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Theorem esumrnmpt2 31004
Description: Rewrite an extended sum into a sum on the range of a mapping function. (Contributed by Thierry Arnoux, 30-May-2020.)
Hypotheses
Ref Expression
esumrnmpt2.1 (𝑦 = 𝐵𝐶 = 𝐷)
esumrnmpt2.2 (𝜑𝐴𝑉)
esumrnmpt2.3 ((𝜑𝑘𝐴) → 𝐷 ∈ (0[,]+∞))
esumrnmpt2.4 ((𝜑𝑘𝐴) → 𝐵𝑊)
esumrnmpt2.5 (((𝜑𝑘𝐴) ∧ 𝐵 = ∅) → 𝐷 = 0)
esumrnmpt2.6 (𝜑Disj 𝑘𝐴 𝐵)
Assertion
Ref Expression
esumrnmpt2 (𝜑 → Σ*𝑦 ∈ ran (𝑘𝐴𝐵)𝐶 = Σ*𝑘𝐴𝐷)
Distinct variable groups:   𝐴,𝑘,𝑦   𝑦,𝐵   𝐶,𝑘   𝑦,𝐷   𝑘,𝑊   𝜑,𝑘,𝑦
Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑦)   𝐷(𝑘)   𝑉(𝑦,𝑘)   𝑊(𝑦)

Proof of Theorem esumrnmpt2
StepHypRef Expression
1 nfrab1 3319 . . . . 5 𝑘{𝑘𝐴 ∣ ¬ 𝐵 = ∅}
2 esumrnmpt2.1 . . . . 5 (𝑦 = 𝐵𝐶 = 𝐷)
3 esumrnmpt2.2 . . . . . 6 (𝜑𝐴𝑉)
4 ssrab2 3941 . . . . . . 7 {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴
54a1i 11 . . . . . 6 (𝜑 → {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴)
63, 5ssexd 5081 . . . . 5 (𝜑 → {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ∈ V)
75sselda 3853 . . . . . 6 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → 𝑘𝐴)
8 esumrnmpt2.3 . . . . . 6 ((𝜑𝑘𝐴) → 𝐷 ∈ (0[,]+∞))
97, 8syldan 583 . . . . 5 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
10 esumrnmpt2.4 . . . . . . 7 ((𝜑𝑘𝐴) → 𝐵𝑊)
117, 10syldan 583 . . . . . 6 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵𝑊)
12 rabid 3312 . . . . . . . . 9 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↔ (𝑘𝐴 ∧ ¬ 𝐵 = ∅))
1312simprbi 489 . . . . . . . 8 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} → ¬ 𝐵 = ∅)
1413adantl 474 . . . . . . 7 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → ¬ 𝐵 = ∅)
15 elsng 4450 . . . . . . . 8 (𝐵𝑊 → (𝐵 ∈ {∅} ↔ 𝐵 = ∅))
1611, 15syl 17 . . . . . . 7 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → (𝐵 ∈ {∅} ↔ 𝐵 = ∅))
1714, 16mtbird 317 . . . . . 6 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → ¬ 𝐵 ∈ {∅})
1811, 17eldifd 3835 . . . . 5 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵 ∈ (𝑊 ∖ {∅}))
19 esumrnmpt2.6 . . . . . 6 (𝜑Disj 𝑘𝐴 𝐵)
20 nfcv 2927 . . . . . . 7 𝑘𝐴
211, 20disjss1f 30107 . . . . . 6 ({𝑘𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴 → (Disj 𝑘𝐴 𝐵Disj 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐵))
225, 19, 21sylc 65 . . . . 5 (𝜑Disj 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐵)
231, 2, 6, 9, 18, 22esumrnmpt 30988 . . . 4 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 = Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
24 nfv 1874 . . . . . . . . . . 11 𝑦(𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅)
25 snex 5185 . . . . . . . . . . . 12 {∅} ∈ V
2625a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → {∅} ∈ V)
27 velsn 4452 . . . . . . . . . . . . . . 15 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
2827biimpi 208 . . . . . . . . . . . . . 14 (𝑦 ∈ {∅} → 𝑦 = ∅)
2928adantl 474 . . . . . . . . . . . . 13 (((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝑦 = ∅)
30 nfv 1874 . . . . . . . . . . . . . . . 16 𝑘𝜑
31 nfre1 3246 . . . . . . . . . . . . . . . 16 𝑘𝑘𝐴 𝐵 = ∅
3230, 31nfan 1863 . . . . . . . . . . . . . . 15 𝑘(𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅)
33 nfv 1874 . . . . . . . . . . . . . . 15 𝑘 𝑦 = ∅
3432, 33nfan 1863 . . . . . . . . . . . . . 14 𝑘((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅)
35 nfv 1874 . . . . . . . . . . . . . 14 𝑘 𝐶 = 0
36 simpllr 764 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝑦 = ∅)
37 simpr 477 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝐵 = ∅)
3836, 37eqtr4d 2812 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝑦 = 𝐵)
3938, 2syl 17 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝐶 = 𝐷)
40 simp-4l 771 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝜑)
41 simplr 757 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝑘𝐴)
42 esumrnmpt2.5 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐴) ∧ 𝐵 = ∅) → 𝐷 = 0)
4340, 41, 37, 42syl21anc 826 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝐷 = 0)
4439, 43eqtrd 2809 . . . . . . . . . . . . . 14 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝐶 = 0)
45 simplr 757 . . . . . . . . . . . . . 14 (((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → ∃𝑘𝐴 𝐵 = ∅)
4634, 35, 44, 45r19.29af2 3266 . . . . . . . . . . . . 13 (((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → 𝐶 = 0)
4729, 46syldan 583 . . . . . . . . . . . 12 (((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 = 0)
48 0e0iccpnf 12662 . . . . . . . . . . . 12 0 ∈ (0[,]+∞)
4947, 48syl6eqel 2869 . . . . . . . . . . 11 (((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 ∈ (0[,]+∞))
50 nfcv 2927 . . . . . . . . . . . . . . . . 17 𝑘𝑦
51 nfmpt1 5022 . . . . . . . . . . . . . . . . . 18 𝑘(𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)
5251nfrn 5665 . . . . . . . . . . . . . . . . 17 𝑘ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)
5350, 52nfel 2939 . . . . . . . . . . . . . . . 16 𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)
5430, 53nfan 1863 . . . . . . . . . . . . . . 15 𝑘(𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵))
55 simpr 477 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
56 rabid 3312 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↔ (𝑘𝐴𝐵 = ∅))
5756simprbi 489 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ {𝑘𝐴𝐵 = ∅} → 𝐵 = ∅)
5857ad2antlr 715 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐵 = ∅)
5955, 58eqtrd 2809 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = ∅)
6059, 27sylibr 226 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 ∈ {∅})
61 vex 3413 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
62 eqid 2773 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)
6362elrnmpt 5669 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ V → (𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝑦 = 𝐵))
6461, 63ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝑦 = 𝐵)
6564biimpi 208 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) → ∃𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝑦 = 𝐵)
6665adantl 474 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) → ∃𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝑦 = 𝐵)
6754, 60, 66r19.29af 3267 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) → 𝑦 ∈ {∅})
6867ex 405 . . . . . . . . . . . . 13 (𝜑 → (𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) → 𝑦 ∈ {∅}))
6968ssrdv 3859 . . . . . . . . . . . 12 (𝜑 → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ⊆ {∅})
7069adantr 473 . . . . . . . . . . 11 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ⊆ {∅})
7124, 26, 49, 70esummono 30990 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ Σ*𝑦 ∈ {∅}𝐶)
72 0ex 5065 . . . . . . . . . . . 12 ∅ ∈ V
7372a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → ∅ ∈ V)
7448a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → 0 ∈ (0[,]+∞))
7546, 73, 74esumsn 31001 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → Σ*𝑦 ∈ {∅}𝐶 = 0)
7671, 75breqtrd 4952 . . . . . . . . 9 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
77 simpr 477 . . . . . . . . . 10 ((𝜑 ∧ ¬ ∃𝑘𝐴 𝐵 = ∅) → ¬ ∃𝑘𝐴 𝐵 = ∅)
78 nfv 1874 . . . . . . . . . . . . 13 𝑦 ¬ ∃𝑘𝐴 𝐵 = ∅
7931nfn 1820 . . . . . . . . . . . . . . . . 17 𝑘 ¬ ∃𝑘𝐴 𝐵 = ∅
80 rabn0 4220 . . . . . . . . . . . . . . . . . . 19 ({𝑘𝐴𝐵 = ∅} ≠ ∅ ↔ ∃𝑘𝐴 𝐵 = ∅)
8180biimpi 208 . . . . . . . . . . . . . . . . . 18 ({𝑘𝐴𝐵 = ∅} ≠ ∅ → ∃𝑘𝐴 𝐵 = ∅)
8281necon1bi 2990 . . . . . . . . . . . . . . . . 17 (¬ ∃𝑘𝐴 𝐵 = ∅ → {𝑘𝐴𝐵 = ∅} = ∅)
83 eqid 2773 . . . . . . . . . . . . . . . . . 18 𝐵 = 𝐵
8483a1i 11 . . . . . . . . . . . . . . . . 17 (¬ ∃𝑘𝐴 𝐵 = ∅ → 𝐵 = 𝐵)
8579, 82, 84mpteq12df 5010 . . . . . . . . . . . . . . . 16 (¬ ∃𝑘𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ ∅ ↦ 𝐵))
86 mpt0 6318 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ∅ ↦ 𝐵) = ∅
8785, 86syl6eq 2825 . . . . . . . . . . . . . . 15 (¬ ∃𝑘𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) = ∅)
8887rneqd 5649 . . . . . . . . . . . . . 14 (¬ ∃𝑘𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) = ran ∅)
89 rn0 5674 . . . . . . . . . . . . . 14 ran ∅ = ∅
9088, 89syl6eq 2825 . . . . . . . . . . . . 13 (¬ ∃𝑘𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) = ∅)
9178, 90esumeq1d 30971 . . . . . . . . . . . 12 (¬ ∃𝑘𝐴 𝐵 = ∅ → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 = Σ*𝑦 ∈ ∅𝐶)
92 esumnul 30984 . . . . . . . . . . . 12 Σ*𝑦 ∈ ∅𝐶 = 0
9391, 92syl6eq 2825 . . . . . . . . . . 11 (¬ ∃𝑘𝐴 𝐵 = ∅ → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
94 0le0 11547 . . . . . . . . . . 11 0 ≤ 0
9593, 94syl6eqbr 4965 . . . . . . . . . 10 (¬ ∃𝑘𝐴 𝐵 = ∅ → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
9677, 95syl 17 . . . . . . . . 9 ((𝜑 ∧ ¬ ∃𝑘𝐴 𝐵 = ∅) → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
9776, 96pm2.61dan 801 . . . . . . . 8 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
98 ssrab2 3941 . . . . . . . . . . . . 13 {𝑘𝐴𝐵 = ∅} ⊆ 𝐴
9998a1i 11 . . . . . . . . . . . 12 (𝜑 → {𝑘𝐴𝐵 = ∅} ⊆ 𝐴)
1003, 99ssexd 5081 . . . . . . . . . . 11 (𝜑 → {𝑘𝐴𝐵 = ∅} ∈ V)
101 nfrab1 3319 . . . . . . . . . . . 12 𝑘{𝑘𝐴𝐵 = ∅}
102101mptexgf 6810 . . . . . . . . . . 11 ({𝑘𝐴𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∈ V)
103 rnexg 7428 . . . . . . . . . . 11 ((𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∈ V → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∈ V)
104100, 102, 1033syl 18 . . . . . . . . . 10 (𝜑 → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∈ V)
1052adantl 474 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
106 simplll 763 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝜑)
10799sselda 3853 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝑘𝐴)
108107adantlr 703 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝑘𝐴)
109108adantr 473 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘𝐴)
110106, 109, 8syl2anc 576 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
111105, 110eqeltrd 2861 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
11254, 111, 66r19.29af 3267 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
113112ralrimiva 3127 . . . . . . . . . 10 (𝜑 → ∀𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
114 nfcv 2927 . . . . . . . . . . 11 𝑦ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)
115114esumcl 30966 . . . . . . . . . 10 ((ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∈ V ∧ ∀𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞)) → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
116104, 113, 115syl2anc 576 . . . . . . . . 9 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
117 elxrge0 12660 . . . . . . . . . 10 *𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞) ↔ (Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ* ∧ 0 ≤ Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶))
118117simprbi 489 . . . . . . . . 9 *𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞) → 0 ≤ Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶)
119116, 118syl 17 . . . . . . . 8 (𝜑 → 0 ≤ Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶)
12097, 119jca 504 . . . . . . 7 (𝜑 → (Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0 ∧ 0 ≤ Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶))
121 iccssxr 12634 . . . . . . . . 9 (0[,]+∞) ⊆ ℝ*
122121, 116sseldi 3851 . . . . . . . 8 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ*)
123121, 48sselii 3850 . . . . . . . . 9 0 ∈ ℝ*
124123a1i 11 . . . . . . . 8 (𝜑 → 0 ∈ ℝ*)
125 xrletri3 12363 . . . . . . . 8 ((Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ* ∧ 0 ∈ ℝ*) → (Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 = 0 ↔ (Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0 ∧ 0 ≤ Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶)))
126122, 124, 125syl2anc 576 . . . . . . 7 (𝜑 → (Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 = 0 ↔ (Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0 ∧ 0 ≤ Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶)))
127120, 126mpbird 249 . . . . . 6 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
128127oveq1d 6990 . . . . 5 (𝜑 → (Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 +𝑒 Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶) = (0 +𝑒 Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶))
1299ralrimiva 3127 . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
1301esumcl 30966 . . . . . . . . 9 (({𝑘𝐴 ∣ ¬ 𝐵 = ∅} ∈ V ∧ ∀𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞)) → Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
1316, 129, 130syl2anc 576 . . . . . . . 8 (𝜑 → Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
132121, 131sseldi 3851 . . . . . . 7 (𝜑 → Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ ℝ*)
13323, 132eqeltrd 2861 . . . . . 6 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ*)
134 xaddid2 12451 . . . . . 6 *𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ* → (0 +𝑒 Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶) = Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶)
135133, 134syl 17 . . . . 5 (𝜑 → (0 +𝑒 Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶) = Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶)
136128, 135eqtrd 2809 . . . 4 (𝜑 → (Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 +𝑒 Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶) = Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶)
137 simpl 475 . . . . . . . . . 10 ((𝜑𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝜑)
13857adantl 474 . . . . . . . . . 10 ((𝜑𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝐵 = ∅)
139137, 107, 138, 42syl21anc 826 . . . . . . . . 9 ((𝜑𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝐷 = 0)
140139ralrimiva 3127 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 = 0)
14130, 140esumeq2d 30973 . . . . . . 7 (𝜑 → Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 = Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}0)
142101esum0 30985 . . . . . . . 8 ({𝑘𝐴𝐵 = ∅} ∈ V → Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}0 = 0)
143100, 142syl 17 . . . . . . 7 (𝜑 → Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}0 = 0)
144141, 143eqtrd 2809 . . . . . 6 (𝜑 → Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 = 0)
145144oveq1d 6990 . . . . 5 (𝜑 → (Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = (0 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷))
146 xaddid2 12451 . . . . . 6 *𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ ℝ* → (0 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
147132, 146syl 17 . . . . 5 (𝜑 → (0 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
148145, 147eqtrd 2809 . . . 4 (𝜑 → (Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
14923, 136, 1483eqtr4d 2819 . . 3 (𝜑 → (Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 +𝑒 Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶) = (Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷))
150 nfv 1874 . . . 4 𝑦𝜑
151 nfcv 2927 . . . 4 𝑦ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
1521mptexgf 6810 . . . . 5 ({𝑘𝐴 ∣ ¬ 𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
153 rnexg 7428 . . . . 5 ((𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V → ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
1546, 152, 1533syl 18 . . . 4 (𝜑 → ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
15569ssrind 4094 . . . . . 6 (𝜑 → (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ({∅} ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)))
156 incom 4061 . . . . . . 7 (ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ({∅} ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
15713neqned 2969 . . . . . . . . . . . 12 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} → 𝐵 ≠ ∅)
158157necomd 3017 . . . . . . . . . . 11 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} → ∅ ≠ 𝐵)
159158neneqd 2967 . . . . . . . . . 10 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} → ¬ ∅ = 𝐵)
160159nrex 3209 . . . . . . . . 9 ¬ ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵
161 eqid 2773 . . . . . . . . . . 11 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
162161elrnmpt 5669 . . . . . . . . . 10 (∅ ∈ V → (∅ ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵))
16372, 162ax-mp 5 . . . . . . . . 9 (∅ ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵)
164160, 163mtbir 315 . . . . . . . 8 ¬ ∅ ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
165 disjsn 4518 . . . . . . . 8 ((ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
166164, 165mpbir 223 . . . . . . 7 (ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅
167156, 166eqtr3i 2799 . . . . . 6 ({∅} ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅
168155, 167syl6sseq 3902 . . . . 5 (𝜑 → (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ∅)
169 ss0 4233 . . . . 5 ((ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ∅ → (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅)
170168, 169syl 17 . . . 4 (𝜑 → (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅)
171 nfmpt1 5022 . . . . . . . 8 𝑘(𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
172171nfrn 5665 . . . . . . 7 𝑘ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
17350, 172nfel 2939 . . . . . 6 𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
17430, 173nfan 1863 . . . . 5 𝑘(𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
1752adantl 474 . . . . . 6 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
176 simplll 763 . . . . . . 7 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝜑)
1777adantlr 703 . . . . . . . 8 (((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → 𝑘𝐴)
178177adantr 473 . . . . . . 7 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘𝐴)
179176, 178, 8syl2anc 576 . . . . . 6 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
180175, 179eqeltrd 2861 . . . . 5 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
181161elrnmpt 5669 . . . . . . . 8 (𝑦 ∈ V → (𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵))
18261, 181ax-mp 5 . . . . . . 7 (𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
183182biimpi 208 . . . . . 6 (𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) → ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
184183adantl 474 . . . . 5 ((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
185174, 180, 184r19.29af 3267 . . . 4 ((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
186150, 114, 151, 104, 154, 170, 112, 185esumsplit 30989 . . 3 (𝜑 → Σ*𝑦 ∈ (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))𝐶 = (Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 +𝑒 Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶))
187 rabnc 4222 . . . . 5 ({𝑘𝐴𝐵 = ∅} ∩ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) = ∅
188187a1i 11 . . . 4 (𝜑 → ({𝑘𝐴𝐵 = ∅} ∩ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) = ∅)
189107, 8syldan 583 . . . 4 ((𝜑𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
19030, 101, 1, 100, 6, 188, 189, 9esumsplit 30989 . . 3 (𝜑 → Σ*𝑘 ∈ ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅})𝐷 = (Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷))
191149, 186, 1903eqtr4d 2819 . 2 (𝜑 → Σ*𝑦 ∈ (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))𝐶 = Σ*𝑘 ∈ ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅})𝐷)
192 rabxm 4221 . . . . . . . 8 𝐴 = ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅})
193192, 83mpteq12i 5017 . . . . . . 7 (𝑘𝐴𝐵) = (𝑘 ∈ ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵)
194 mptun 6322 . . . . . . 7 (𝑘 ∈ ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵) = ((𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
195193, 194eqtri 2797 . . . . . 6 (𝑘𝐴𝐵) = ((𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
196195rneqi 5648 . . . . 5 ran (𝑘𝐴𝐵) = ran ((𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
197 rnun 5842 . . . . 5 ran ((𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
198196, 197eqtri 2797 . . . 4 ran (𝑘𝐴𝐵) = (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
199198a1i 11 . . 3 (𝜑 → ran (𝑘𝐴𝐵) = (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)))
200150, 199esumeq1d 30971 . 2 (𝜑 → Σ*𝑦 ∈ ran (𝑘𝐴𝐵)𝐶 = Σ*𝑦 ∈ (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))𝐶)
201192a1i 11 . . 3 (𝜑𝐴 = ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}))
20230, 201esumeq1d 30971 . 2 (𝜑 → Σ*𝑘𝐴𝐷 = Σ*𝑘 ∈ ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅})𝐷)
203191, 200, 2023eqtr4d 2819 1 (𝜑 → Σ*𝑦 ∈ ran (𝑘𝐴𝐵)𝐶 = Σ*𝑘𝐴𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387   = wceq 1508  wcel 2051  wne 2962  wral 3083  wrex 3084  {crab 3087  Vcvv 3410  cun 3822  cin 3823  wss 3824  c0 4173  {csn 4436  Disj wdisj 4894   class class class wbr 4926  cmpt 5005  ran crn 5405  (class class class)co 6975  0cc0 10334  +∞cpnf 10470  *cxr 10472  cle 10474   +𝑒 cxad 12321  [,]cicc 12556  Σ*cesum 30963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-inf2 8897  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411  ax-pre-sup 10412  ax-addf 10413  ax-mulf 10414
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-int 4747  df-iun 4791  df-iin 4792  df-disj 4895  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-se 5364  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-isom 6195  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-of 7226  df-om 7396  df-1st 7500  df-2nd 7501  df-supp 7633  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-1o 7904  df-2o 7905  df-oadd 7908  df-er 8088  df-map 8207  df-pm 8208  df-ixp 8259  df-en 8306  df-dom 8307  df-sdom 8308  df-fin 8309  df-fsupp 8628  df-fi 8669  df-sup 8700  df-inf 8701  df-oi 8768  df-card 9161  df-cda 9387  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-div 11098  df-nn 11439  df-2 11502  df-3 11503  df-4 11504  df-5 11505  df-6 11506  df-7 11507  df-8 11508  df-9 11509  df-n0 11707  df-z 11793  df-dec 11911  df-uz 12058  df-q 12162  df-rp 12204  df-xneg 12323  df-xadd 12324  df-xmul 12325  df-ioo 12557  df-ioc 12558  df-ico 12559  df-icc 12560  df-fz 12708  df-fzo 12849  df-fl 12976  df-mod 13052  df-seq 13184  df-exp 13244  df-fac 13448  df-bc 13477  df-hash 13505  df-shft 14286  df-cj 14318  df-re 14319  df-im 14320  df-sqrt 14454  df-abs 14455  df-limsup 14688  df-clim 14705  df-rlim 14706  df-sum 14903  df-ef 15280  df-sin 15282  df-cos 15283  df-pi 15285  df-struct 16340  df-ndx 16341  df-slot 16342  df-base 16344  df-sets 16345  df-ress 16346  df-plusg 16433  df-mulr 16434  df-starv 16435  df-sca 16436  df-vsca 16437  df-ip 16438  df-tset 16439  df-ple 16440  df-ds 16442  df-unif 16443  df-hom 16444  df-cco 16445  df-rest 16551  df-topn 16552  df-0g 16570  df-gsum 16571  df-topgen 16572  df-pt 16573  df-prds 16576  df-ordt 16629  df-xrs 16630  df-qtop 16635  df-imas 16636  df-xps 16638  df-mre 16728  df-mrc 16729  df-acs 16731  df-ps 17681  df-tsr 17682  df-plusf 17722  df-mgm 17723  df-sgrp 17765  df-mnd 17776  df-mhm 17816  df-submnd 17817  df-grp 17907  df-minusg 17908  df-sbg 17909  df-mulg 18025  df-subg 18073  df-cntz 18231  df-cmn 18681  df-abl 18682  df-mgp 18976  df-ur 18988  df-ring 19035  df-cring 19036  df-subrg 19269  df-abv 19323  df-lmod 19371  df-scaf 19372  df-sra 19679  df-rgmod 19680  df-psmet 20255  df-xmet 20256  df-met 20257  df-bl 20258  df-mopn 20259  df-fbas 20260  df-fg 20261  df-cnfld 20264  df-top 21222  df-topon 21239  df-topsp 21261  df-bases 21274  df-cld 21347  df-ntr 21348  df-cls 21349  df-nei 21426  df-lp 21464  df-perf 21465  df-cn 21555  df-cnp 21556  df-haus 21643  df-tx 21890  df-hmeo 22083  df-fil 22174  df-fm 22266  df-flim 22267  df-flf 22268  df-tmd 22400  df-tgp 22401  df-tsms 22454  df-trg 22487  df-xms 22649  df-ms 22650  df-tms 22651  df-nm 22911  df-ngp 22912  df-nrg 22914  df-nlm 22915  df-ii 23204  df-cncf 23205  df-limc 24183  df-dv 24184  df-log 24857  df-esum 30964
This theorem is referenced by:  carsggect  31254  carsgclctunlem2  31255  pmeasadd  31261
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