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Theorem sibfof 29522
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 𝑏 𝑝 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 20500 . . . . . . . . . . 11 (𝑊 ∈ TopSp → 𝐵 = 𝐽)
62, 5syl 17 . . . . . . . . . 10 (𝜑𝐵 = 𝐽)
76sqxpeqd 5054 . . . . . . . . 9 (𝜑 → (𝐵 × 𝐵) = ( 𝐽 × 𝐽))
87feq2d 5929 . . . . . . . 8 (𝜑 → ( + :(𝐵 × 𝐵)⟶𝐶+ :( 𝐽 × 𝐽)⟶𝐶))
91, 8mpbid 220 . . . . . . 7 (𝜑+ :( 𝐽 × 𝐽)⟶𝐶)
109fovrnda 6680 . . . . . 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 29518 . . . . . 6 (𝜑𝐹: dom 𝑀 𝐽)
19 sibfof.2 . . . . . . 7 (𝜑𝐺 ∈ dom (𝑊sitg𝑀))
203, 4, 11, 12, 13, 14, 15, 16, 19sibff 29518 . . . . . 6 (𝜑𝐺: dom 𝑀 𝐽)
21 dmexg 6966 . . . . . . 7 (𝑀 ran measures → dom 𝑀 ∈ V)
22 uniexg 6830 . . . . . . 7 (dom 𝑀 ∈ V → dom 𝑀 ∈ V)
2316, 21, 223syl 18 . . . . . 6 (𝜑 dom 𝑀 ∈ V)
24 inidm 3783 . . . . . 6 ( dom 𝑀 dom 𝑀) = dom 𝑀
2510, 18, 20, 23, 23, 24off 6787 . . . . 5 (𝜑 → (𝐹𝑓 + 𝐺): dom 𝑀𝐶)
26 sibfof.3 . . . . . . . 8 (𝜑𝐾 ∈ TopSp)
27 sibfof.c . . . . . . . . 9 𝐶 = (Base‘𝐾)
28 eqid 2609 . . . . . . . . 9 (TopOpen‘𝐾) = (TopOpen‘𝐾)
2927, 28tpsuni 20500 . . . . . . . 8 (𝐾 ∈ TopSp → 𝐶 = (TopOpen‘𝐾))
3026, 29syl 17 . . . . . . 7 (𝜑𝐶 = (TopOpen‘𝐾))
31 fvex 6097 . . . . . . . 8 (TopOpen‘𝐾) ∈ V
32 unisg 29326 . . . . . . . 8 ((TopOpen‘𝐾) ∈ V → (sigaGen‘(TopOpen‘𝐾)) = (TopOpen‘𝐾))
3331, 32ax-mp 5 . . . . . . 7 (sigaGen‘(TopOpen‘𝐾)) = (TopOpen‘𝐾)
3430, 33syl6eqr 2661 . . . . . 6 (𝜑𝐶 = (sigaGen‘(TopOpen‘𝐾)))
3534feq3d 5930 . . . . 5 (𝜑 → ((𝐹𝑓 + 𝐺): dom 𝑀𝐶 ↔ (𝐹𝑓 + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾))))
3625, 35mpbid 220 . . . 4 (𝜑 → (𝐹𝑓 + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾)))
3731a1i 11 . . . . . . 7 (𝜑 → (TopOpen‘𝐾) ∈ V)
3837sgsiga 29325 . . . . . 6 (𝜑 → (sigaGen‘(TopOpen‘𝐾)) ∈ ran sigAlgebra)
39 uniexg 6830 . . . . . 6 ((sigaGen‘(TopOpen‘𝐾)) ∈ ran sigAlgebra → (sigaGen‘(TopOpen‘𝐾)) ∈ V)
4038, 39syl 17 . . . . 5 (𝜑 (sigaGen‘(TopOpen‘𝐾)) ∈ V)
4140, 23elmapd 7735 . . . 4 (𝜑 → ((𝐹𝑓 + 𝐺) ∈ ( (sigaGen‘(TopOpen‘𝐾)) ↑𝑚 dom 𝑀) ↔ (𝐹𝑓 + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾))))
4236, 41mpbird 245 . . 3 (𝜑 → (𝐹𝑓 + 𝐺) ∈ ( (sigaGen‘(TopOpen‘𝐾)) ↑𝑚 dom 𝑀))
43 inundif 3997 . . . . . . 7 ((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ∪ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) = 𝑏
4443imaeq2i 5369 . . . . . 6 ((𝐹𝑓 + 𝐺) “ ((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ∪ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))) = ((𝐹𝑓 + 𝐺) “ 𝑏)
45 ffun 5946 . . . . . . . 8 ((𝐹𝑓 + 𝐺): dom 𝑀𝐶 → Fun (𝐹𝑓 + 𝐺))
46 unpreima 6233 . . . . . . . 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 479 . . . . . 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 2657 . . . . 5 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ 𝑏) = (((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∪ ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))))
50 dmmeas 29384 . . . . . . . 8 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
5116, 50syl 17 . . . . . . 7 (𝜑 → dom 𝑀 ran sigAlgebra)
5251adantr 479 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → dom 𝑀 ran sigAlgebra)
53 imaiun 6384 . . . . . . . 8 ((𝐹𝑓 + 𝐺) “ 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)){𝑧}) = 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧})
54 iunid 4505 . . . . . . . . 9 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)){𝑧} = (𝑏 ∩ ran (𝐹𝑓 + 𝐺))
5554imaeq2i 5369 . . . . . . . 8 ((𝐹𝑓 + 𝐺) “ 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)){𝑧}) = ((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)))
5653, 55eqtr3i 2633 . . . . . . 7 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) = ((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)))
57 inss2 3795 . . . . . . . . . 10 (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ⊆ ran (𝐹𝑓 + 𝐺)
586feq3d 5930 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹: dom 𝑀𝐵𝐹: dom 𝑀 𝐽))
5918, 58mpbird 245 . . . . . . . . . . . . . 14 (𝜑𝐹: dom 𝑀𝐵)
606feq3d 5930 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺: dom 𝑀𝐵𝐺: dom 𝑀 𝐽))
6120, 60mpbird 245 . . . . . . . . . . . . . 14 (𝜑𝐺: dom 𝑀𝐵)
62 ffn 5943 . . . . . . . . . . . . . . 15 ( + :(𝐵 × 𝐵)⟶𝐶+ Fn (𝐵 × 𝐵))
631, 62syl 17 . . . . . . . . . . . . . 14 (𝜑+ Fn (𝐵 × 𝐵))
6459, 61, 23, 63ofpreima2 28642 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝑓 + 𝐺) “ {𝑧}) = 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
6564adantr 479 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → ((𝐹𝑓 + 𝐺) “ {𝑧}) = 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
6651adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → dom 𝑀 ran sigAlgebra)
6751ad2antrr 757 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → dom 𝑀 ran sigAlgebra)
68 simpll 785 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
69 inss1 3794 . . . . . . . . . . . . . . . . . 18 (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ ( + “ {𝑧})
70 cnvimass 5390 . . . . . . . . . . . . . . . . . . . 20 ( + “ {𝑧}) ⊆ dom +
71 fdm 5949 . . . . . . . . . . . . . . . . . . . . 21 ( + :(𝐵 × 𝐵)⟶𝐶 → dom + = (𝐵 × 𝐵))
721, 71syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom + = (𝐵 × 𝐵))
7370, 72syl5sseq 3615 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ( + “ {𝑧}) ⊆ (𝐵 × 𝐵))
7473adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → ( + “ {𝑧}) ⊆ (𝐵 × 𝐵))
7569, 74syl5ss 3578 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐵))
7675sselda 3567 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐵))
7751adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → dom 𝑀 ran sigAlgebra)
78 sibfof.4 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ Fre)
7978sgsiga 29325 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
8011, 79syl5eqel 2691 . . . . . . . . . . . . . . . . . 18 (𝜑𝑆 ran sigAlgebra)
8180adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝑆 ran sigAlgebra)
823, 4, 11, 12, 13, 14, 15, 16, 17sibfmbl 29517 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
8382adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
844tpstop 20501 . . . . . . . . . . . . . . . . . . . . 21 (𝑊 ∈ TopSp → 𝐽 ∈ Top)
85 cldssbrsiga 29370 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
862, 84, 853syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
8786adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
8878adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐽 ∈ Fre)
89 xp1st 7066 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ (𝐵 × 𝐵) → (1st𝑝) ∈ 𝐵)
9089adantl 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (1st𝑝) ∈ 𝐵)
916adantr 479 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐵 = 𝐽)
9290, 91eleqtrd 2689 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (1st𝑝) ∈ 𝐽)
93 eqid 2609 . . . . . . . . . . . . . . . . . . . . 21 𝐽 = 𝐽
9493t1sncld 20887 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ Fre ∧ (1st𝑝) ∈ 𝐽) → {(1st𝑝)} ∈ (Clsd‘𝐽))
9588, 92, 94syl2anc 690 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ (Clsd‘𝐽))
9687, 95sseldd 3568 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ (sigaGen‘𝐽))
9796, 11syl6eleqr 2698 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ 𝑆)
9877, 81, 83, 97mbfmcnvima 29439 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀)
9968, 76, 98syl2anc 690 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀)
1003, 4, 11, 12, 13, 14, 15, 16, 19sibfmbl 29517 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 ∈ (dom 𝑀MblFnM𝑆))
101100adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
102 xp2nd 7067 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ (𝐵 × 𝐵) → (2nd𝑝) ∈ 𝐵)
103102adantl 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (2nd𝑝) ∈ 𝐵)
104103, 91eleqtrd 2689 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (2nd𝑝) ∈ 𝐽)
10593t1sncld 20887 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ Fre ∧ (2nd𝑝) ∈ 𝐽) → {(2nd𝑝)} ∈ (Clsd‘𝐽))
10688, 104, 105syl2anc 690 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ (Clsd‘𝐽))
10787, 106sseldd 3568 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ (sigaGen‘𝐽))
108107, 11syl6eleqr 2698 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ 𝑆)
10977, 81, 101, 108mbfmcnvima 29439 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀)
11068, 76, 109syl2anc 690 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀)
111 inelsiga 29318 . . . . . . . . . . . . . . 15 ((dom 𝑀 ran sigAlgebra ∧ (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀 ∧ (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
11267, 99, 110, 111syl3anc 1317 . . . . . . . . . . . . . 14 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
113112ralrimiva 2948 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → ∀𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
1143, 4, 11, 12, 13, 14, 15, 16, 17sibfrn 29519 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝐹 ∈ Fin)
1153, 4, 11, 12, 13, 14, 15, 16, 19sibfrn 29519 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝐺 ∈ Fin)
116 xpfi 8093 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
117114, 115, 116syl2anc 690 . . . . . . . . . . . . . . . 16 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
118 inss2 3795 . . . . . . . . . . . . . . . 16 (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)
119 ssdomg 7864 . . . . . . . . . . . . . . . 16 ((ran 𝐹 × ran 𝐺) ∈ Fin → ((( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺)))
120117, 118, 119mpisyl 21 . . . . . . . . . . . . . . 15 (𝜑 → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺))
121 isfinite 8409 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 × ran 𝐺) ∈ Fin ↔ (ran 𝐹 × ran 𝐺) ≺ ω)
122121biimpi 204 . . . . . . . . . . . . . . . 16 ((ran 𝐹 × ran 𝐺) ∈ Fin → (ran 𝐹 × ran 𝐺) ≺ ω)
123 sdomdom 7846 . . . . . . . . . . . . . . . 16 ((ran 𝐹 × ran 𝐺) ≺ ω → (ran 𝐹 × ran 𝐺) ≼ ω)
124117, 122, 1233syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (ran 𝐹 × ran 𝐺) ≼ ω)
125 domtr 7872 . . . . . . . . . . . . . . 15 (((( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺) ∧ (ran 𝐹 × ran 𝐺) ≼ ω) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
126120, 124, 125syl2anc 690 . . . . . . . . . . . . . 14 (𝜑 → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
127126adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
128 nfcv 2750 . . . . . . . . . . . . . 14 𝑝(( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))
129128sigaclcuni 29301 . . . . . . . . . . . . 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)
13066, 113, 127, 129syl3anc 1317 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
13165, 130eqeltrd 2687 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → ((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
132131ralrimiva 2948 . . . . . . . . . 10 (𝜑 → ∀𝑧 ∈ ran (𝐹𝑓 + 𝐺)((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
133 ssralv 3628 . . . . . . . . . 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))
13457, 132, 133mpsyl 65 . . . . . . . . 9 (𝜑 → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
135134adantr 479 . . . . . . . 8 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
136 ffun 5946 . . . . . . . . . . . . . 14 ( + :(𝐵 × 𝐵)⟶𝐶 → Fun + )
1371, 136syl 17 . . . . . . . . . . . . 13 (𝜑 → Fun + )
138 imafi 8119 . . . . . . . . . . . . 13 ((Fun + ∧ (ran 𝐹 × ran 𝐺) ∈ Fin) → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
139137, 117, 138syl2anc 690 . . . . . . . . . . . 12 (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
14018, 20, 9, 23ofrn2 28615 . . . . . . . . . . . 12 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
141 ssfi 8042 . . . . . . . . . . . 12 ((( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin ∧ ran (𝐹𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺))) → ran (𝐹𝑓 + 𝐺) ∈ Fin)
142139, 140, 141syl2anc 690 . . . . . . . . . . 11 (𝜑 → ran (𝐹𝑓 + 𝐺) ∈ Fin)
143 ssdomg 7864 . . . . . . . . . . 11 (ran (𝐹𝑓 + 𝐺) ∈ Fin → ((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ⊆ ran (𝐹𝑓 + 𝐺) → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ran (𝐹𝑓 + 𝐺)))
144142, 57, 143mpisyl 21 . . . . . . . . . 10 (𝜑 → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ran (𝐹𝑓 + 𝐺))
145 isfinite 8409 . . . . . . . . . . . 12 (ran (𝐹𝑓 + 𝐺) ∈ Fin ↔ ran (𝐹𝑓 + 𝐺) ≺ ω)
146142, 145sylib 206 . . . . . . . . . . 11 (𝜑 → ran (𝐹𝑓 + 𝐺) ≺ ω)
147 sdomdom 7846 . . . . . . . . . . 11 (ran (𝐹𝑓 + 𝐺) ≺ ω → ran (𝐹𝑓 + 𝐺) ≼ ω)
148146, 147syl 17 . . . . . . . . . 10 (𝜑 → ran (𝐹𝑓 + 𝐺) ≼ ω)
149 domtr 7872 . . . . . . . . . 10 (((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ran (𝐹𝑓 + 𝐺) ∧ ran (𝐹𝑓 + 𝐺) ≼ ω) → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ω)
150144, 148, 149syl2anc 690 . . . . . . . . 9 (𝜑 → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ω)
151150adantr 479 . . . . . . . 8 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ω)
152 nfcv 2750 . . . . . . . . 9 𝑧(𝑏 ∩ ran (𝐹𝑓 + 𝐺))
153152sigaclcuni 29301 . . . . . . . 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)
15452, 135, 151, 153syl3anc 1317 . . . . . . 7 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
15556, 154syl5eqelr 2692 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∈ dom 𝑀)
156 difpreima 6235 . . . . . . . . . 10 (Fun (𝐹𝑓 + 𝐺) → ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) = (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺))))
15725, 45, 1563syl 18 . . . . . . . . 9 (𝜑 → ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) = (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺))))
158 cnvimarndm 5391 . . . . . . . . . . 11 ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺)) = dom (𝐹𝑓 + 𝐺)
159158difeq2i 3686 . . . . . . . . . 10 (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺))) = (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ dom (𝐹𝑓 + 𝐺))
160 cnvimass 5390 . . . . . . . . . . 11 ((𝐹𝑓 + 𝐺) “ 𝑏) ⊆ dom (𝐹𝑓 + 𝐺)
161 ssdif0 3895 . . . . . . . . . . 11 (((𝐹𝑓 + 𝐺) “ 𝑏) ⊆ dom (𝐹𝑓 + 𝐺) ↔ (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ dom (𝐹𝑓 + 𝐺)) = ∅)
162160, 161mpbi 218 . . . . . . . . . 10 (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ dom (𝐹𝑓 + 𝐺)) = ∅
163159, 162eqtri 2631 . . . . . . . . 9 (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺))) = ∅
164157, 163syl6eq 2659 . . . . . . . 8 (𝜑 → ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) = ∅)
165 0elsiga 29297 . . . . . . . . 9 (dom 𝑀 ran sigAlgebra → ∅ ∈ dom 𝑀)
16616, 50, 1653syl 18 . . . . . . . 8 (𝜑 → ∅ ∈ dom 𝑀)
167164, 166eqeltrd 2687 . . . . . . 7 (𝜑 → ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) ∈ dom 𝑀)
168167adantr 479 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) ∈ dom 𝑀)
169 unelsiga 29317 . . . . . 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)
17052, 155, 168, 169syl3anc 1317 . . . . 5 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∪ ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))) ∈ dom 𝑀)
17149, 170eqeltrd 2687 . . . 4 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ 𝑏) ∈ dom 𝑀)
172171ralrimiva 2948 . . 3 (𝜑 → ∀𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))((𝐹𝑓 + 𝐺) “ 𝑏) ∈ dom 𝑀)
17351, 38ismbfm 29434 . . 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)))
17442, 172, 173mpbir2and 958 . 2 (𝜑 → (𝐹𝑓 + 𝐺) ∈ (dom 𝑀MblFnM(sigaGen‘(TopOpen‘𝐾))))
17564adantr 479 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → ((𝐹𝑓 + 𝐺) “ {𝑧}) = 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
176175fveq2d 6091 . . . . . 6 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) = (𝑀 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
177 measbasedom 29385 . . . . . . . . 9 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
17816, 177sylib 206 . . . . . . . 8 (𝜑𝑀 ∈ (measures‘dom 𝑀))
179178adantr 479 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → 𝑀 ∈ (measures‘dom 𝑀))
180 eldifi 3693 . . . . . . . 8 (𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)}) → 𝑧 ∈ ran (𝐹𝑓 + 𝐺))
181180, 113sylan2 489 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → ∀𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
182126adantr 479 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
183 sneq 4134 . . . . . . . . . . 11 (𝑥 = (1st𝑝) → {𝑥} = {(1st𝑝)})
184183imaeq2d 5371 . . . . . . . . . 10 (𝑥 = (1st𝑝) → (𝐹 “ {𝑥}) = (𝐹 “ {(1st𝑝)}))
185 sneq 4134 . . . . . . . . . . 11 (𝑦 = (2nd𝑝) → {𝑦} = {(2nd𝑝)})
186185imaeq2d 5371 . . . . . . . . . 10 (𝑦 = (2nd𝑝) → (𝐺 “ {𝑦}) = (𝐺 “ {(2nd𝑝)}))
187 ffun 5946 . . . . . . . . . . . 12 (𝐹: dom 𝑀 𝐽 → Fun 𝐹)
18818, 187syl 17 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
189 sndisj 4568 . . . . . . . . . . 11 Disj 𝑥 ∈ ran 𝐹{𝑥}
190 disjpreima 28572 . . . . . . . . . . 11 ((Fun 𝐹Disj 𝑥 ∈ ran 𝐹{𝑥}) → Disj 𝑥 ∈ ran 𝐹(𝐹 “ {𝑥}))
191188, 189, 190sylancl 692 . . . . . . . . . 10 (𝜑Disj 𝑥 ∈ ran 𝐹(𝐹 “ {𝑥}))
192 ffun 5946 . . . . . . . . . . . 12 (𝐺: dom 𝑀 𝐽 → Fun 𝐺)
19320, 192syl 17 . . . . . . . . . . 11 (𝜑 → Fun 𝐺)
194 sndisj 4568 . . . . . . . . . . 11 Disj 𝑦 ∈ ran 𝐺{𝑦}
195 disjpreima 28572 . . . . . . . . . . 11 ((Fun 𝐺Disj 𝑦 ∈ ran 𝐺{𝑦}) → Disj 𝑦 ∈ ran 𝐺(𝐺 “ {𝑦}))
196193, 194, 195sylancl 692 . . . . . . . . . 10 (𝜑Disj 𝑦 ∈ ran 𝐺(𝐺 “ {𝑦}))
197184, 186, 191, 196disjxpin 28576 . . . . . . . . 9 (𝜑Disj 𝑝 ∈ (ran 𝐹 × ran 𝐺)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
198 disjss1 4553 . . . . . . . . 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)}))))
199118, 197, 198mpsyl 65 . . . . . . . 8 (𝜑Disj 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
200199adantr 479 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → Disj 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
201 measvuni 29397 . . . . . . 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)}))))
202179, 181, 182, 200, 201syl112anc 1321 . . . . . 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)}))))
203 ssfi 8042 . . . . . . . . 9 (((ran 𝐹 × ran 𝐺) ∈ Fin ∧ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
204117, 118, 203sylancl 692 . . . . . . . 8 (𝜑 → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
205204adantr 479 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
206 simpll 785 . . . . . . . 8 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
207 simpr 475 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)))
208118, 207sseldi 3565 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (ran 𝐹 × ran 𝐺))
209 xp1st 7066 . . . . . . . . 9 (𝑝 ∈ (ran 𝐹 × ran 𝐺) → (1st𝑝) ∈ ran 𝐹)
210208, 209syl 17 . . . . . . . 8 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (1st𝑝) ∈ ran 𝐹)
211 xp2nd 7067 . . . . . . . . 9 (𝑝 ∈ (ran 𝐹 × ran 𝐺) → (2nd𝑝) ∈ ran 𝐺)
212208, 211syl 17 . . . . . . . 8 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (2nd𝑝) ∈ ran 𝐺)
213 oveq12 6535 . . . . . . . . . . . . . . . 16 ((𝑥 = 0𝑦 = 0 ) → (𝑥 + 𝑦) = ( 0 + 0 ))
214 sibfof.5 . . . . . . . . . . . . . . . 16 (𝜑 → ( 0 + 0 ) = (0g𝐾))
215213, 214sylan9eqr 2665 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 = 0𝑦 = 0 )) → (𝑥 + 𝑦) = (0g𝐾))
216215ex 448 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥 = 0𝑦 = 0 ) → (𝑥 + 𝑦) = (0g𝐾)))
217216necon3ad 2794 . . . . . . . . . . . . 13 (𝜑 → ((𝑥 + 𝑦) ≠ (0g𝐾) → ¬ (𝑥 = 0𝑦 = 0 )))
218 neorian 2875 . . . . . . . . . . . . 13 ((𝑥0𝑦0 ) ↔ ¬ (𝑥 = 0𝑦 = 0 ))
219217, 218syl6ibr 240 . . . . . . . . . . . 12 (𝜑 → ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
220219adantr 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
221220ralrimivva 2953 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
222206, 221syl 17 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
22369a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ ( + “ {𝑧}))
224223sselda 3567 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ( + “ {𝑧}))
225 fniniseg 6230 . . . . . . . . . . . . 13 ( + Fn (𝐵 × 𝐵) → (𝑝 ∈ ( + “ {𝑧}) ↔ (𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧)))
226206, 63, 2253syl 18 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑝 ∈ ( + “ {𝑧}) ↔ (𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧)))
227224, 226mpbid 220 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧))
228 simpr 475 . . . . . . . . . . . 12 ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧) → ( +𝑝) = 𝑧)
229 1st2nd2 7073 . . . . . . . . . . . . . . 15 (𝑝 ∈ (𝐵 × 𝐵) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
230229fveq2d 6091 . . . . . . . . . . . . . 14 (𝑝 ∈ (𝐵 × 𝐵) → ( +𝑝) = ( + ‘⟨(1st𝑝), (2nd𝑝)⟩))
231 df-ov 6529 . . . . . . . . . . . . . 14 ((1st𝑝) + (2nd𝑝)) = ( + ‘⟨(1st𝑝), (2nd𝑝)⟩)
232230, 231syl6eqr 2661 . . . . . . . . . . . . 13 (𝑝 ∈ (𝐵 × 𝐵) → ( +𝑝) = ((1st𝑝) + (2nd𝑝)))
233232adantr 479 . . . . . . . . . . . 12 ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧) → ( +𝑝) = ((1st𝑝) + (2nd𝑝)))
234228, 233eqtr3d 2645 . . . . . . . . . . 11 ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧) → 𝑧 = ((1st𝑝) + (2nd𝑝)))
235227, 234syl 17 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 = ((1st𝑝) + (2nd𝑝)))
236 simplr 787 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)}))
237236eldifbd 3552 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ¬ 𝑧 ∈ {(0g𝐾)})
238 velsn 4140 . . . . . . . . . . . 12 (𝑧 ∈ {(0g𝐾)} ↔ 𝑧 = (0g𝐾))
239238necon3bbii 2828 . . . . . . . . . . 11 𝑧 ∈ {(0g𝐾)} ↔ 𝑧 ≠ (0g𝐾))
240237, 239sylib 206 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ≠ (0g𝐾))
241235, 240eqnetrrd 2849 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾))
242180, 76sylanl2 680 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐵))
243242, 89syl 17 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (1st𝑝) ∈ 𝐵)
244242, 102syl 17 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (2nd𝑝) ∈ 𝐵)
245 oveq1 6533 . . . . . . . . . . . . 13 (𝑥 = (1st𝑝) → (𝑥 + 𝑦) = ((1st𝑝) + 𝑦))
246245neeq1d 2840 . . . . . . . . . . . 12 (𝑥 = (1st𝑝) → ((𝑥 + 𝑦) ≠ (0g𝐾) ↔ ((1st𝑝) + 𝑦) ≠ (0g𝐾)))
247 neeq1 2843 . . . . . . . . . . . . 13 (𝑥 = (1st𝑝) → (𝑥0 ↔ (1st𝑝) ≠ 0 ))
248247orbi1d 734 . . . . . . . . . . . 12 (𝑥 = (1st𝑝) → ((𝑥0𝑦0 ) ↔ ((1st𝑝) ≠ 0𝑦0 )))
249246, 248imbi12d 332 . . . . . . . . . . 11 (𝑥 = (1st𝑝) → (((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )) ↔ (((1st𝑝) + 𝑦) ≠ (0g𝐾) → ((1st𝑝) ≠ 0𝑦0 ))))
250 oveq2 6534 . . . . . . . . . . . . 13 (𝑦 = (2nd𝑝) → ((1st𝑝) + 𝑦) = ((1st𝑝) + (2nd𝑝)))
251250neeq1d 2840 . . . . . . . . . . . 12 (𝑦 = (2nd𝑝) → (((1st𝑝) + 𝑦) ≠ (0g𝐾) ↔ ((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾)))
252 neeq1 2843 . . . . . . . . . . . . 13 (𝑦 = (2nd𝑝) → (𝑦0 ↔ (2nd𝑝) ≠ 0 ))
253252orbi2d 733 . . . . . . . . . . . 12 (𝑦 = (2nd𝑝) → (((1st𝑝) ≠ 0𝑦0 ) ↔ ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 )))
254251, 253imbi12d 332 . . . . . . . . . . 11 (𝑦 = (2nd𝑝) → ((((1st𝑝) + 𝑦) ≠ (0g𝐾) → ((1st𝑝) ≠ 0𝑦0 )) ↔ (((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))))
255249, 254rspc2v 3292 . . . . . . . . . 10 (((1st𝑝) ∈ 𝐵 ∧ (2nd𝑝) ∈ 𝐵) → (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )) → (((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))))
256243, 244, 255syl2anc 690 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )) → (((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))))
257222, 241, 256mp2d 46 . . . . . . . 8 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))
2583, 4, 11, 12, 13, 14, 15, 16, 17, 19, 2, 78sibfinima 29521 . . . . . . . 8 (((𝜑 ∧ (1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) ∧ ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 )) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ (0[,)+∞))
259206, 210, 212, 257, 258syl31anc 1320 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ (0[,)+∞))
260205, 259esumpfinval 29257 . . . . . 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)}))))
261176, 202, 2603eqtrd 2647 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) = Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
262 rge0ssre 12109 . . . . . . 7 (0[,)+∞) ⊆ ℝ
263262, 259sseldi 3565 . . . . . 6 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ ℝ)
264205, 263fsumrecl 14260 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ ℝ)
265261, 264eqeltrd 2687 . . . 4 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ ℝ)
266179adantr 479 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑀 ∈ (measures‘dom 𝑀))
267180, 112sylanl2 680 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
268 measge0 29390 . . . . . . 7 ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀) → 0 ≤ (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
269266, 267, 268syl2anc 690 . . . . . 6 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 0 ≤ (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
270205, 263, 269fsumge0 14316 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → 0 ≤ Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
271270, 261breqtrrd 4605 . . . 4 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → 0 ≤ (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})))
272 elrege0 12107 . . . 4 ((𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞) ↔ ((𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ ℝ ∧ 0 ≤ (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧}))))
273265, 271, 272sylanbrc 694 . . 3 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
274273ralrimiva 2948 . 2 (𝜑 → ∀𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})(𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
275 eqid 2609 . . 3 (sigaGen‘(TopOpen‘𝐾)) = (sigaGen‘(TopOpen‘𝐾))
276 eqid 2609 . . 3 (0g𝐾) = (0g𝐾)
277 eqid 2609 . . 3 ( ·𝑠𝐾) = ( ·𝑠𝐾)
278 eqid 2609 . . 3 (ℝHom‘(Scalar‘𝐾)) = (ℝHom‘(Scalar‘𝐾))
27927, 28, 275, 276, 277, 278, 26, 16issibf 29515 . 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[,)+∞))))
280174, 142, 274, 279mpbir3and 1237 1 (𝜑 → (𝐹𝑓 + 𝐺) ∈ dom (𝐾sitg𝑀))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  Vcvv 3172  cdif 3536  cun 3537  cin 3538  wss 3539  c0 3873  {csn 4124  cop 4130   cuni 4366   ciun 4449  Disj wdisj 4547   class class class wbr 4577   × cxp 5025  ccnv 5026  dom cdm 5027  ran crn 5028  cima 5030  Fun wfun 5783   Fn wfn 5784  wf 5785  cfv 5789  (class class class)co 6526  𝑓 cof 6770  ωcom 6934  1st c1st 7034  2nd c2nd 7035  𝑚 cmap 7721  cdom 7816  csdm 7817  Fincfn 7818  cr 9791  0cc0 9792  +∞cpnf 9927  cle 9931  [,)cico 12006  Σcsu 14212  Basecbs 15643  Scalarcsca 15719   ·𝑠 cvsca 15720  TopOpenctopn 15853  0gc0g 15871  Topctop 20464  TopSpctps 20466  Clsdccld 20577  Frect1 20868  ℝHomcrrh 29158  Σ*cesum 29209  sigAlgebracsiga 29290  sigaGencsigagen 29321  measurescmeas 29378  MblFnMcmbfm 29432  sitgcsitg 29511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-inf2 8398  ax-ac2 9145  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870  ax-addf 9871  ax-mulf 9872
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-disj 4548  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-se 4987  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-isom 5798  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-fi 8177  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-acn 8628  df-ac 8799  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10536  df-nn 10870  df-2 10928  df-3 10929  df-4 10930  df-5 10931  df-6 10932  df-7 10933  df-8 10934  df-9 10935  df-n0 11142  df-z 11213  df-dec 11328  df-uz 11522  df-q 11623  df-rp 11667  df-xneg 11780  df-xadd 11781  df-xmul 11782  df-ioo 12008  df-ioc 12009  df-ico 12010  df-icc 12011  df-fz 12155  df-fzo 12292  df-fl 12412  df-mod 12488  df-seq 12621  df-exp 12680  df-fac 12880  df-bc 12909  df-hash 12937  df-shft 13603  df-cj 13635  df-re 13636  df-im 13637  df-sqrt 13771  df-abs 13772  df-limsup 13998  df-clim 14015  df-rlim 14016  df-sum 14213  df-ef 14585  df-sin 14587  df-cos 14588  df-pi 14590  df-struct 15645  df-ndx 15646  df-slot 15647  df-base 15648  df-sets 15649  df-ress 15650  df-plusg 15729  df-mulr 15730  df-starv 15731  df-sca 15732  df-vsca 15733  df-ip 15734  df-tset 15735  df-ple 15736  df-ds 15739  df-unif 15740  df-hom 15741  df-cco 15742  df-rest 15854  df-topn 15855  df-0g 15873  df-gsum 15874  df-topgen 15875  df-pt 15876  df-prds 15879  df-ordt 15932  df-xrs 15933  df-qtop 15938  df-imas 15939  df-xps 15941  df-mre 16017  df-mrc 16018  df-acs 16020  df-ps 16971  df-tsr 16972  df-plusf 17012  df-mgm 17013  df-sgrp 17055  df-mnd 17066  df-mhm 17106  df-submnd 17107  df-grp 17196  df-minusg 17197  df-sbg 17198  df-mulg 17312  df-subg 17362  df-cntz 17521  df-cmn 17966  df-abl 17967  df-mgp 18261  df-ur 18273  df-ring 18320  df-cring 18321  df-subrg 18549  df-abv 18588  df-lmod 18636  df-scaf 18637  df-sra 18941  df-rgmod 18942  df-psmet 19507  df-xmet 19508  df-met 19509  df-bl 19510  df-mopn 19511  df-fbas 19512  df-fg 19513  df-cnfld 19516  df-top 20468  df-bases 20469  df-topon 20470  df-topsp 20471  df-cld 20580  df-ntr 20581  df-cls 20582  df-nei 20659  df-lp 20697  df-perf 20698  df-cn 20788  df-cnp 20789  df-t1 20875  df-haus 20876  df-tx 21122  df-hmeo 21315  df-fil 21407  df-fm 21499  df-flim 21500  df-flf 21501  df-tmd 21633  df-tgp 21634  df-tsms 21687  df-trg 21720  df-xms 21882  df-ms 21883  df-tms 21884  df-nm 22144  df-ngp 22145  df-nrg 22147  df-nlm 22148  df-ii 22435  df-cncf 22436  df-limc 23380  df-dv 23381  df-log 24051  df-esum 29210  df-siga 29291  df-sigagen 29322  df-meas 29379  df-mbfm 29433  df-sitg 29512
This theorem is referenced by:  sitmcl  29533
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