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Theorem sibfof 32940
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 22285 . . . . . . . . . . 11 (𝑊 ∈ TopSp → 𝐵 = 𝐽)
62, 5syl 17 . . . . . . . . . 10 (𝜑𝐵 = 𝐽)
76sqxpeqd 5665 . . . . . . . . 9 (𝜑 → (𝐵 × 𝐵) = ( 𝐽 × 𝐽))
87feq2d 6654 . . . . . . . 8 (𝜑 → ( + :(𝐵 × 𝐵)⟶𝐶+ :( 𝐽 × 𝐽)⟶𝐶))
91, 8mpbid 231 . . . . . . 7 (𝜑+ :( 𝐽 × 𝐽)⟶𝐶)
109fovcdmda 7525 . . . . . 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 32936 . . . . . 6 (𝜑𝐹: dom 𝑀 𝐽)
19 sibfof.2 . . . . . . 7 (𝜑𝐺 ∈ dom (𝑊sitg𝑀))
203, 4, 11, 12, 13, 14, 15, 16, 19sibff 32936 . . . . . 6 (𝜑𝐺: dom 𝑀 𝐽)
21 dmexg 7840 . . . . . . 7 (𝑀 ran measures → dom 𝑀 ∈ V)
22 uniexg 7677 . . . . . . 7 (dom 𝑀 ∈ V → dom 𝑀 ∈ V)
2316, 21, 223syl 18 . . . . . 6 (𝜑 dom 𝑀 ∈ V)
24 inidm 4178 . . . . . 6 ( dom 𝑀 dom 𝑀) = dom 𝑀
2510, 18, 20, 23, 23, 24off 7635 . . . . 5 (𝜑 → (𝐹f + 𝐺): dom 𝑀𝐶)
26 sibfof.3 . . . . . . . 8 (𝜑𝐾 ∈ TopSp)
27 sibfof.c . . . . . . . . 9 𝐶 = (Base‘𝐾)
28 eqid 2736 . . . . . . . . 9 (TopOpen‘𝐾) = (TopOpen‘𝐾)
2927, 28tpsuni 22285 . . . . . . . 8 (𝐾 ∈ TopSp → 𝐶 = (TopOpen‘𝐾))
3026, 29syl 17 . . . . . . 7 (𝜑𝐶 = (TopOpen‘𝐾))
31 fvex 6855 . . . . . . . 8 (TopOpen‘𝐾) ∈ V
32 unisg 32742 . . . . . . . 8 ((TopOpen‘𝐾) ∈ V → (sigaGen‘(TopOpen‘𝐾)) = (TopOpen‘𝐾))
3331, 32ax-mp 5 . . . . . . 7 (sigaGen‘(TopOpen‘𝐾)) = (TopOpen‘𝐾)
3430, 33eqtr4di 2794 . . . . . 6 (𝜑𝐶 = (sigaGen‘(TopOpen‘𝐾)))
3534feq3d 6655 . . . . 5 (𝜑 → ((𝐹f + 𝐺): dom 𝑀𝐶 ↔ (𝐹f + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾))))
3625, 35mpbid 231 . . . 4 (𝜑 → (𝐹f + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾)))
3731a1i 11 . . . . . . 7 (𝜑 → (TopOpen‘𝐾) ∈ V)
3837sgsiga 32741 . . . . . 6 (𝜑 → (sigaGen‘(TopOpen‘𝐾)) ∈ ran sigAlgebra)
3938uniexd 7679 . . . . 5 (𝜑 (sigaGen‘(TopOpen‘𝐾)) ∈ V)
4039, 23elmapd 8779 . . . 4 (𝜑 → ((𝐹f + 𝐺) ∈ ( (sigaGen‘(TopOpen‘𝐾)) ↑m dom 𝑀) ↔ (𝐹f + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾))))
4136, 40mpbird 256 . . 3 (𝜑 → (𝐹f + 𝐺) ∈ ( (sigaGen‘(TopOpen‘𝐾)) ↑m dom 𝑀))
42 inundif 4438 . . . . . . 7 ((𝑏 ∩ ran (𝐹f + 𝐺)) ∪ (𝑏 ∖ ran (𝐹f + 𝐺))) = 𝑏
4342imaeq2i 6011 . . . . . 6 ((𝐹f + 𝐺) “ ((𝑏 ∩ ran (𝐹f + 𝐺)) ∪ (𝑏 ∖ ran (𝐹f + 𝐺)))) = ((𝐹f + 𝐺) “ 𝑏)
44 ffun 6671 . . . . . . . 8 ((𝐹f + 𝐺): dom 𝑀𝐶 → Fun (𝐹f + 𝐺))
45 unpreima 7013 . . . . . . . 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 481 . . . . . 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, 47eqtr3id 2790 . . . . 5 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹f + 𝐺) “ 𝑏) = (((𝐹f + 𝐺) “ (𝑏 ∩ ran (𝐹f + 𝐺))) ∪ ((𝐹f + 𝐺) “ (𝑏 ∖ ran (𝐹f + 𝐺)))))
49 dmmeas 32800 . . . . . . . 8 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
5016, 49syl 17 . . . . . . 7 (𝜑 → dom 𝑀 ran sigAlgebra)
5150adantr 481 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → dom 𝑀 ran sigAlgebra)
52 imaiun 7192 . . . . . . . 8 ((𝐹f + 𝐺) “ 𝑧 ∈ (𝑏 ∩ ran (𝐹f + 𝐺)){𝑧}) = 𝑧 ∈ (𝑏 ∩ ran (𝐹f + 𝐺))((𝐹f + 𝐺) “ {𝑧})
53 iunid 5020 . . . . . . . . 9 𝑧 ∈ (𝑏 ∩ ran (𝐹f + 𝐺)){𝑧} = (𝑏 ∩ ran (𝐹f + 𝐺))
5453imaeq2i 6011 . . . . . . . 8 ((𝐹f + 𝐺) “ 𝑧 ∈ (𝑏 ∩ ran (𝐹f + 𝐺)){𝑧}) = ((𝐹f + 𝐺) “ (𝑏 ∩ ran (𝐹f + 𝐺)))
5552, 54eqtr3i 2766 . . . . . . 7 𝑧 ∈ (𝑏 ∩ ran (𝐹f + 𝐺))((𝐹f + 𝐺) “ {𝑧}) = ((𝐹f + 𝐺) “ (𝑏 ∩ ran (𝐹f + 𝐺)))
56 inss2 4189 . . . . . . . . . 10 (𝑏 ∩ ran (𝐹f + 𝐺)) ⊆ ran (𝐹f + 𝐺)
576feq3d 6655 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹: dom 𝑀𝐵𝐹: dom 𝑀 𝐽))
5818, 57mpbird 256 . . . . . . . . . . . . . 14 (𝜑𝐹: dom 𝑀𝐵)
596feq3d 6655 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺: dom 𝑀𝐵𝐺: dom 𝑀 𝐽))
6020, 59mpbird 256 . . . . . . . . . . . . . 14 (𝜑𝐺: dom 𝑀𝐵)
611ffnd 6669 . . . . . . . . . . . . . 14 (𝜑+ Fn (𝐵 × 𝐵))
6258, 60, 23, 61ofpreima2 31582 . . . . . . . . . . . . 13 (𝜑 → ((𝐹f + 𝐺) “ {𝑧}) = 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
6362adantr 481 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) → ((𝐹f + 𝐺) “ {𝑧}) = 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
6450adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) → dom 𝑀 ran sigAlgebra)
6550ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → dom 𝑀 ran sigAlgebra)
66 simpll 765 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
67 inss1 4188 . . . . . . . . . . . . . . . . . 18 (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ ( + “ {𝑧})
68 cnvimass 6033 . . . . . . . . . . . . . . . . . . . 20 ( + “ {𝑧}) ⊆ dom +
6968, 1fssdm 6688 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ( + “ {𝑧}) ⊆ (𝐵 × 𝐵))
7069adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) → ( + “ {𝑧}) ⊆ (𝐵 × 𝐵))
7167, 70sstrid 3955 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐵))
7271sselda 3944 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐵))
7350adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → dom 𝑀 ran sigAlgebra)
74 sibfof.4 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ Fre)
7574sgsiga 32741 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
7611, 75eqeltrid 2842 . . . . . . . . . . . . . . . . . 18 (𝜑𝑆 ran sigAlgebra)
7776adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝑆 ran sigAlgebra)
783, 4, 11, 12, 13, 14, 15, 16, 17sibfmbl 32935 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
7978adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
804tpstop 22286 . . . . . . . . . . . . . . . . . . . . 21 (𝑊 ∈ TopSp → 𝐽 ∈ Top)
81 cldssbrsiga 32786 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
822, 80, 813syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
8382adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
8474adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐽 ∈ Fre)
85 xp1st 7953 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ (𝐵 × 𝐵) → (1st𝑝) ∈ 𝐵)
8685adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (1st𝑝) ∈ 𝐵)
876adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐵 = 𝐽)
8886, 87eleqtrd 2840 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (1st𝑝) ∈ 𝐽)
89 eqid 2736 . . . . . . . . . . . . . . . . . . . . 21 𝐽 = 𝐽
9089t1sncld 22677 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ Fre ∧ (1st𝑝) ∈ 𝐽) → {(1st𝑝)} ∈ (Clsd‘𝐽))
9184, 88, 90syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ (Clsd‘𝐽))
9283, 91sseldd 3945 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ (sigaGen‘𝐽))
9392, 11eleqtrrdi 2849 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ 𝑆)
9473, 77, 79, 93mbfmcnvima 32855 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀)
9566, 72, 94syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀)
963, 4, 11, 12, 13, 14, 15, 16, 19sibfmbl 32935 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 ∈ (dom 𝑀MblFnM𝑆))
9796adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
98 xp2nd 7954 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ (𝐵 × 𝐵) → (2nd𝑝) ∈ 𝐵)
9998adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (2nd𝑝) ∈ 𝐵)
10099, 87eleqtrd 2840 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (2nd𝑝) ∈ 𝐽)
10189t1sncld 22677 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ Fre ∧ (2nd𝑝) ∈ 𝐽) → {(2nd𝑝)} ∈ (Clsd‘𝐽))
10284, 100, 101syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ (Clsd‘𝐽))
10383, 102sseldd 3945 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ (sigaGen‘𝐽))
104103, 11eleqtrrdi 2849 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ 𝑆)
10573, 77, 97, 104mbfmcnvima 32855 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀)
10666, 72, 105syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀)
107 inelsiga 32734 . . . . . . . . . . . . . . 15 ((dom 𝑀 ran sigAlgebra ∧ (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀 ∧ (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
10865, 95, 106, 107syl3anc 1371 . . . . . . . . . . . . . 14 (((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
109108ralrimiva 3143 . . . . . . . . . . . . 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 32937 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝐹 ∈ Fin)
1113, 4, 11, 12, 13, 14, 15, 16, 19sibfrn 32937 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝐺 ∈ Fin)
112 xpfi 9261 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
113110, 111, 112syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
114 inss2 4189 . . . . . . . . . . . . . . . 16 (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)
115 ssdomg 8940 . . . . . . . . . . . . . . . 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 9588 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 × ran 𝐺) ∈ Fin ↔ (ran 𝐹 × ran 𝐺) ≺ ω)
118117biimpi 215 . . . . . . . . . . . . . . . 16 ((ran 𝐹 × ran 𝐺) ∈ Fin → (ran 𝐹 × ran 𝐺) ≺ ω)
119 sdomdom 8920 . . . . . . . . . . . . . . . 16 ((ran 𝐹 × ran 𝐺) ≺ ω → (ran 𝐹 × ran 𝐺) ≼ ω)
120113, 118, 1193syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (ran 𝐹 × ran 𝐺) ≼ ω)
121 domtr 8947 . . . . . . . . . . . . . . 15 (((( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺) ∧ (ran 𝐹 × ran 𝐺) ≼ ω) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
122116, 120, 121syl2anc 584 . . . . . . . . . . . . . 14 (𝜑 → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
123122adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
124 nfcv 2907 . . . . . . . . . . . . . 14 𝑝(( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))
125124sigaclcuni 32717 . . . . . . . . . . . . 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 1371 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) → 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
12763, 126eqeltrd 2838 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran (𝐹f + 𝐺)) → ((𝐹f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
128127ralrimiva 3143 . . . . . . . . . 10 (𝜑 → ∀𝑧 ∈ ran (𝐹f + 𝐺)((𝐹f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
129 ssralv 4010 . . . . . . . . . 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 481 . . . . . . . 8 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹f + 𝐺))((𝐹f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
1321ffund 6672 . . . . . . . . . . . . 13 (𝜑 → Fun + )
133 imafi 9119 . . . . . . . . . . . . 13 ((Fun + ∧ (ran 𝐹 × ran 𝐺) ∈ Fin) → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
134132, 113, 133syl2anc 584 . . . . . . . . . . . 12 (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
13518, 20, 9, 23ofrn2 31556 . . . . . . . . . . . 12 (𝜑 → ran (𝐹f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
136 ssfi 9117 . . . . . . . . . . . 12 ((( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin ∧ ran (𝐹f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺))) → ran (𝐹f + 𝐺) ∈ Fin)
137134, 135, 136syl2anc 584 . . . . . . . . . . 11 (𝜑 → ran (𝐹f + 𝐺) ∈ Fin)
138 ssdomg 8940 . . . . . . . . . . 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 9588 . . . . . . . . . . . 12 (ran (𝐹f + 𝐺) ∈ Fin ↔ ran (𝐹f + 𝐺) ≺ ω)
141137, 140sylib 217 . . . . . . . . . . 11 (𝜑 → ran (𝐹f + 𝐺) ≺ ω)
142 sdomdom 8920 . . . . . . . . . . 11 (ran (𝐹f + 𝐺) ≺ ω → ran (𝐹f + 𝐺) ≼ ω)
143141, 142syl 17 . . . . . . . . . 10 (𝜑 → ran (𝐹f + 𝐺) ≼ ω)
144 domtr 8947 . . . . . . . . . 10 (((𝑏 ∩ ran (𝐹f + 𝐺)) ≼ ran (𝐹f + 𝐺) ∧ ran (𝐹f + 𝐺) ≼ ω) → (𝑏 ∩ ran (𝐹f + 𝐺)) ≼ ω)
145139, 143, 144syl2anc 584 . . . . . . . . 9 (𝜑 → (𝑏 ∩ ran (𝐹f + 𝐺)) ≼ ω)
146145adantr 481 . . . . . . . 8 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (𝑏 ∩ ran (𝐹f + 𝐺)) ≼ ω)
147 nfcv 2907 . . . . . . . . 9 𝑧(𝑏 ∩ ran (𝐹f + 𝐺))
148147sigaclcuni 32717 . . . . . . . 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 1371 . . . . . . 7 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → 𝑧 ∈ (𝑏 ∩ ran (𝐹f + 𝐺))((𝐹f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
15055, 149eqeltrrid 2843 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹f + 𝐺) “ (𝑏 ∩ ran (𝐹f + 𝐺))) ∈ dom 𝑀)
151 difpreima 7015 . . . . . . . . . 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 6034 . . . . . . . . . . 11 ((𝐹f + 𝐺) “ ran (𝐹f + 𝐺)) = dom (𝐹f + 𝐺)
154153difeq2i 4079 . . . . . . . . . 10 (((𝐹f + 𝐺) “ 𝑏) ∖ ((𝐹f + 𝐺) “ ran (𝐹f + 𝐺))) = (((𝐹f + 𝐺) “ 𝑏) ∖ dom (𝐹f + 𝐺))
155 cnvimass 6033 . . . . . . . . . . 11 ((𝐹f + 𝐺) “ 𝑏) ⊆ dom (𝐹f + 𝐺)
156 ssdif0 4323 . . . . . . . . . . 11 (((𝐹f + 𝐺) “ 𝑏) ⊆ dom (𝐹f + 𝐺) ↔ (((𝐹f + 𝐺) “ 𝑏) ∖ dom (𝐹f + 𝐺)) = ∅)
157155, 156mpbi 229 . . . . . . . . . 10 (((𝐹f + 𝐺) “ 𝑏) ∖ dom (𝐹f + 𝐺)) = ∅
158154, 157eqtri 2764 . . . . . . . . 9 (((𝐹f + 𝐺) “ 𝑏) ∖ ((𝐹f + 𝐺) “ ran (𝐹f + 𝐺))) = ∅
159152, 158eqtrdi 2792 . . . . . . . 8 (𝜑 → ((𝐹f + 𝐺) “ (𝑏 ∖ ran (𝐹f + 𝐺))) = ∅)
160 0elsiga 32713 . . . . . . . . 9 (dom 𝑀 ran sigAlgebra → ∅ ∈ dom 𝑀)
16116, 49, 1603syl 18 . . . . . . . 8 (𝜑 → ∅ ∈ dom 𝑀)
162159, 161eqeltrd 2838 . . . . . . 7 (𝜑 → ((𝐹f + 𝐺) “ (𝑏 ∖ ran (𝐹f + 𝐺))) ∈ dom 𝑀)
163162adantr 481 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹f + 𝐺) “ (𝑏 ∖ ran (𝐹f + 𝐺))) ∈ dom 𝑀)
164 unelsiga 32733 . . . . . 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 1371 . . . . 5 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (((𝐹f + 𝐺) “ (𝑏 ∩ ran (𝐹f + 𝐺))) ∪ ((𝐹f + 𝐺) “ (𝑏 ∖ ran (𝐹f + 𝐺)))) ∈ dom 𝑀)
16648, 165eqeltrd 2838 . . . 4 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹f + 𝐺) “ 𝑏) ∈ dom 𝑀)
167166ralrimiva 3143 . . 3 (𝜑 → ∀𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))((𝐹f + 𝐺) “ 𝑏) ∈ dom 𝑀)
16850, 38ismbfm 32850 . . 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 711 . 2 (𝜑 → (𝐹f + 𝐺) ∈ (dom 𝑀MblFnM(sigaGen‘(TopOpen‘𝐾))))
17062adantr 481 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → ((𝐹f + 𝐺) “ {𝑧}) = 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
171170fveq2d 6846 . . . . . 6 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹f + 𝐺) “ {𝑧})) = (𝑀 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
172 measbasedom 32801 . . . . . . . . 9 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
17316, 172sylib 217 . . . . . . . 8 (𝜑𝑀 ∈ (measures‘dom 𝑀))
174173adantr 481 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → 𝑀 ∈ (measures‘dom 𝑀))
175 eldifi 4086 . . . . . . . 8 (𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)}) → 𝑧 ∈ ran (𝐹f + 𝐺))
176175, 109sylan2 593 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → ∀𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
177122adantr 481 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
178 sneq 4596 . . . . . . . . . . 11 (𝑥 = (1st𝑝) → {𝑥} = {(1st𝑝)})
179178imaeq2d 6013 . . . . . . . . . 10 (𝑥 = (1st𝑝) → (𝐹 “ {𝑥}) = (𝐹 “ {(1st𝑝)}))
180 sneq 4596 . . . . . . . . . . 11 (𝑦 = (2nd𝑝) → {𝑦} = {(2nd𝑝)})
181180imaeq2d 6013 . . . . . . . . . 10 (𝑦 = (2nd𝑝) → (𝐺 “ {𝑦}) = (𝐺 “ {(2nd𝑝)}))
18218ffund 6672 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
183 sndisj 5096 . . . . . . . . . . 11 Disj 𝑥 ∈ ran 𝐹{𝑥}
184 disjpreima 31502 . . . . . . . . . . 11 ((Fun 𝐹Disj 𝑥 ∈ ran 𝐹{𝑥}) → Disj 𝑥 ∈ ran 𝐹(𝐹 “ {𝑥}))
185182, 183, 184sylancl 586 . . . . . . . . . 10 (𝜑Disj 𝑥 ∈ ran 𝐹(𝐹 “ {𝑥}))
18620ffund 6672 . . . . . . . . . . 11 (𝜑 → Fun 𝐺)
187 sndisj 5096 . . . . . . . . . . 11 Disj 𝑦 ∈ ran 𝐺{𝑦}
188 disjpreima 31502 . . . . . . . . . . 11 ((Fun 𝐺Disj 𝑦 ∈ ran 𝐺{𝑦}) → Disj 𝑦 ∈ ran 𝐺(𝐺 “ {𝑦}))
189186, 187, 188sylancl 586 . . . . . . . . . 10 (𝜑Disj 𝑦 ∈ ran 𝐺(𝐺 “ {𝑦}))
190179, 181, 185, 189disjxpin 31506 . . . . . . . . 9 (𝜑Disj 𝑝 ∈ (ran 𝐹 × ran 𝐺)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
191 disjss1 5076 . . . . . . . . 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 481 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → Disj 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
194 measvuni 32813 . . . . . . 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 1374 . . . . . 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 9117 . . . . . . . . 9 (((ran 𝐹 × ran 𝐺) ∈ Fin ∧ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
197113, 114, 196sylancl 586 . . . . . . . 8 (𝜑 → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
198197adantr 481 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
199 simpll 765 . . . . . . . 8 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
200 simpr 485 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)))
201114, 200sselid 3942 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (ran 𝐹 × ran 𝐺))
202 xp1st 7953 . . . . . . . . 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 7954 . . . . . . . . 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 7366 . . . . . . . . . . . . . . . 16 ((𝑥 = 0𝑦 = 0 ) → (𝑥 + 𝑦) = ( 0 + 0 ))
207 sibfof.5 . . . . . . . . . . . . . . . 16 (𝜑 → ( 0 + 0 ) = (0g𝐾))
208206, 207sylan9eqr 2798 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 = 0𝑦 = 0 )) → (𝑥 + 𝑦) = (0g𝐾))
209208ex 413 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥 = 0𝑦 = 0 ) → (𝑥 + 𝑦) = (0g𝐾)))
210209necon3ad 2956 . . . . . . . . . . . . 13 (𝜑 → ((𝑥 + 𝑦) ≠ (0g𝐾) → ¬ (𝑥 = 0𝑦 = 0 )))
211 neorian 3039 . . . . . . . . . . . . 13 ((𝑥0𝑦0 ) ↔ ¬ (𝑥 = 0𝑦 = 0 ))
212210, 211syl6ibr 251 . . . . . . . . . . . 12 (𝜑 → ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
213212adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
214213ralrimivva 3197 . . . . . . . . . 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 3944 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ( + “ {𝑧}))
218 fniniseg 7010 . . . . . . . . . . . . 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 231 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧))
221 simpr 485 . . . . . . . . . . . 12 ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧) → ( +𝑝) = 𝑧)
222 1st2nd2 7960 . . . . . . . . . . . . . . 15 (𝑝 ∈ (𝐵 × 𝐵) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
223222fveq2d 6846 . . . . . . . . . . . . . 14 (𝑝 ∈ (𝐵 × 𝐵) → ( +𝑝) = ( + ‘⟨(1st𝑝), (2nd𝑝)⟩))
224 df-ov 7360 . . . . . . . . . . . . . 14 ((1st𝑝) + (2nd𝑝)) = ( + ‘⟨(1st𝑝), (2nd𝑝)⟩)
225223, 224eqtr4di 2794 . . . . . . . . . . . . 13 (𝑝 ∈ (𝐵 × 𝐵) → ( +𝑝) = ((1st𝑝) + (2nd𝑝)))
226225adantr 481 . . . . . . . . . . . 12 ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧) → ( +𝑝) = ((1st𝑝) + (2nd𝑝)))
227221, 226eqtr3d 2778 . . . . . . . . . . 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 767 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)}))
230229eldifbd 3923 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ¬ 𝑧 ∈ {(0g𝐾)})
231 velsn 4602 . . . . . . . . . . . 12 (𝑧 ∈ {(0g𝐾)} ↔ 𝑧 = (0g𝐾))
232231necon3bbii 2991 . . . . . . . . . . 11 𝑧 ∈ {(0g𝐾)} ↔ 𝑧 ≠ (0g𝐾))
233230, 232sylib 217 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ≠ (0g𝐾))
234228, 233eqnetrrd 3012 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾))
235175, 72sylanl2 679 . . . . . . . . . . 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 7364 . . . . . . . . . . . . 13 (𝑥 = (1st𝑝) → (𝑥 + 𝑦) = ((1st𝑝) + 𝑦))
239238neeq1d 3003 . . . . . . . . . . . 12 (𝑥 = (1st𝑝) → ((𝑥 + 𝑦) ≠ (0g𝐾) ↔ ((1st𝑝) + 𝑦) ≠ (0g𝐾)))
240 neeq1 3006 . . . . . . . . . . . . 13 (𝑥 = (1st𝑝) → (𝑥0 ↔ (1st𝑝) ≠ 0 ))
241240orbi1d 915 . . . . . . . . . . . 12 (𝑥 = (1st𝑝) → ((𝑥0𝑦0 ) ↔ ((1st𝑝) ≠ 0𝑦0 )))
242239, 241imbi12d 344 . . . . . . . . . . 11 (𝑥 = (1st𝑝) → (((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )) ↔ (((1st𝑝) + 𝑦) ≠ (0g𝐾) → ((1st𝑝) ≠ 0𝑦0 ))))
243 oveq2 7365 . . . . . . . . . . . . 13 (𝑦 = (2nd𝑝) → ((1st𝑝) + 𝑦) = ((1st𝑝) + (2nd𝑝)))
244243neeq1d 3003 . . . . . . . . . . . 12 (𝑦 = (2nd𝑝) → (((1st𝑝) + 𝑦) ≠ (0g𝐾) ↔ ((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾)))
245 neeq1 3006 . . . . . . . . . . . . 13 (𝑦 = (2nd𝑝) → (𝑦0 ↔ (2nd𝑝) ≠ 0 ))
246245orbi2d 914 . . . . . . . . . . . 12 (𝑦 = (2nd𝑝) → (((1st𝑝) ≠ 0𝑦0 ) ↔ ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 )))
247244, 246imbi12d 344 . . . . . . . . . . 11 (𝑦 = (2nd𝑝) → ((((1st𝑝) + 𝑦) ≠ (0g𝐾) → ((1st𝑝) ≠ 0𝑦0 )) ↔ (((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))))
248242, 247rspc2v 3590 . . . . . . . . . 10 (((1st𝑝) ∈ 𝐵 ∧ (2nd𝑝) ∈ 𝐵) → (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )) → (((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))))
249236, 237, 248syl2anc 584 . . . . . . . . 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 32939 . . . . . . . 8 (((𝜑 ∧ (1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) ∧ ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 )) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ (0[,)+∞))
252199, 203, 205, 250, 251syl31anc 1373 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ (0[,)+∞))
253198, 252esumpfinval 32674 . . . . . 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 2780 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹f + 𝐺) “ {𝑧})) = Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
255 rge0ssre 13373 . . . . . . 7 (0[,)+∞) ⊆ ℝ
256255, 252sselid 3942 . . . . . 6 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ ℝ)
257198, 256fsumrecl 15619 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ ℝ)
258254, 257eqeltrd 2838 . . . 4 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹f + 𝐺) “ {𝑧})) ∈ ℝ)
259174adantr 481 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑀 ∈ (measures‘dom 𝑀))
260175, 108sylanl2 679 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
261 measge0 32806 . . . . . . 7 ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀) → 0 ≤ (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
262259, 260, 261syl2anc 584 . . . . . 6 (((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 0 ≤ (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
263198, 256, 262fsumge0 15680 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → 0 ≤ Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
264263, 254breqtrrd 5133 . . . 4 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → 0 ≤ (𝑀‘((𝐹f + 𝐺) “ {𝑧})))
265 elrege0 13371 . . . 4 ((𝑀‘((𝐹f + 𝐺) “ {𝑧})) ∈ (0[,)+∞) ↔ ((𝑀‘((𝐹f + 𝐺) “ {𝑧})) ∈ ℝ ∧ 0 ≤ (𝑀‘((𝐹f + 𝐺) “ {𝑧}))))
266258, 264, 265sylanbrc 583 . . 3 ((𝜑𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹f + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
267266ralrimiva 3143 . 2 (𝜑 → ∀𝑧 ∈ (ran (𝐹f + 𝐺) ∖ {(0g𝐾)})(𝑀‘((𝐹f + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
268 eqid 2736 . . 3 (sigaGen‘(TopOpen‘𝐾)) = (sigaGen‘(TopOpen‘𝐾))
269 eqid 2736 . . 3 (0g𝐾) = (0g𝐾)
270 eqid 2736 . . 3 ( ·𝑠𝐾) = ( ·𝑠𝐾)
271 eqid 2736 . . 3 (ℝHom‘(Scalar‘𝐾)) = (ℝHom‘(Scalar‘𝐾))
27227, 28, 268, 269, 270, 271, 26, 16issibf 32933 . 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 1342 1 (𝜑 → (𝐹f + 𝐺) ∈ dom (𝐾sitg𝑀))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2943  wral 3064  Vcvv 3445  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  {csn 4586  cop 4592   cuni 4865   ciun 4954  Disj wdisj 5070   class class class wbr 5105   × cxp 5631  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  ωcom 7802  1st c1st 7919  2nd c2nd 7920  m cmap 8765  cdom 8881  csdm 8882  Fincfn 8883  cr 11050  0cc0 11051  +∞cpnf 11186  cle 11190  [,)cico 13266  Σcsu 15570  Basecbs 17083  Scalarcsca 17136   ·𝑠 cvsca 17137  TopOpenctopn 17303  0gc0g 17321  Topctop 22242  TopSpctps 22281  Clsdccld 22367  Frect1 22658  ℝHomcrrh 32574  Σ*cesum 32626  sigAlgebracsiga 32707  sigaGencsigagen 32737  measurescmeas 32794  MblFnMcmbfm 32848  sitgcsitg 32929
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950  df-sin 15952  df-cos 15953  df-pi 15955  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-ordt 17383  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-ps 18455  df-tsr 18456  df-plusf 18496  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-subrg 20220  df-abv 20276  df-lmod 20324  df-scaf 20325  df-sra 20633  df-rgmod 20634  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-t1 22665  df-haus 22666  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-tmd 23423  df-tgp 23424  df-tsms 23478  df-trg 23511  df-xms 23673  df-ms 23674  df-tms 23675  df-nm 23938  df-ngp 23939  df-nrg 23941  df-nlm 23942  df-ii 24240  df-cncf 24241  df-limc 25230  df-dv 25231  df-log 25912  df-esum 32627  df-siga 32708  df-sigagen 32738  df-meas 32795  df-mbfm 32849  df-sitg 32930
This theorem is referenced by:  sitmcl  32951
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