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Theorem sibfof 30530
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 20788 . . . . . . . . . . 11 (𝑊 ∈ TopSp → 𝐵 = 𝐽)
62, 5syl 17 . . . . . . . . . 10 (𝜑𝐵 = 𝐽)
76sqxpeqd 5175 . . . . . . . . 9 (𝜑 → (𝐵 × 𝐵) = ( 𝐽 × 𝐽))
87feq2d 6069 . . . . . . . 8 (𝜑 → ( + :(𝐵 × 𝐵)⟶𝐶+ :( 𝐽 × 𝐽)⟶𝐶))
91, 8mpbid 222 . . . . . . 7 (𝜑+ :( 𝐽 × 𝐽)⟶𝐶)
109fovrnda 6847 . . . . . 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 30526 . . . . . 6 (𝜑𝐹: dom 𝑀 𝐽)
19 sibfof.2 . . . . . . 7 (𝜑𝐺 ∈ dom (𝑊sitg𝑀))
203, 4, 11, 12, 13, 14, 15, 16, 19sibff 30526 . . . . . 6 (𝜑𝐺: dom 𝑀 𝐽)
21 dmexg 7139 . . . . . . 7 (𝑀 ran measures → dom 𝑀 ∈ V)
22 uniexg 6997 . . . . . . 7 (dom 𝑀 ∈ V → dom 𝑀 ∈ V)
2316, 21, 223syl 18 . . . . . 6 (𝜑 dom 𝑀 ∈ V)
24 inidm 3855 . . . . . 6 ( dom 𝑀 dom 𝑀) = dom 𝑀
2510, 18, 20, 23, 23, 24off 6954 . . . . 5 (𝜑 → (𝐹𝑓 + 𝐺): dom 𝑀𝐶)
26 sibfof.3 . . . . . . . 8 (𝜑𝐾 ∈ TopSp)
27 sibfof.c . . . . . . . . 9 𝐶 = (Base‘𝐾)
28 eqid 2651 . . . . . . . . 9 (TopOpen‘𝐾) = (TopOpen‘𝐾)
2927, 28tpsuni 20788 . . . . . . . 8 (𝐾 ∈ TopSp → 𝐶 = (TopOpen‘𝐾))
3026, 29syl 17 . . . . . . 7 (𝜑𝐶 = (TopOpen‘𝐾))
31 fvex 6239 . . . . . . . 8 (TopOpen‘𝐾) ∈ V
32 unisg 30334 . . . . . . . 8 ((TopOpen‘𝐾) ∈ V → (sigaGen‘(TopOpen‘𝐾)) = (TopOpen‘𝐾))
3331, 32ax-mp 5 . . . . . . 7 (sigaGen‘(TopOpen‘𝐾)) = (TopOpen‘𝐾)
3430, 33syl6eqr 2703 . . . . . 6 (𝜑𝐶 = (sigaGen‘(TopOpen‘𝐾)))
3534feq3d 6070 . . . . 5 (𝜑 → ((𝐹𝑓 + 𝐺): dom 𝑀𝐶 ↔ (𝐹𝑓 + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾))))
3625, 35mpbid 222 . . . 4 (𝜑 → (𝐹𝑓 + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾)))
3731a1i 11 . . . . . . 7 (𝜑 → (TopOpen‘𝐾) ∈ V)
3837sgsiga 30333 . . . . . 6 (𝜑 → (sigaGen‘(TopOpen‘𝐾)) ∈ ran sigAlgebra)
39 uniexg 6997 . . . . . 6 ((sigaGen‘(TopOpen‘𝐾)) ∈ ran sigAlgebra → (sigaGen‘(TopOpen‘𝐾)) ∈ V)
4038, 39syl 17 . . . . 5 (𝜑 (sigaGen‘(TopOpen‘𝐾)) ∈ V)
4140, 23elmapd 7913 . . . 4 (𝜑 → ((𝐹𝑓 + 𝐺) ∈ ( (sigaGen‘(TopOpen‘𝐾)) ↑𝑚 dom 𝑀) ↔ (𝐹𝑓 + 𝐺): dom 𝑀 (sigaGen‘(TopOpen‘𝐾))))
4236, 41mpbird 247 . . 3 (𝜑 → (𝐹𝑓 + 𝐺) ∈ ( (sigaGen‘(TopOpen‘𝐾)) ↑𝑚 dom 𝑀))
43 inundif 4079 . . . . . . 7 ((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ∪ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) = 𝑏
4443imaeq2i 5499 . . . . . 6 ((𝐹𝑓 + 𝐺) “ ((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ∪ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))) = ((𝐹𝑓 + 𝐺) “ 𝑏)
45 ffun 6086 . . . . . . . 8 ((𝐹𝑓 + 𝐺): dom 𝑀𝐶 → Fun (𝐹𝑓 + 𝐺))
46 unpreima 6381 . . . . . . . 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 2699 . . . . 5 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ 𝑏) = (((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∪ ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))))
50 dmmeas 30392 . . . . . . . 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 6543 . . . . . . . 8 ((𝐹𝑓 + 𝐺) “ 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)){𝑧}) = 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧})
54 iunid 4607 . . . . . . . . 9 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)){𝑧} = (𝑏 ∩ ran (𝐹𝑓 + 𝐺))
5554imaeq2i 5499 . . . . . . . 8 ((𝐹𝑓 + 𝐺) “ 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)){𝑧}) = ((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)))
5653, 55eqtr3i 2675 . . . . . . 7 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) = ((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺)))
57 inss2 3867 . . . . . . . . . 10 (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ⊆ ran (𝐹𝑓 + 𝐺)
586feq3d 6070 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹: dom 𝑀𝐵𝐹: dom 𝑀 𝐽))
5918, 58mpbird 247 . . . . . . . . . . . . . 14 (𝜑𝐹: dom 𝑀𝐵)
606feq3d 6070 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺: dom 𝑀𝐵𝐺: dom 𝑀 𝐽))
6120, 60mpbird 247 . . . . . . . . . . . . . 14 (𝜑𝐺: dom 𝑀𝐵)
62 ffn 6083 . . . . . . . . . . . . . . 15 ( + :(𝐵 × 𝐵)⟶𝐶+ Fn (𝐵 × 𝐵))
631, 62syl 17 . . . . . . . . . . . . . 14 (𝜑+ Fn (𝐵 × 𝐵))
6459, 61, 23, 63ofpreima2 29594 . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → dom 𝑀 ran sigAlgebra)
68 simpll 805 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
69 inss1 3866 . . . . . . . . . . . . . . . . . 18 (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ ( + “ {𝑧})
70 cnvimass 5520 . . . . . . . . . . . . . . . . . . . 20 ( + “ {𝑧}) ⊆ dom +
71 fdm 6089 . . . . . . . . . . . . . . . . . . . . 21 ( + :(𝐵 × 𝐵)⟶𝐶 → dom + = (𝐵 × 𝐵))
721, 71syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom + = (𝐵 × 𝐵))
7370, 72syl5sseq 3686 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ( + “ {𝑧}) ⊆ (𝐵 × 𝐵))
7473adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → ( + “ {𝑧}) ⊆ (𝐵 × 𝐵))
7569, 74syl5ss 3647 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐵))
7675sselda 3636 . . . . . . . . . . . . . . . 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 30333 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
8011, 79syl5eqel 2734 . . . . . . . . . . . . . . . . . 18 (𝜑𝑆 ran sigAlgebra)
8180adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝑆 ran sigAlgebra)
823, 4, 11, 12, 13, 14, 15, 16, 17sibfmbl 30525 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
8382adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
844tpstop 20789 . . . . . . . . . . . . . . . . . . . . 21 (𝑊 ∈ TopSp → 𝐽 ∈ Top)
85 cldssbrsiga 30378 . . . . . . . . . . . . . . . . . . . . 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 7242 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ (𝐵 × 𝐵) → (1st𝑝) ∈ 𝐵)
9089adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (1st𝑝) ∈ 𝐵)
916adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐵 = 𝐽)
9290, 91eleqtrd 2732 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (1st𝑝) ∈ 𝐽)
93 eqid 2651 . . . . . . . . . . . . . . . . . . . . 21 𝐽 = 𝐽
9493t1sncld 21178 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ Fre ∧ (1st𝑝) ∈ 𝐽) → {(1st𝑝)} ∈ (Clsd‘𝐽))
9588, 92, 94syl2anc 694 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ (Clsd‘𝐽))
9687, 95sseldd 3637 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ (sigaGen‘𝐽))
9796, 11syl6eleqr 2741 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(1st𝑝)} ∈ 𝑆)
9877, 81, 83, 97mbfmcnvima 30447 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀)
9968, 76, 98syl2anc 694 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀)
1003, 4, 11, 12, 13, 14, 15, 16, 19sibfmbl 30525 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 ∈ (dom 𝑀MblFnM𝑆))
101100adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
102 xp2nd 7243 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ (𝐵 × 𝐵) → (2nd𝑝) ∈ 𝐵)
103102adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (2nd𝑝) ∈ 𝐵)
104103, 91eleqtrd 2732 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (2nd𝑝) ∈ 𝐽)
10593t1sncld 21178 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ Fre ∧ (2nd𝑝) ∈ 𝐽) → {(2nd𝑝)} ∈ (Clsd‘𝐽))
10688, 104, 105syl2anc 694 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ (Clsd‘𝐽))
10787, 106sseldd 3637 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ (sigaGen‘𝐽))
108107, 11syl6eleqr 2741 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → {(2nd𝑝)} ∈ 𝑆)
10977, 81, 101, 108mbfmcnvima 30447 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (𝐵 × 𝐵)) → (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀)
11068, 76, 109syl2anc 694 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀)
111 inelsiga 30326 . . . . . . . . . . . . . . 15 ((dom 𝑀 ran sigAlgebra ∧ (𝐹 “ {(1st𝑝)}) ∈ dom 𝑀 ∧ (𝐺 “ {(2nd𝑝)}) ∈ dom 𝑀) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
11267, 99, 110, 111syl3anc 1366 . . . . . . . . . . . . . 14 (((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
113112ralrimiva 2995 . . . . . . . . . . . . 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 30527 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝐹 ∈ Fin)
1153, 4, 11, 12, 13, 14, 15, 16, 19sibfrn 30527 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝐺 ∈ Fin)
116 xpfi 8272 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
117114, 115, 116syl2anc 694 . . . . . . . . . . . . . . . 16 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
118 inss2 3867 . . . . . . . . . . . . . . . 16 (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)
119 ssdomg 8043 . . . . . . . . . . . . . . . 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 8587 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 × ran 𝐺) ∈ Fin ↔ (ran 𝐹 × ran 𝐺) ≺ ω)
122121biimpi 206 . . . . . . . . . . . . . . . 16 ((ran 𝐹 × ran 𝐺) ∈ Fin → (ran 𝐹 × ran 𝐺) ≺ ω)
123 sdomdom 8025 . . . . . . . . . . . . . . . 16 ((ran 𝐹 × ran 𝐺) ≺ ω → (ran 𝐹 × ran 𝐺) ≼ ω)
124117, 122, 1233syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (ran 𝐹 × ran 𝐺) ≼ ω)
125 domtr 8050 . . . . . . . . . . . . . . 15 (((( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺) ∧ (ran 𝐹 × ran 𝐺) ≼ ω) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
126120, 124, 125syl2anc 694 . . . . . . . . . . . . . 14 (𝜑 → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
127126adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
128 nfcv 2793 . . . . . . . . . . . . . 14 𝑝(( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))
129128sigaclcuni 30309 . . . . . . . . . . . . 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 1366 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
13165, 130eqeltrd 2730 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran (𝐹𝑓 + 𝐺)) → ((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
132131ralrimiva 2995 . . . . . . . . . 10 (𝜑 → ∀𝑧 ∈ ran (𝐹𝑓 + 𝐺)((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
133 ssralv 3699 . . . . . . . . . 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 68 . . . . . . . . 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 6086 . . . . . . . . . . . . . 14 ( + :(𝐵 × 𝐵)⟶𝐶 → Fun + )
1371, 136syl 17 . . . . . . . . . . . . 13 (𝜑 → Fun + )
138 imafi 8300 . . . . . . . . . . . . 13 ((Fun + ∧ (ran 𝐹 × ran 𝐺) ∈ Fin) → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
139137, 117, 138syl2anc 694 . . . . . . . . . . . 12 (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
14018, 20, 9, 23ofrn2 29570 . . . . . . . . . . . 12 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
141 ssfi 8221 . . . . . . . . . . . 12 ((( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin ∧ ran (𝐹𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺))) → ran (𝐹𝑓 + 𝐺) ∈ Fin)
142139, 140, 141syl2anc 694 . . . . . . . . . . 11 (𝜑 → ran (𝐹𝑓 + 𝐺) ∈ Fin)
143 ssdomg 8043 . . . . . . . . . . 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 8587 . . . . . . . . . . . 12 (ran (𝐹𝑓 + 𝐺) ∈ Fin ↔ ran (𝐹𝑓 + 𝐺) ≺ ω)
146142, 145sylib 208 . . . . . . . . . . 11 (𝜑 → ran (𝐹𝑓 + 𝐺) ≺ ω)
147 sdomdom 8025 . . . . . . . . . . 11 (ran (𝐹𝑓 + 𝐺) ≺ ω → ran (𝐹𝑓 + 𝐺) ≼ ω)
148146, 147syl 17 . . . . . . . . . 10 (𝜑 → ran (𝐹𝑓 + 𝐺) ≼ ω)
149 domtr 8050 . . . . . . . . . 10 (((𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ran (𝐹𝑓 + 𝐺) ∧ ran (𝐹𝑓 + 𝐺) ≼ ω) → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ω)
150144, 148, 149syl2anc 694 . . . . . . . . 9 (𝜑 → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ω)
151150adantr 480 . . . . . . . 8 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (𝑏 ∩ ran (𝐹𝑓 + 𝐺)) ≼ ω)
152 nfcv 2793 . . . . . . . . 9 𝑧(𝑏 ∩ ran (𝐹𝑓 + 𝐺))
153152sigaclcuni 30309 . . . . . . . 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 1366 . . . . . . 7 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → 𝑧 ∈ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))((𝐹𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
15556, 154syl5eqelr 2735 . . . . . 6 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∈ dom 𝑀)
156 difpreima 6383 . . . . . . . . . 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 5521 . . . . . . . . . . 11 ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺)) = dom (𝐹𝑓 + 𝐺)
159158difeq2i 3758 . . . . . . . . . 10 (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺))) = (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ dom (𝐹𝑓 + 𝐺))
160 cnvimass 5520 . . . . . . . . . . 11 ((𝐹𝑓 + 𝐺) “ 𝑏) ⊆ dom (𝐹𝑓 + 𝐺)
161 ssdif0 3975 . . . . . . . . . . 11 (((𝐹𝑓 + 𝐺) “ 𝑏) ⊆ dom (𝐹𝑓 + 𝐺) ↔ (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ dom (𝐹𝑓 + 𝐺)) = ∅)
162160, 161mpbi 220 . . . . . . . . . 10 (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ dom (𝐹𝑓 + 𝐺)) = ∅
163159, 162eqtri 2673 . . . . . . . . 9 (((𝐹𝑓 + 𝐺) “ 𝑏) ∖ ((𝐹𝑓 + 𝐺) “ ran (𝐹𝑓 + 𝐺))) = ∅
164157, 163syl6eq 2701 . . . . . . . 8 (𝜑 → ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺))) = ∅)
165 0elsiga 30305 . . . . . . . . 9 (dom 𝑀 ran sigAlgebra → ∅ ∈ dom 𝑀)
16616, 50, 1653syl 18 . . . . . . . 8 (𝜑 → ∅ ∈ dom 𝑀)
167164, 166eqeltrd 2730 . . . . . . 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 30325 . . . . . 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 1366 . . . . 5 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (((𝐹𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹𝑓 + 𝐺))) ∪ ((𝐹𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹𝑓 + 𝐺)))) ∈ dom 𝑀)
17149, 170eqeltrd 2730 . . . 4 ((𝜑𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((𝐹𝑓 + 𝐺) “ 𝑏) ∈ dom 𝑀)
172171ralrimiva 2995 . . 3 (𝜑 → ∀𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))((𝐹𝑓 + 𝐺) “ 𝑏) ∈ dom 𝑀)
17351, 38ismbfm 30442 . . 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 977 . 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 6233 . . . . . 6 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) = (𝑀 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
177 measbasedom 30393 . . . . . . . . 9 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
17816, 177sylib 208 . . . . . . . 8 (𝜑𝑀 ∈ (measures‘dom 𝑀))
179178adantr 480 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → 𝑀 ∈ (measures‘dom 𝑀))
180 eldifi 3765 . . . . . . . 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 4220 . . . . . . . . . . 11 (𝑥 = (1st𝑝) → {𝑥} = {(1st𝑝)})
184183imaeq2d 5501 . . . . . . . . . 10 (𝑥 = (1st𝑝) → (𝐹 “ {𝑥}) = (𝐹 “ {(1st𝑝)}))
185 sneq 4220 . . . . . . . . . . 11 (𝑦 = (2nd𝑝) → {𝑦} = {(2nd𝑝)})
186185imaeq2d 5501 . . . . . . . . . 10 (𝑦 = (2nd𝑝) → (𝐺 “ {𝑦}) = (𝐺 “ {(2nd𝑝)}))
187 ffun 6086 . . . . . . . . . . . 12 (𝐹: dom 𝑀 𝐽 → Fun 𝐹)
18818, 187syl 17 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
189 sndisj 4676 . . . . . . . . . . 11 Disj 𝑥 ∈ ran 𝐹{𝑥}
190 disjpreima 29523 . . . . . . . . . . 11 ((Fun 𝐹Disj 𝑥 ∈ ran 𝐹{𝑥}) → Disj 𝑥 ∈ ran 𝐹(𝐹 “ {𝑥}))
191188, 189, 190sylancl 695 . . . . . . . . . 10 (𝜑Disj 𝑥 ∈ ran 𝐹(𝐹 “ {𝑥}))
192 ffun 6086 . . . . . . . . . . . 12 (𝐺: dom 𝑀 𝐽 → Fun 𝐺)
19320, 192syl 17 . . . . . . . . . . 11 (𝜑 → Fun 𝐺)
194 sndisj 4676 . . . . . . . . . . 11 Disj 𝑦 ∈ ran 𝐺{𝑦}
195 disjpreima 29523 . . . . . . . . . . 11 ((Fun 𝐺Disj 𝑦 ∈ ran 𝐺{𝑦}) → Disj 𝑦 ∈ ran 𝐺(𝐺 “ {𝑦}))
196193, 194, 195sylancl 695 . . . . . . . . . 10 (𝜑Disj 𝑦 ∈ ran 𝐺(𝐺 “ {𝑦}))
197184, 186, 191, 196disjxpin 29527 . . . . . . . . 9 (𝜑Disj 𝑝 ∈ (ran 𝐹 × ran 𝐺)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
198 disjss1 4658 . . . . . . . . 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 68 . . . . . . . 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 30405 . . . . . . 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 1370 . . . . . 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 8221 . . . . . . . . 9 (((ran 𝐹 × ran 𝐺) ∈ Fin ∧ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
204117, 118, 203sylancl 695 . . . . . . . 8 (𝜑 → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
205204adantr 480 . . . . . . 7 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
206 simpll 805 . . . . . . . 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 3634 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (ran 𝐹 × ran 𝐺))
209 xp1st 7242 . . . . . . . . 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 7243 . . . . . . . . 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 6699 . . . . . . . . . . . . . . . 16 ((𝑥 = 0𝑦 = 0 ) → (𝑥 + 𝑦) = ( 0 + 0 ))
214 sibfof.5 . . . . . . . . . . . . . . . 16 (𝜑 → ( 0 + 0 ) = (0g𝐾))
215213, 214sylan9eqr 2707 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 = 0𝑦 = 0 )) → (𝑥 + 𝑦) = (0g𝐾))
216215ex 449 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥 = 0𝑦 = 0 ) → (𝑥 + 𝑦) = (0g𝐾)))
217216necon3ad 2836 . . . . . . . . . . . . 13 (𝜑 → ((𝑥 + 𝑦) ≠ (0g𝐾) → ¬ (𝑥 = 0𝑦 = 0 )))
218 neorian 2917 . . . . . . . . . . . . 13 ((𝑥0𝑦0 ) ↔ ¬ (𝑥 = 0𝑦 = 0 ))
219217, 218syl6ibr 242 . . . . . . . . . . . 12 (𝜑 → ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
220219adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )))
221220ralrimivva 3000 . . . . . . . . . 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 3636 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ( + “ {𝑧}))
225 fniniseg 6378 . . . . . . . . . . . . 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 222 . . . . . . . . . . 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 7249 . . . . . . . . . . . . . . 15 (𝑝 ∈ (𝐵 × 𝐵) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
230229fveq2d 6233 . . . . . . . . . . . . . 14 (𝑝 ∈ (𝐵 × 𝐵) → ( +𝑝) = ( + ‘⟨(1st𝑝), (2nd𝑝)⟩))
231 df-ov 6693 . . . . . . . . . . . . . 14 ((1st𝑝) + (2nd𝑝)) = ( + ‘⟨(1st𝑝), (2nd𝑝)⟩)
232230, 231syl6eqr 2703 . . . . . . . . . . . . 13 (𝑝 ∈ (𝐵 × 𝐵) → ( +𝑝) = ((1st𝑝) + (2nd𝑝)))
233232adantr 480 . . . . . . . . . . . 12 ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( +𝑝) = 𝑧) → ( +𝑝) = ((1st𝑝) + (2nd𝑝)))
234228, 233eqtr3d 2687 . . . . . . . . . . 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 807 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)}))
237236eldifbd 3620 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ¬ 𝑧 ∈ {(0g𝐾)})
238 velsn 4226 . . . . . . . . . . . 12 (𝑧 ∈ {(0g𝐾)} ↔ 𝑧 = (0g𝐾))
239238necon3bbii 2870 . . . . . . . . . . 11 𝑧 ∈ {(0g𝐾)} ↔ 𝑧 ≠ (0g𝐾))
240237, 239sylib 208 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ≠ (0g𝐾))
241235, 240eqnetrrd 2891 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾))
242180, 76sylanl2 684 . . . . . . . . . . 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 6697 . . . . . . . . . . . . 13 (𝑥 = (1st𝑝) → (𝑥 + 𝑦) = ((1st𝑝) + 𝑦))
246245neeq1d 2882 . . . . . . . . . . . 12 (𝑥 = (1st𝑝) → ((𝑥 + 𝑦) ≠ (0g𝐾) ↔ ((1st𝑝) + 𝑦) ≠ (0g𝐾)))
247 neeq1 2885 . . . . . . . . . . . . 13 (𝑥 = (1st𝑝) → (𝑥0 ↔ (1st𝑝) ≠ 0 ))
248247orbi1d 739 . . . . . . . . . . . 12 (𝑥 = (1st𝑝) → ((𝑥0𝑦0 ) ↔ ((1st𝑝) ≠ 0𝑦0 )))
249246, 248imbi12d 333 . . . . . . . . . . 11 (𝑥 = (1st𝑝) → (((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )) ↔ (((1st𝑝) + 𝑦) ≠ (0g𝐾) → ((1st𝑝) ≠ 0𝑦0 ))))
250 oveq2 6698 . . . . . . . . . . . . 13 (𝑦 = (2nd𝑝) → ((1st𝑝) + 𝑦) = ((1st𝑝) + (2nd𝑝)))
251250neeq1d 2882 . . . . . . . . . . . 12 (𝑦 = (2nd𝑝) → (((1st𝑝) + 𝑦) ≠ (0g𝐾) ↔ ((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾)))
252 neeq1 2885 . . . . . . . . . . . . 13 (𝑦 = (2nd𝑝) → (𝑦0 ↔ (2nd𝑝) ≠ 0 ))
253252orbi2d 738 . . . . . . . . . . . 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 3353 . . . . . . . . . 10 (((1st𝑝) ∈ 𝐵 ∧ (2nd𝑝) ∈ 𝐵) → (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ≠ (0g𝐾) → (𝑥0𝑦0 )) → (((1st𝑝) + (2nd𝑝)) ≠ (0g𝐾) → ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 ))))
256243, 244, 255syl2anc 694 . . . . . . . . 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 49 . . . . . . . 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 30529 . . . . . . . 8 (((𝜑 ∧ (1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) ∧ ((1st𝑝) ≠ 0 ∨ (2nd𝑝) ≠ 0 )) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ (0[,)+∞))
259206, 210, 212, 257, 258syl31anc 1369 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ (0[,)+∞))
260205, 259esumpfinval 30265 . . . . . 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 2689 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) = Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
262 rge0ssre 12318 . . . . . . 7 (0[,)+∞) ⊆ ℝ
263262, 259sseldi 3634 . . . . . 6 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ ℝ)
264205, 263fsumrecl 14509 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) ∈ ℝ)
265261, 264eqeltrd 2730 . . . 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 684 . . . . . . 7 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀)
268 measge0 30398 . . . . . . 7 ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∈ dom 𝑀) → 0 ≤ (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
269266, 267, 268syl2anc 694 . . . . . 6 (((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) ∧ 𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 0 ≤ (𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
270205, 263, 269fsumge0 14571 . . . . 5 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → 0 ≤ Σ𝑝 ∈ (( + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
271270, 261breqtrrd 4713 . . . 4 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → 0 ≤ (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})))
272 elrege0 12316 . . . 4 ((𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞) ↔ ((𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ ℝ ∧ 0 ≤ (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧}))))
273265, 271, 272sylanbrc 699 . . 3 ((𝜑𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})) → (𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
274273ralrimiva 2995 . 2 (𝜑 → ∀𝑧 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {(0g𝐾)})(𝑀‘((𝐹𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
275 eqid 2651 . . 3 (sigaGen‘(TopOpen‘𝐾)) = (sigaGen‘(TopOpen‘𝐾))
276 eqid 2651 . . 3 (0g𝐾) = (0g𝐾)
277 eqid 2651 . . 3 ( ·𝑠𝐾) = ( ·𝑠𝐾)
278 eqid 2651 . . 3 (ℝHom‘(Scalar‘𝐾)) = (ℝHom‘(Scalar‘𝐾))
27927, 28, 275, 276, 277, 278, 26, 16issibf 30523 . 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 1264 1 (𝜑 → (𝐹𝑓 + 𝐺) ∈ dom (𝐾sitg𝑀))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210  cop 4216   cuni 4468   ciun 4552  Disj wdisj 4652   class class class wbr 4685   × cxp 5141  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑓 cof 6937  ωcom 7107  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  cdom 7995  csdm 7996  Fincfn 7997  cr 9973  0cc0 9974  +∞cpnf 10109  cle 10113  [,)cico 12215  Σcsu 14460  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  TopOpenctopn 16129  0gc0g 16147  Topctop 20746  TopSpctps 20784  Clsdccld 20868  Frect1 21159  ℝHomcrrh 30165  Σ*cesum 30217  sigAlgebracsiga 30298  sigaGencsigagen 30329  measurescmeas 30386  MblFnMcmbfm 30440  sitgcsitg 30519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-ac2 9323  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-ac 8977  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-ef 14842  df-sin 14844  df-cos 14845  df-pi 14847  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-ordt 16208  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-ps 17247  df-tsr 17248  df-plusf 17288  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-cring 18596  df-subrg 18826  df-abv 18865  df-lmod 18913  df-scaf 18914  df-sra 19220  df-rgmod 19221  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-t1 21166  df-haus 21167  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-tmd 21923  df-tgp 21924  df-tsms 21977  df-trg 22010  df-xms 22172  df-ms 22173  df-tms 22174  df-nm 22434  df-ngp 22435  df-nrg 22437  df-nlm 22438  df-ii 22727  df-cncf 22728  df-limc 23675  df-dv 23676  df-log 24348  df-esum 30218  df-siga 30299  df-sigagen 30330  df-meas 30387  df-mbfm 30441  df-sitg 30520
This theorem is referenced by:  sitmcl  30541
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