon3ad 3024 . . . . . . . . . . . . 13 (𝜑 → ((𝑥 + 𝑦) ≠ (0g𝐾) → ¬ (𝑥 = 0𝑦 = 0 )))
211 neorian 3105 . . . . . . . . . . . . 13 ((𝑥0𝑦0 ) ↔ ¬ (𝑥 = 0𝑦 = 0 ))
212210, 211syl6ibr 255 . . . . . . . . . . . 12 (𝜑 → ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
213212adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
214213ralrimivva 3181 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
215199, 214syl 17 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
21667a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ ( + “ {𝑧}))
217216sselda 3942 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ( + “ {𝑧}))
218 fniniseg 6812 . . . . . . . . . . . . 13 ( + Fn (𝐵 × 𝐵) → (𝑝 ∈ ( + “ {𝑧}) ↔ (𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧)))
219199, 61, 2183syl 18 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑝 ∈ ( + “ {𝑧}) ↔ (𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧)))
220217, 219mpbid 235 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧))
221 simpr 488 . . . . . . . . . . . 12 ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧) → ( +𝑝) = 𝑧)
222 1st2nd2 7714 . . . . . . . . . . . . . . 15 (𝑝 ∈ (𝐵 × 𝐵) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
223222fveq2d 6656 . . . . . . . . . . . . . 14 (𝑝 ∈ (𝐵 × 𝐵) → ( +𝑝) = ( + ‘⟨(1st𝑝), (2nd𝑝)⟩))
224 df-ov 7143 . . . . . . . . . . . . . 14 ((1st𝑝) + (2nd𝑝)) = ( + ‘⟨(1st𝑝), (2nd𝑝)⟩)
225223, 224eqtr4di 2875 . . . . . . . . . . . . 13 (𝑝 ∈ (𝐵 × 𝐵) → ( +𝑝) = ((1st𝑝) + (2nd𝑝)))
226225adantr 484 . . . . . . . . . . . 12 ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧) → ( +𝑝) = ((1st𝑝) + (2nd𝑝)))
227221, 226eqtr3d 2859 . . . . . . . . . . 11 ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧) → 𝑧 = ((1st𝑝) + (2nd𝑝)))
228220, 227syl 17 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 = ((1st𝑝) + (2nd𝑝)))
229 simplr 768 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)}))
230229eldifbd 3921 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ¬ 𝑧 ∈ {(0g𝐾)})
231 velsn 4555 . . . . . . . . . . . 12 (𝑧 ∈ {(0g𝐾)} ↔ 𝑧 = (0g𝐾))
232231necon3bbii 3058 . . . . . . . . . . 11 𝑧 ∈ {(0g𝐾)} ↔ 𝑧 ≠ (0g𝐾))
233230, 232sylib 221 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ≠ (0g𝐾))
234228, 233eqnetrrd 3079 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾))
235175, 72sylanl2 680 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐵))
236235, 85syl 17 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (1st𝑝) ∈ 𝐵)
237235, 98syl 17 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (2nd𝑝) ∈ 𝐵)
238 oveq1 7147 . . . . . . . . . . . . 13 (𝑥 = (1st𝑝) → (𝑥 + 𝑦) = ((1st𝑝) + 𝑦))
239238neeq1d 3070 . . . . . . . . . . . 12 (𝑥 = (1st𝑝) → ((𝑥 + 𝑦) ≠ (0g𝐾) ↔ ((1st𝑝) + 𝑦) ≠ (0g𝐾)))
240 neeq1 3073 . . . . . . . . . . . . 13 (𝑥 = (1st𝑝) → (𝑥0 ↔ (1st𝑝) ≠ 0 ))
241240orbi1d 914 . . . . . . . . . . . 12 (𝑥 = (1st𝑝) → ((𝑥0𝑦0 ) ↔ ((1st𝑝) ≠ 0𝑦0 )))
242239, 241imbi12d 348 . . . . . . . . . . 11 (𝑥 = (1st𝑝) → (((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )) ↔ (((1st𝑝) + 𝑦) ≠ (0g𝐾) → ((1st𝑝) ≠ 0𝑦0 ))))
243 oveq2 7148 . . . . . . . . . . . . 13 (𝑦 = (2nd𝑝) → ((1st𝑝) + 𝑦) = ((1st𝑝) + (2nd𝑝)))
244243neeq1d 3070 . . . . . . . . . . . 12 (𝑦 = (2nd𝑝) → (((1st𝑝) + 𝑦) ≠ (0g𝐾) ↔ ((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾)))
245 neeq1 3073 . . . . . . . . . . . . 13 (𝑦 = (2nd𝑝) → (𝑦0 ↔ (2nd𝑝) ≠ 0 ))
246245orbi2d 913 . . . . . . . . . . . 12 (𝑦 = (2nd𝑝) → (((1st𝑝) ≠ 0𝑦0 ) ↔ ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 )))
247244, 246imbi12d 348 . . . . . . . . . . 11 (𝑦 = (2nd𝑝) → ((((1st𝑝) + 𝑦) ≠ (0g𝐾) → ((1st𝑝) ≠ 0𝑦0 )) ↔ (((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))))
248242, 247rspc2v 3608 . . . . . . . . . 10 (((1st𝑝) ∈ 𝐵 ∧ (2nd𝑝) ∈ 𝐵) → (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )) → (((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))))
249236, 237, 248syl2anc 587 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )) → (((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))))
250215, 234, 249mp2d 49 . . . . . . . 8 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))
2513, 4, 11, 12, 13, 14, 15, 16, 17, 19, 2, 74sibfinima 31671 . . . . . . . 8 (((𝜑 ∧ (1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) ∧ ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 )) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ (0[,)+∞))
252199, 203, 205, 250, 251syl31anc 1370 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ (0[,)+∞))
253198, 252esumpfinval 31408 . . . . . 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)}))))
254171, 195, 2533eqtrd 2861 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹f + 𝐺) “ {𝑧})) = Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
255 rge0ssre 12834 . . . . . . 7 (0[,)+∞) ⊆ ℝ
256255, 252sseldi 3940 . . . . . 6 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ ℝ)
257198, 256fsumrecl 15082 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ ℝ)
258254, 257eqeltrd 2914 . . . 4 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹f + 𝐺) “ {𝑧})) ∈ ℝ)
259174adantr 484 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑀 ∈ (measures‘dom 𝑀))
260175, 108sylanl2 680 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
261 measge0 31540 . . . . . . 7 ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀) → 0 ≤ (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
262259, 260, 261syl2anc 587 . . . . . 6 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 0 ≤ (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
263198, 256, 262fsumge0 15141 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → 0 ≤ Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
264263, 254breqtrrd 5070 . . . 4 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → 0 ≤ (𝑀‘((𝐹f + 𝐺) “ {𝑧})))
265 elrege0 12832 . . . 4 ((𝑀‘((𝐹f + 𝐺) “ {𝑧})) ∈ (0[,)+∞) ↔ ((𝑀‘((𝐹f + 𝐺) “ {𝑧})) ∈ ℝ ∧ 0 ≤ (𝑀‘((𝐹f + 𝐺) “ {𝑧}))))
266258, 264, 265sylanbrc 586 . . 3 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹f + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
267266ralrimiva 3174 . 2 (𝜑 → ∀𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})(𝑀‘((𝐹f + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
268 eqid 2822 . . 3 (sigaGen‘(TopOpen‘𝐾)) = (sigaGen‘(TopOpen‘𝐾))
269 eqid 2822 . . 3 (0g𝐾) = (0g𝐾)
270 eqid 2822 . . 3 ( ·𝑠𝐾) = ( ·𝑠𝐾)
271 eqid 2822 . . 3 (ℝHom‘(Scalar‘𝐾)) = (ℝHom‘(Scalar‘𝐾))
27227, 28, 268, 269, 270, 271, 26, 16issibf 31665 . 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[,)+∞))))
273169, 137, 267, 272mpbir3and 1339 1 (𝜑 → (𝐹f + 𝐺) ∈ dom (𝐾sitg𝑀))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2114   ≠ wne 3011  ∀wral 3130  Vcvv 3469   ∖ cdif 3905   ∪ cun 3906   ∩ cin 3907   ⊆ wss 3908  ∅c0 4265  {csn 4539  ⟨cop 4545  ∪ cuni 4813  ∪ ciun 4894  Disj wdisj 5007   class class class wbr 5042   × cxp 5530  ◡ccnv 5531  dom cdm 5532  ran crn 5533   “ cima 5535  Fun wfun 6328   Fn wfn 6329  ⟶wf 6330  ‘cfv 6334  (class class class)co 7140   ∘f cof 7392  ωcom 7565  1st c1st 7673  2nd c2nd 7674   ↑m cmap 8393   ≼ cdom 8494   ≺ csdm 8495  Fincfn 8496  ℝcr 10525  0cc0 10526  +∞cpnf 10661   ≤ cle 10665  [,)cico 12728  Σcsu 15033  Basecbs 16474  Scalarcsca 16559   ·𝑠 cvsca 16560  TopOpenctopn 16686  0gc0g 16704  Topctop 21496  TopSpctps 21535  Clsdccld 21619  Frect1 21910  ℝHomcrrh 31308  Σ*cesum 31360  sigAlgebracsiga 31441  sigaGencsigagen 31471  measurescmeas 31528  MblFnMcmbfm 31582  sitgcsitg 31661 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092  ax-ac2 9874  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605  ax-mulf 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-iin 4897  df-disj 5008  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-of 7394  df-om 7566  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-acn 9359  df-ac 9531  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12886  df-fzo 13029  df-fl 13157  df-mod 13233  df-seq 13365  df-exp 13426  df-fac 13630  df-bc 13659  df-hash 13687  df-shft 14417  df-cj 14449  df-re 14450  df-im 14451  df-sqrt 14585  df-abs 14586  df-limsup 14819  df-clim 14836  df-rlim 14837  df-sum 15034  df-ef 15412  df-sin 15414  df-cos 15415  df-pi 15417  df-struct 16476  df-ndx 16477  df-slot 16478  df-base 16480  df-sets 16481  df-ress 16482  df-plusg 16569  df-mulr 16570  df-starv 16571  df-sca 16572  df-vsca 16573  df-ip 16574  df-tset 16575  df-ple 16576  df-ds 16578  df-unif 16579  df-hom 16580  df-cco 16581  df-rest 16687  df-topn 16688  df-0g 16706  df-gsum 16707  df-topgen 16708  df-pt 16709  df-prds 16712  df-ordt 16765  df-xrs 16766  df-qtop 16771  df-imas 16772  df-xps 16774  df-mre 16848  df-mrc 16849  df-acs 16851  df-ps 17801  df-tsr 17802  df-plusf 17842  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-mhm 17947  df-submnd 17948  df-grp 18097  df-minusg 18098  df-sbg 18099  df-mulg 18216  df-subg 18267  df-cntz 18438  df-cmn 18899  df-abl 18900  df-mgp 19231  df-ur 19243  df-ring 19290  df-cring 19291  df-subrg 19524  df-abv 19579  df-lmod 19627  df-scaf 19628  df-sra 19935  df-rgmod 19936  df-psmet 20081  df-xmet 20082  df-met 20083  df-bl 20084  df-mopn 20085  df-fbas 20086  df-fg 20087  df-cnfld 20090  df-top 21497  df-topon 21514  df-topsp 21536  df-bases 21549  df-cld 21622  df-ntr 21623  df-cls 21624  df-nei 21701  df-lp 21739  df-perf 21740  df-cn 21830  df-cnp 21831  df-t1 21917  df-haus 21918  df-tx 22165  df-hmeo 22358  df-fil 22449  df-fm 22541  df-flim 22542  df-flf 22543  df-tmd 22675  df-tgp 22676  df-tsms 22730  df-trg 22763  df-xms 22925  df-ms 22926  df-tms 22927  df-nm 23187  df-ngp 23188  df-nrg 23190  df-nlm 23191  df-ii 23480  df-cncf 23481  df-limc 24467  df-dv 24468  df-log 25146  df-esum 31361  df-siga 31442  df-sigagen 31472  df-meas 31529  df-mbfm 31583  df-sitg 31662 This theorem is referenced by:  sitmcl  31683
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