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

Theorem sibfof 29729
 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 20553 . . . . . . . . . . 11 (𝑊 ∈ TopSp → 𝐵 = 𝐽)
62, 5syl 17 . . . . . . . . . 10 (𝜑𝐵 = 𝐽)
76sqxpeqd 5065 . . . . . . . . 9 (𝜑 → (𝐵 × 𝐵) = ( 𝐽 × 𝐽))
87feq2d 5944 . . . . . . . 8 (𝜑 → ( + :(𝐵 × 𝐵)⟶𝐶+ :( 𝐽 × 𝐽)⟶𝐶))
91, 8mpbid 221 . . . . . . 7 (𝜑+ :( 𝐽 × 𝐽)⟶𝐶)
109fovrnda 6703 . . . . . 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 29725 . . . . . 6 (𝜑𝐹: dom 𝑀 𝐽)
19 sibfof.2 . . . . . . 7 (𝜑𝐺 ∈ dom (𝑊sitg𝑀))
203, 4, 11, 12, 13, 14, 15, 16, 19sibff 29725 . . . . . 6 (𝜑𝐺: dom 𝑀 𝐽)
21 dmexg 6989 . . . . . . 7 (𝑀 ran measures → dom 𝑀 ∈ V)
22 uniexg 6853 . . . . . . 7 (dom 𝑀 ∈ V → dom 𝑀 ∈ V)
2316, 21, 223syl 18 . . . . . 6 (𝜑 dom 𝑀 ∈ V)
24 inidm 3784 . . . . . 6 ( dom 𝑀 dom 𝑀) = dom 𝑀
2510, 18, 20, 23, 23, 24off 6810 . . . . 5 (𝜑 → (𝐹𝑓 + 𝐺): dom 𝑀𝐶)
26 sibfof.3 . . . . . . . 8 (𝜑𝐾 ∈ TopSp)
27 sibfof.c . . . . . . . . 9 𝐶 = (Base‘𝐾)
28 eqid 2610 . . . . . . . . 9 (TopOpen‘𝐾) = (TopOpen‘𝐾)
2927, 28tpsuni 20553 . . . . . . . 8 (𝐾 ∈ TopSp → 𝐶 = (TopOpen‘𝐾))
3026, 29syl 17 . . . . . . 7 (𝜑𝐶 = (TopOpen‘𝐾))
31 fvex 6113 . . . . . . . 8 (TopOpen‘𝐾) ∈ V
32 unisg 29533 . . . . . . . 8 ((TopOpen‘𝐾) ∈ V → (sigaGen‘(TopOpen‘𝐾)) = (TopOpen‘𝐾))
3331, 32ax-mp 5 . . . . . . 7 (sigaGen‘(TopOpen‘𝐾)) = (TopOpen‘𝐾)
3430, 33syl6eqr 2662 . . . . . 6 (𝜑𝐶 = (sigaGen‘(TopOpen‘𝐾)))
3534feq3d 5945 . . . . 5 (𝜑 → ((𝐹𝑓 + 𝐺): dom 𝑀𝐶 ↔ (𝐹𝑓 + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾))))
3625, 35mpbid 221 . . . 4 (𝜑 → (𝐹𝑓 + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾)))
3731a1i 11 . . . . . . 7 (𝜑 → (TopOpen‘𝐾) ∈ V)
3837sgsiga 29532 . . . . . 6 (𝜑 → (sigaGen‘(TopOpen‘𝐾)) ∈ ran sigAlgebra)
39 uniexg 6853 . . . . . 6 ((sigaGen‘(TopOpen‘𝐾)) ∈ ran sigAlgebra → (sigaGen‘(TopOpen‘𝐾)) ∈ V)
4038, 39syl 17 . . . . 5 (𝜑 (sigaGen‘(TopOpen‘𝐾)) ∈ V)
4140, 23elmapd 7758 . . . 4 (𝜑 → ((𝐹𝑓 + 𝐺) ∈ ( (sigaGen‘(TopOpen‘𝐾)) ↑𝑚 dom 𝑀) ↔ (𝐹𝑓 + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾))))
4236, 41mpbird 246 . . 3 (𝜑 → (𝐹𝑓 + 𝐺) ∈ ( (sigaGen‘(TopOpen‘𝐾)) ↑𝑚 dom 𝑀))
43 inundif 3998 . . . . . . 7 ((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ∪ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) = 𝑏
4443imaeq2i 5383 . . . . . 6 ((𝐹𝑓 + 𝐺) “ ((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ∪ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))) = ((𝐹𝑓 + 𝐺) “ 𝑏)
45 ffun 5961 . . . . . . . 8 ((𝐹𝑓 + 𝐺): dom 𝑀𝐶 → Fun (𝐹𝑓 + 𝐺))
46 unpreima 6249 . . . . . . . 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 480 . . . . . 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 2658 . . . . 5 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ 𝑏) = (((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∪ ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))))
50 dmmeas 29591 . . . . . . . 8 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
5116, 50syl 17 . . . . . . 7 (𝜑 → dom 𝑀 ran sigAlgebra)
5251adantr 480 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → dom 𝑀 ran sigAlgebra)
53 imaiun 6407 . . . . . . . 8 ((𝐹𝑓 + 𝐺) “ 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)){𝑧}) = 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧})
54 iunid 4511 . . . . . . . . 9 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)){𝑧} = (𝑏 ∩ ran (𝐹𝑓 + 𝐺))
5554imaeq2i 5383 . . . . . . . 8 ((𝐹𝑓 + 𝐺) “ 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)){𝑧}) = ((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)))
5653, 55eqtr3i 2634 . . . . . . 7 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) = ((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)))
57 inss2 3796 . . . . . . . . . 10 (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ⊆ ran (𝐹𝑓 + 𝐺)
586feq3d 5945 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹: dom 𝑀𝐵𝐹: dom 𝑀 𝐽))
5918, 58mpbird 246 . . . . . . . . . . . . . 14 (𝜑𝐹: dom 𝑀𝐵)
606feq3d 5945 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺: dom 𝑀𝐵𝐺: dom 𝑀 𝐽))
6120, 60mpbird 246 . . . . . . . . . . . . . 14 (𝜑𝐺: dom 𝑀𝐵)
62 ffn 5958 . . . . . . . . . . . . . . 15 ( + :(𝐵 × 𝐵)⟶𝐶+ Fn (𝐵 × 𝐵))
631, 62syl 17 . . . . . . . . . . . . . 14 (𝜑+ Fn (𝐵 × 𝐵))
6459, 61, 23, 63ofpreima2 28849 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝑓 + 𝐺) “ {𝑧}) = 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
6564adantr 480 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → ((𝐹𝑓 + 𝐺) “ {𝑧}) = 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
6651adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → dom 𝑀 ran sigAlgebra)
6751ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → dom 𝑀 ran sigAlgebra)
68 simpll 786 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
69 inss1 3795 . . . . . . . . . . . . . . . . . 18 (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ ( + “ {𝑧})
70 cnvimass 5404 . . . . . . . . . . . . . . . . . . . 20 ( + “ {𝑧}) ⊆ dom +
71 fdm 5964 . . . . . . . . . . . . . . . . . . . . 21 ( + :(𝐵 × 𝐵)⟶𝐶 → dom + = (𝐵 × 𝐵))
721, 71syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom + = (𝐵 × 𝐵))
7370, 72syl5sseq 3616 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ( + “ {𝑧}) ⊆ (𝐵 × 𝐵))
7473adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → ( + “ {𝑧}) ⊆ (𝐵 × 𝐵))
7569, 74syl5ss 3579 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐵))
7675sselda 3568 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐵))
7751adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → dom 𝑀 ran sigAlgebra)
78 sibfof.4 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ Fre)
7978sgsiga 29532 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
8011, 79syl5eqel 2692 . . . . . . . . . . . . . . . . . 18 (𝜑𝑆 ran sigAlgebra)
8180adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝑆 ran sigAlgebra)
823, 4, 11, 12, 13, 14, 15, 16, 17sibfmbl 29724 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
8382adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
844tpstop 20554 . . . . . . . . . . . . . . . . . . . . 21 (𝑊 ∈ TopSp → 𝐽 ∈ Top)
85 cldssbrsiga 29577 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
862, 84, 853syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
8786adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
8878adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐽 ∈ Fre)
89 xp1st 7089 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ (𝐵 × 𝐵) → (1st𝑝) ∈ 𝐵)
9089adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (1st𝑝) ∈ 𝐵)
916adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐵 = 𝐽)
9290, 91eleqtrd 2690 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (1st𝑝) ∈ 𝐽)
93 eqid 2610 . . . . . . . . . . . . . . . . . . . . 21 𝐽 = 𝐽
9493t1sncld 20940 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ Fre ∧ (1st𝑝) ∈ 𝐽) → {(1st𝑝)} ∈ (Clsd‘𝐽))
9588, 92, 94syl2anc 691 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ (Clsd‘𝐽))
9687, 95sseldd 3569 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ (sigaGen‘𝐽))
9796, 11syl6eleqr 2699 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ 𝑆)
9877, 81, 83, 97mbfmcnvima 29646 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀)
9968, 76, 98syl2anc 691 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀)
1003, 4, 11, 12, 13, 14, 15, 16, 19sibfmbl 29724 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 ∈ (dom 𝑀MblFnM𝑆))
101100adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
102 xp2nd 7090 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ (𝐵 × 𝐵) → (2nd𝑝) ∈ 𝐵)
103102adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (2nd𝑝) ∈ 𝐵)
104103, 91eleqtrd 2690 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (2nd𝑝) ∈ 𝐽)
10593t1sncld 20940 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ Fre ∧ (2nd𝑝) ∈ 𝐽) → {(2nd𝑝)} ∈ (Clsd‘𝐽))
10688, 104, 105syl2anc 691 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ (Clsd‘𝐽))
10787, 106sseldd 3569 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ (sigaGen‘𝐽))
108107, 11syl6eleqr 2699 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ 𝑆)
10977, 81, 101, 108mbfmcnvima 29646 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀)
11068, 76, 109syl2anc 691 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀)
111 inelsiga 29525 . . . . . . . . . . . . . . 15 ((dom 𝑀 ran sigAlgebra ∧ (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀 ∧ (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
11267, 99, 110, 111syl3anc 1318 . . . . . . . . . . . . . 14 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
113112ralrimiva 2949 . . . . . . . . . . . . 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 29726 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝐹 ∈ Fin)
1153, 4, 11, 12, 13, 14, 15, 16, 19sibfrn 29726 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝐺 ∈ Fin)
116 xpfi 8116 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
117114, 115, 116syl2anc 691 . . . . . . . . . . . . . . . 16 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
118 inss2 3796 . . . . . . . . . . . . . . . 16 (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)
119 ssdomg 7887 . . . . . . . . . . . . . . . 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 8432 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 × ran 𝐺) ∈ Fin ↔ (ran 𝐹 × ran 𝐺) ≺ ω)
122121biimpi 205 . . . . . . . . . . . . . . . 16 ((ran 𝐹 × ran 𝐺) ∈ Fin → (ran 𝐹 × ran 𝐺) ≺ ω)
123 sdomdom 7869 . . . . . . . . . . . . . . . 16 ((ran 𝐹 × ran 𝐺) ≺ ω → (ran 𝐹 × ran 𝐺) ≼ ω)
124117, 122, 1233syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (ran 𝐹 × ran 𝐺) ≼ ω)
125 domtr 7895 . . . . . . . . . . . . . . 15 (((( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺) ∧ (ran 𝐹 × ran 𝐺) ≼ ω) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
126120, 124, 125syl2anc 691 . . . . . . . . . . . . . 14 (𝜑 → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
127126adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
128 nfcv 2751 . . . . . . . . . . . . . 14 𝑝(( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))
129128sigaclcuni 29508 . . . . . . . . . . . . 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 1318 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
13165, 130eqeltrd 2688 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → ((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
132131ralrimiva 2949 . . . . . . . . . 10 (𝜑 → ∀𝑧 ∈ ran (𝐹𝑓 + 𝐺)((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
133 ssralv 3629 . . . . . . . . . 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 66 . . . . . . . . 9 (𝜑 → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
135134adantr 480 . . . . . . . 8 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
136 ffun 5961 . . . . . . . . . . . . . 14 ( + :(𝐵 × 𝐵)⟶𝐶 → Fun + )
1371, 136syl 17 . . . . . . . . . . . . 13 (𝜑 → Fun + )
138 imafi 8142 . . . . . . . . . . . . 13 ((Fun + ∧ (ran 𝐹 × ran 𝐺) ∈ Fin) → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
139137, 117, 138syl2anc 691 . . . . . . . . . . . 12 (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
14018, 20, 9, 23ofrn2 28822 . . . . . . . . . . . 12 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
141 ssfi 8065 . . . . . . . . . . . 12 ((( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin ∧ ran (𝐹𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺))) → ran (𝐹𝑓 + 𝐺) ∈ Fin)
142139, 140, 141syl2anc 691 . . . . . . . . . . 11 (𝜑 → ran (𝐹𝑓 + 𝐺) ∈ Fin)
143 ssdomg 7887 . . . . . . . . . . 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 8432 . . . . . . . . . . . 12 (ran (𝐹𝑓 + 𝐺) ∈ Fin ↔ ran (𝐹𝑓 + 𝐺) ≺ ω)
146142, 145sylib 207 . . . . . . . . . . 11 (𝜑 → ran (𝐹𝑓 + 𝐺) ≺ ω)
147 sdomdom 7869 . . . . . . . . . . 11 (ran (𝐹𝑓 + 𝐺) ≺ ω → ran (𝐹𝑓 + 𝐺) ≼ ω)
148146, 147syl 17 . . . . . . . . . 10 (𝜑 → ran (𝐹𝑓 + 𝐺) ≼ ω)
149 domtr 7895 . . . . . . . . . 10 (((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ran (𝐹𝑓 + 𝐺) ∧ ran (𝐹𝑓 + 𝐺) ≼ ω) → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ω)
150144, 148, 149syl2anc 691 . . . . . . . . 9 (𝜑 → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ω)
151150adantr 480 . . . . . . . 8 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ω)
152 nfcv 2751 . . . . . . . . 9 𝑧(𝑏 ∩ ran (𝐹𝑓 + 𝐺))
153152sigaclcuni 29508 . . . . . . . 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 1318 . . . . . . 7 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
15556, 154syl5eqelr 2693 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∈ dom 𝑀)
156 difpreima 6251 . . . . . . . . . 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 5405 . . . . . . . . . . 11 ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺)) = dom (𝐹𝑓 + 𝐺)
159158difeq2i 3687 . . . . . . . . . 10 (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺))) = (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ dom (𝐹𝑓 + 𝐺))
160 cnvimass 5404 . . . . . . . . . . 11 ((𝐹𝑓 + 𝐺) “ 𝑏) ⊆ dom (𝐹𝑓 + 𝐺)
161 ssdif0 3896 . . . . . . . . . . 11 (((𝐹𝑓 + 𝐺) “ 𝑏) ⊆ dom (𝐹𝑓 + 𝐺) ↔ (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ dom (𝐹𝑓 + 𝐺)) = ∅)
162160, 161mpbi 219 . . . . . . . . . 10 (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ dom (𝐹𝑓 + 𝐺)) = ∅
163159, 162eqtri 2632 . . . . . . . . 9 (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺))) = ∅
164157, 163syl6eq 2660 . . . . . . . 8 (𝜑 → ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) = ∅)
165 0elsiga 29504 . . . . . . . . 9 (dom 𝑀 ran sigAlgebra → ∅ ∈ dom 𝑀)
16616, 50, 1653syl 18 . . . . . . . 8 (𝜑 → ∅ ∈ dom 𝑀)
167164, 166eqeltrd 2688 . . . . . . 7 (𝜑 → ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) ∈ dom 𝑀)
168167adantr 480 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) ∈ dom 𝑀)
169 unelsiga 29524 . . . . . 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 1318 . . . . 5 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∪ ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))) ∈ dom 𝑀)
17149, 170eqeltrd 2688 . . . 4 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ 𝑏) ∈ dom 𝑀)
172171ralrimiva 2949 . . 3 (𝜑 → ∀𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))((𝐹𝑓 + 𝐺) “ 𝑏) ∈ dom 𝑀)
17351, 38ismbfm 29641 . . 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 959 . 2 (𝜑 → (𝐹𝑓 + 𝐺) ∈ (dom 𝑀MblFnM(sigaGen‘(TopOpen‘𝐾))))
17564adantr 480 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → ((𝐹𝑓 + 𝐺) “ {𝑧}) = 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
176175fveq2d 6107 . . . . . 6 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) = (𝑀 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
177 measbasedom 29592 . . . . . . . . 9 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
17816, 177sylib 207 . . . . . . . 8 (𝜑𝑀 ∈ (measures‘dom 𝑀))
179178adantr 480 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → 𝑀 ∈ (measures‘dom 𝑀))
180 eldifi 3694 . . . . . . . 8 (𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)}) → 𝑧 ∈ ran (𝐹𝑓 + 𝐺))
181180, 113sylan2 490 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → ∀𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
182126adantr 480 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
183 sneq 4135 . . . . . . . . . . 11 (𝑥 = (1st𝑝) → {𝑥} = {(1st𝑝)})
184183imaeq2d 5385 . . . . . . . . . 10 (𝑥 = (1st𝑝) → (𝐹 “ {𝑥}) = (𝐹 “ {(1st𝑝)}))
185 sneq 4135 . . . . . . . . . . 11 (𝑦 = (2nd𝑝) → {𝑦} = {(2nd𝑝)})
186185imaeq2d 5385 . . . . . . . . . 10 (𝑦 = (2nd𝑝) → (𝐺 “ {𝑦}) = (𝐺 “ {(2nd𝑝)}))
187 ffun 5961 . . . . . . . . . . . 12 (𝐹: dom 𝑀 𝐽 → Fun 𝐹)
18818, 187syl 17 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
189 sndisj 4574 . . . . . . . . . . 11 Disj 𝑥 ∈ ran 𝐹{𝑥}
190 disjpreima 28779 . . . . . . . . . . 11 ((Fun 𝐹Disj 𝑥 ∈ ran 𝐹{𝑥}) → Disj 𝑥 ∈ ran 𝐹(𝐹 “ {𝑥}))
191188, 189, 190sylancl 693 . . . . . . . . . 10 (𝜑Disj 𝑥 ∈ ran 𝐹(𝐹 “ {𝑥}))
192 ffun 5961 . . . . . . . . . . . 12 (𝐺: dom 𝑀 𝐽 → Fun 𝐺)
19320, 192syl 17 . . . . . . . . . . 11 (𝜑 → Fun 𝐺)
194 sndisj 4574 . . . . . . . . . . 11 Disj 𝑦 ∈ ran 𝐺{𝑦}
195 disjpreima 28779 . . . . . . . . . . 11 ((Fun 𝐺Disj 𝑦 ∈ ran 𝐺{𝑦}) → Disj 𝑦 ∈ ran 𝐺(𝐺 “ {𝑦}))
196193, 194, 195sylancl 693 . . . . . . . . . 10 (𝜑Disj 𝑦 ∈ ran 𝐺(𝐺 “ {𝑦}))
197184, 186, 191, 196disjxpin 28783 . . . . . . . . 9 (𝜑Disj 𝑝 ∈ (ran 𝐹 × ran 𝐺)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
198 disjss1 4559 . . . . . . . . 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 66 . . . . . . . 8 (𝜑Disj 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
200199adantr 480 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → Disj 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
201 measvuni 29604 . . . . . . 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 1322 . . . . . 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 8065 . . . . . . . . 9 (((ran 𝐹 × ran 𝐺) ∈ Fin ∧ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
204117, 118, 203sylancl 693 . . . . . . . 8 (𝜑 → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
205204adantr 480 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
206 simpll 786 . . . . . . . 8 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
207 simpr 476 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)))
208118, 207sseldi 3566 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (ran 𝐹 × ran 𝐺))
209 xp1st 7089 . . . . . . . . 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 7090 . . . . . . . . 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 6558 . . . . . . . . . . . . . . . 16 ((𝑥 = 0𝑦 = 0 ) → (𝑥 + 𝑦) = ( 0 + 0 ))
214 sibfof.5 . . . . . . . . . . . . . . . 16 (𝜑 → ( 0 + 0 ) = (0g𝐾))
215213, 214sylan9eqr 2666 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 = 0𝑦 = 0 )) → (𝑥 + 𝑦) = (0g𝐾))
216215ex 449 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥 = 0𝑦 = 0 ) → (𝑥 + 𝑦) = (0g𝐾)))
217216necon3ad 2795 . . . . . . . . . . . . 13 (𝜑 → ((𝑥 + 𝑦) ≠ (0g𝐾) → ¬ (𝑥 = 0𝑦 = 0 )))
218 neorian 2876 . . . . . . . . . . . . 13 ((𝑥0𝑦0 ) ↔ ¬ (𝑥 = 0𝑦 = 0 ))
219217, 218syl6ibr 241 . . . . . . . . . . . 12 (𝜑 → ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
220219adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
221220ralrimivva 2954 . . . . . . . . . 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 3568 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ( + “ {𝑧}))
225 fniniseg 6246 . . . . . . . . . . . . 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 221 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧))
228 simpr 476 . . . . . . . . . . . 12 ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧) → ( +𝑝) = 𝑧)
229 1st2nd2 7096 . . . . . . . . . . . . . . 15 (𝑝 ∈ (𝐵 × 𝐵) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
230229fveq2d 6107 . . . . . . . . . . . . . 14 (𝑝 ∈ (𝐵 × 𝐵) → ( +𝑝) = ( + ‘⟨(1st𝑝), (2nd𝑝)⟩))
231 df-ov 6552 . . . . . . . . . . . . . 14 ((1st𝑝) + (2nd𝑝)) = ( + ‘⟨(1st𝑝), (2nd𝑝)⟩)
232230, 231syl6eqr 2662 . . . . . . . . . . . . 13 (𝑝 ∈ (𝐵 × 𝐵) → ( +𝑝) = ((1st𝑝) + (2nd𝑝)))
233232adantr 480 . . . . . . . . . . . 12 ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧) → ( +𝑝) = ((1st𝑝) + (2nd𝑝)))
234228, 233eqtr3d 2646 . . . . . . . . . . 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 788 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)}))
237236eldifbd 3553 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ¬ 𝑧 ∈ {(0g𝐾)})
238 velsn 4141 . . . . . . . . . . . 12 (𝑧 ∈ {(0g𝐾)} ↔ 𝑧 = (0g𝐾))
239238necon3bbii 2829 . . . . . . . . . . 11 𝑧 ∈ {(0g𝐾)} ↔ 𝑧 ≠ (0g𝐾))
240237, 239sylib 207 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ≠ (0g𝐾))
241235, 240eqnetrrd 2850 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾))
242180, 76sylanl2 681 . . . . . . . . . . 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 6556 . . . . . . . . . . . . 13 (𝑥 = (1st𝑝) → (𝑥 + 𝑦) = ((1st𝑝) + 𝑦))
246245neeq1d 2841 . . . . . . . . . . . 12 (𝑥 = (1st𝑝) → ((𝑥 + 𝑦) ≠ (0g𝐾) ↔ ((1st𝑝) + 𝑦) ≠ (0g𝐾)))
247 neeq1 2844 . . . . . . . . . . . . 13 (𝑥 = (1st𝑝) → (𝑥0 ↔ (1st𝑝) ≠ 0 ))
248247orbi1d 735 . . . . . . . . . . . 12 (𝑥 = (1st𝑝) → ((𝑥0𝑦0 ) ↔ ((1st𝑝) ≠ 0𝑦0 )))
249246, 248imbi12d 333 . . . . . . . . . . 11 (𝑥 = (1st𝑝) → (((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )) ↔ (((1st𝑝) + 𝑦) ≠ (0g𝐾) → ((1st𝑝) ≠ 0𝑦0 ))))
250 oveq2 6557 . . . . . . . . . . . . 13 (𝑦 = (2nd𝑝) → ((1st𝑝) + 𝑦) = ((1st𝑝) + (2nd𝑝)))
251250neeq1d 2841 . . . . . . . . . . . 12 (𝑦 = (2nd𝑝) → (((1st𝑝) + 𝑦) ≠ (0g𝐾) ↔ ((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾)))
252 neeq1 2844 . . . . . . . . . . . . 13 (𝑦 = (2nd𝑝) → (𝑦0 ↔ (2nd𝑝) ≠ 0 ))
253252orbi2d 734 . . . . . . . . . . . 12 (𝑦 = (2nd𝑝) → (((1st𝑝) ≠ 0𝑦0 ) ↔ ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 )))
254251, 253imbi12d 333 . . . . . . . . . . 11 (𝑦 = (2nd𝑝) → ((((1st𝑝) + 𝑦) ≠ (0g𝐾) → ((1st𝑝) ≠ 0𝑦0 )) ↔ (((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))))
255249, 254rspc2v 3293 . . . . . . . . . 10 (((1st𝑝) ∈ 𝐵 ∧ (2nd𝑝) ∈ 𝐵) → (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )) → (((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))))
256243, 244, 255syl2anc 691 . . . . . . . . 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 47 . . . . . . . 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 29728 . . . . . . . 8 (((𝜑 ∧ (1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) ∧ ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 )) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ (0[,)+∞))
259206, 210, 212, 257, 258syl31anc 1321 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ (0[,)+∞))
260205, 259esumpfinval 29464 . . . . . 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 2648 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) = Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
262 rge0ssre 12151 . . . . . . 7 (0[,)+∞) ⊆ ℝ
263262, 259sseldi 3566 . . . . . 6 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ ℝ)
264205, 263fsumrecl 14312 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ ℝ)
265261, 264eqeltrd 2688 . . . 4 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ ℝ)
266179adantr 480 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑀 ∈ (measures‘dom 𝑀))
267180, 112sylanl2 681 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
268 measge0 29597 . . . . . . 7 ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀) → 0 ≤ (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
269266, 267, 268syl2anc 691 . . . . . 6 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 0 ≤ (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
270205, 263, 269fsumge0 14368 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → 0 ≤ Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
271270, 261breqtrrd 4611 . . . 4 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → 0 ≤ (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})))
272 elrege0 12149 . . . 4 ((𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞) ↔ ((𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ ℝ ∧ 0 ≤ (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧}))))
273265, 271, 272sylanbrc 695 . . 3 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
274273ralrimiva 2949 . 2 (𝜑 → ∀𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})(𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
275 eqid 2610 . . 3 (sigaGen‘(TopOpen‘𝐾)) = (sigaGen‘(TopOpen‘𝐾))
276 eqid 2610 . . 3 (0g𝐾) = (0g𝐾)
277 eqid 2610 . . 3 ( ·𝑠𝐾) = ( ·𝑠𝐾)
278 eqid 2610 . . 3 (ℝHom‘(Scalar‘𝐾)) = (ℝHom‘(Scalar‘𝐾))
27927, 28, 275, 276, 277, 278, 26, 16issibf 29722 . 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 1238 1 (𝜑 → (𝐹𝑓 + 𝐺) ∈ dom (𝐾sitg𝑀))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455  Disj wdisj 4553   class class class wbr 4583   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793  ωcom 6957  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744   ≼ cdom 7839   ≺ csdm 7840  Fincfn 7841  ℝcr 9814  0cc0 9815  +∞cpnf 9950   ≤ cle 9954  [,)cico 12048  Σcsu 14264  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  TopOpenctopn 15905  0gc0g 15923  Topctop 20517  TopSpctps 20519  Clsdccld 20630  Frect1 20921  ℝHomcrrh 29365  Σ*cesum 29416  sigAlgebracsiga 29497  sigaGencsigagen 29528  measurescmeas 29585  MblFnMcmbfm 29639  sitgcsitg 29718 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-ac 8822  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-pi 14642  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-ordt 15984  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-ps 17023  df-tsr 17024  df-plusf 17064  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-subrg 18601  df-abv 18640  df-lmod 18688  df-scaf 18689  df-sra 18993  df-rgmod 18994  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-t1 20928  df-haus 20929  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-tmd 21686  df-tgp 21687  df-tsms 21740  df-trg 21773  df-xms 21935  df-ms 21936  df-tms 21937  df-nm 22197  df-ngp 22198  df-nrg 22200  df-nlm 22201  df-ii 22488  df-cncf 22489  df-limc 23436  df-dv 23437  df-log 24107  df-esum 29417  df-siga 29498  df-sigagen 29529  df-meas 29586  df-mbfm 29640  df-sitg 29719 This theorem is referenced by:  sitmcl  29740
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