Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  esumrnmpt2 Structured version   Visualization version   GIF version

Theorem esumrnmpt2 32667
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 3426 . . . . 5 𝑘{𝑘𝐴 ∣ ¬ 𝐵 = ∅}
2 esumrnmpt2.1 . . . . 5 (𝑦 = 𝐵𝐶 = 𝐷)
3 esumrnmpt2.2 . . . . . 6 (𝜑𝐴𝑉)
4 ssrab2 4037 . . . . . . 7 {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴
54a1i 11 . . . . . 6 (𝜑 → {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴)
63, 5ssexd 5281 . . . . 5 (𝜑 → {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ∈ V)
75sselda 3944 . . . . . 6 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → 𝑘𝐴)
8 esumrnmpt2.3 . . . . . 6 ((𝜑𝑘𝐴) → 𝐷 ∈ (0[,]+∞))
97, 8syldan 591 . . . . 5 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
10 esumrnmpt2.4 . . . . . . 7 ((𝜑𝑘𝐴) → 𝐵𝑊)
117, 10syldan 591 . . . . . 6 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵𝑊)
12 rabid 3427 . . . . . . . . 9 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↔ (𝑘𝐴 ∧ ¬ 𝐵 = ∅))
1312simprbi 497 . . . . . . . 8 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} → ¬ 𝐵 = ∅)
1413adantl 482 . . . . . . 7 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → ¬ 𝐵 = ∅)
15 elsng 4600 . . . . . . . 8 (𝐵𝑊 → (𝐵 ∈ {∅} ↔ 𝐵 = ∅))
1611, 15syl 17 . . . . . . 7 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → (𝐵 ∈ {∅} ↔ 𝐵 = ∅))
1714, 16mtbird 324 . . . . . 6 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → ¬ 𝐵 ∈ {∅})
1811, 17eldifd 3921 . . . . 5 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵 ∈ (𝑊 ∖ {∅}))
19 esumrnmpt2.6 . . . . . 6 (𝜑Disj 𝑘𝐴 𝐵)
20 nfcv 2907 . . . . . . 7 𝑘𝐴
211, 20disjss1f 31490 . . . . . 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 32651 . . . 4 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 = Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
24 nfv 1917 . . . . . . . . . . 11 𝑦(𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅)
25 snex 5388 . . . . . . . . . . . 12 {∅} ∈ V
2625a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → {∅} ∈ V)
27 velsn 4602 . . . . . . . . . . . . . . 15 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
2827biimpi 215 . . . . . . . . . . . . . 14 (𝑦 ∈ {∅} → 𝑦 = ∅)
2928adantl 482 . . . . . . . . . . . . 13 (((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝑦 = ∅)
30 nfv 1917 . . . . . . . . . . . . . . . 16 𝑘𝜑
31 nfre1 3268 . . . . . . . . . . . . . . . 16 𝑘𝑘𝐴 𝐵 = ∅
3230, 31nfan 1902 . . . . . . . . . . . . . . 15 𝑘(𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅)
33 nfv 1917 . . . . . . . . . . . . . . 15 𝑘 𝑦 = ∅
3432, 33nfan 1902 . . . . . . . . . . . . . 14 𝑘((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅)
35 nfv 1917 . . . . . . . . . . . . . 14 𝑘 𝐶 = 0
36 simpllr 774 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝑦 = ∅)
37 simpr 485 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝐵 = ∅)
3836, 37eqtr4d 2779 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝑦 = 𝐵)
3938, 2syl 17 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝐶 = 𝐷)
40 simp-4l 781 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝜑)
41 simplr 767 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝑘𝐴)
42 esumrnmpt2.5 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐴) ∧ 𝐵 = ∅) → 𝐷 = 0)
4340, 41, 37, 42syl21anc 836 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝐷 = 0)
4439, 43eqtrd 2776 . . . . . . . . . . . . . 14 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝐶 = 0)
45 simplr 767 . . . . . . . . . . . . . 14 (((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → ∃𝑘𝐴 𝐵 = ∅)
4634, 35, 44, 45r19.29af2 3250 . . . . . . . . . . . . 13 (((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → 𝐶 = 0)
4729, 46syldan 591 . . . . . . . . . . . 12 (((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 = 0)
48 0e0iccpnf 13376 . . . . . . . . . . . 12 0 ∈ (0[,]+∞)
4947, 48eqeltrdi 2846 . . . . . . . . . . 11 (((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 ∈ (0[,]+∞))
50 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑘𝑦
51 nfmpt1 5213 . . . . . . . . . . . . . . . . . 18 𝑘(𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)
5251nfrn 5907 . . . . . . . . . . . . . . . . 17 𝑘ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)
5350, 52nfel 2921 . . . . . . . . . . . . . . . 16 𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)
5430, 53nfan 1902 . . . . . . . . . . . . . . 15 𝑘(𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵))
55 simpr 485 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
56 rabid 3427 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↔ (𝑘𝐴𝐵 = ∅))
5756simprbi 497 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ {𝑘𝐴𝐵 = ∅} → 𝐵 = ∅)
5857ad2antlr 725 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐵 = ∅)
5955, 58eqtrd 2776 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = ∅)
6059, 27sylibr 233 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 ∈ {∅})
61 vex 3449 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
62 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)
6362elrnmpt 5911 . . . . . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) → ∃𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝑦 = 𝐵)
6665adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) → ∃𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝑦 = 𝐵)
6754, 60, 66r19.29af 3251 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) → 𝑦 ∈ {∅})
6867ex 413 . . . . . . . . . . . . 13 (𝜑 → (𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) → 𝑦 ∈ {∅}))
6968ssrdv 3950 . . . . . . . . . . . 12 (𝜑 → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ⊆ {∅})
7069adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ⊆ {∅})
7124, 26, 49, 70esummono 32653 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ Σ*𝑦 ∈ {∅}𝐶)
72 0ex 5264 . . . . . . . . . . . 12 ∅ ∈ V
7372a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → ∅ ∈ V)
7448a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → 0 ∈ (0[,]+∞))
7546, 73, 74esumsn 32664 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → Σ*𝑦 ∈ {∅}𝐶 = 0)
7671, 75breqtrd 5131 . . . . . . . . 9 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
77 simpr 485 . . . . . . . . . 10 ((𝜑 ∧ ¬ ∃𝑘𝐴 𝐵 = ∅) → ¬ ∃𝑘𝐴 𝐵 = ∅)
78 nfv 1917 . . . . . . . . . . . . 13 𝑦 ¬ ∃𝑘𝐴 𝐵 = ∅
7931nfn 1860 . . . . . . . . . . . . . . . . 17 𝑘 ¬ ∃𝑘𝐴 𝐵 = ∅
80 rabn0 4345 . . . . . . . . . . . . . . . . . . 19 ({𝑘𝐴𝐵 = ∅} ≠ ∅ ↔ ∃𝑘𝐴 𝐵 = ∅)
8180biimpi 215 . . . . . . . . . . . . . . . . . 18 ({𝑘𝐴𝐵 = ∅} ≠ ∅ → ∃𝑘𝐴 𝐵 = ∅)
8281necon1bi 2972 . . . . . . . . . . . . . . . . 17 (¬ ∃𝑘𝐴 𝐵 = ∅ → {𝑘𝐴𝐵 = ∅} = ∅)
83 eqid 2736 . . . . . . . . . . . . . . . . . 18 𝐵 = 𝐵
8483a1i 11 . . . . . . . . . . . . . . . . 17 (¬ ∃𝑘𝐴 𝐵 = ∅ → 𝐵 = 𝐵)
8579, 82, 84mpteq12df 5191 . . . . . . . . . . . . . . . 16 (¬ ∃𝑘𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ ∅ ↦ 𝐵))
86 mpt0 6643 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ∅ ↦ 𝐵) = ∅
8785, 86eqtrdi 2792 . . . . . . . . . . . . . . 15 (¬ ∃𝑘𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) = ∅)
8887rneqd 5893 . . . . . . . . . . . . . 14 (¬ ∃𝑘𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) = ran ∅)
89 rn0 5881 . . . . . . . . . . . . . 14 ran ∅ = ∅
9088, 89eqtrdi 2792 . . . . . . . . . . . . 13 (¬ ∃𝑘𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) = ∅)
9178, 90esumeq1d 32634 . . . . . . . . . . . 12 (¬ ∃𝑘𝐴 𝐵 = ∅ → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 = Σ*𝑦 ∈ ∅𝐶)
92 esumnul 32647 . . . . . . . . . . . 12 Σ*𝑦 ∈ ∅𝐶 = 0
9391, 92eqtrdi 2792 . . . . . . . . . . 11 (¬ ∃𝑘𝐴 𝐵 = ∅ → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
94 0le0 12254 . . . . . . . . . . 11 0 ≤ 0
9593, 94eqbrtrdi 5144 . . . . . . . . . 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 811 . . . . . . . 8 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
98 ssrab2 4037 . . . . . . . . . . . . 13 {𝑘𝐴𝐵 = ∅} ⊆ 𝐴
9998a1i 11 . . . . . . . . . . . 12 (𝜑 → {𝑘𝐴𝐵 = ∅} ⊆ 𝐴)
1003, 99ssexd 5281 . . . . . . . . . . 11 (𝜑 → {𝑘𝐴𝐵 = ∅} ∈ V)
101 nfrab1 3426 . . . . . . . . . . . 12 𝑘{𝑘𝐴𝐵 = ∅}
102101mptexgf 7172 . . . . . . . . . . 11 ({𝑘𝐴𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∈ V)
103 rnexg 7841 . . . . . . . . . . 11 ((𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∈ V → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∈ V)
104100, 102, 1033syl 18 . . . . . . . . . 10 (𝜑 → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∈ V)
1052adantl 482 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
106 simplll 773 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝜑)
10799sselda 3944 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝑘𝐴)
108107adantlr 713 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝑘𝐴)
109108adantr 481 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘𝐴)
110106, 109, 8syl2anc 584 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
111105, 110eqeltrd 2838 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
11254, 111, 66r19.29af 3251 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
113112ralrimiva 3143 . . . . . . . . . 10 (𝜑 → ∀𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
114 nfcv 2907 . . . . . . . . . . 11 𝑦ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)
115114esumcl 32629 . . . . . . . . . 10 ((ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∈ V ∧ ∀𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞)) → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
116104, 113, 115syl2anc 584 . . . . . . . . 9 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
117 elxrge0 13374 . . . . . . . . . 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 497 . . . . . . . . 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 512 . . . . . . 7 (𝜑 → (Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0 ∧ 0 ≤ Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶))
121 iccssxr 13347 . . . . . . . . 9 (0[,]+∞) ⊆ ℝ*
122121, 116sselid 3942 . . . . . . . 8 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ*)
123121, 48sselii 3941 . . . . . . . . 9 0 ∈ ℝ*
124123a1i 11 . . . . . . . 8 (𝜑 → 0 ∈ ℝ*)
125 xrletri3 13073 . . . . . . . 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 584 . . . . . . 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 256 . . . . . 6 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
128127oveq1d 7372 . . . . 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 3143 . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
1301esumcl 32629 . . . . . . . . 9 (({𝑘𝐴 ∣ ¬ 𝐵 = ∅} ∈ V ∧ ∀𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞)) → Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
1316, 129, 130syl2anc 584 . . . . . . . 8 (𝜑 → Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
132121, 131sselid 3942 . . . . . . 7 (𝜑 → Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ ℝ*)
13323, 132eqeltrd 2838 . . . . . 6 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ*)
134 xaddid2 13161 . . . . . 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 2776 . . . 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 483 . . . . . . . . . 10 ((𝜑𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝜑)
13857adantl 482 . . . . . . . . . 10 ((𝜑𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝐵 = ∅)
139137, 107, 138, 42syl21anc 836 . . . . . . . . 9 ((𝜑𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝐷 = 0)
140139ralrimiva 3143 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 = 0)
14130, 140esumeq2d 32636 . . . . . . 7 (𝜑 → Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 = Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}0)
142101esum0 32648 . . . . . . . 8 ({𝑘𝐴𝐵 = ∅} ∈ V → Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}0 = 0)
143100, 142syl 17 . . . . . . 7 (𝜑 → Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}0 = 0)
144141, 143eqtrd 2776 . . . . . 6 (𝜑 → Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 = 0)
145144oveq1d 7372 . . . . 5 (𝜑 → (Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = (0 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷))
146 xaddid2 13161 . . . . . 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 2776 . . . 4 (𝜑 → (Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
14923, 136, 1483eqtr4d 2786 . . 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 1917 . . . 4 𝑦𝜑
151 nfcv 2907 . . . 4 𝑦ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
1521mptexgf 7172 . . . . 5 ({𝑘𝐴 ∣ ¬ 𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
153 rnexg 7841 . . . . 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 4195 . . . . . 6 (𝜑 → (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ({∅} ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)))
156 incom 4161 . . . . . . 7 (ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ({∅} ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
15713neqned 2950 . . . . . . . . . . . 12 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} → 𝐵 ≠ ∅)
158157necomd 2999 . . . . . . . . . . 11 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} → ∅ ≠ 𝐵)
159158neneqd 2948 . . . . . . . . . 10 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} → ¬ ∅ = 𝐵)
160159nrex 3077 . . . . . . . . 9 ¬ ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵
161 eqid 2736 . . . . . . . . . . 11 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
162161elrnmpt 5911 . . . . . . . . . 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 322 . . . . . . . 8 ¬ ∅ ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
165 disjsn 4672 . . . . . . . 8 ((ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
166164, 165mpbir 230 . . . . . . 7 (ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅
167156, 166eqtr3i 2766 . . . . . 6 ({∅} ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅
168155, 167sseqtrdi 3994 . . . . 5 (𝜑 → (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ∅)
169 ss0 4358 . . . . 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 5213 . . . . . . . 8 𝑘(𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
172171nfrn 5907 . . . . . . 7 𝑘ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
17350, 172nfel 2921 . . . . . 6 𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
17430, 173nfan 1902 . . . . 5 𝑘(𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
1752adantl 482 . . . . . 6 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
176 simplll 773 . . . . . . 7 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝜑)
1777adantlr 713 . . . . . . . 8 (((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → 𝑘𝐴)
178177adantr 481 . . . . . . 7 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘𝐴)
179176, 178, 8syl2anc 584 . . . . . 6 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
180175, 179eqeltrd 2838 . . . . 5 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
181161elrnmpt 5911 . . . . . . . 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 215 . . . . . 6 (𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) → ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
184183adantl 482 . . . . 5 ((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
185174, 180, 184r19.29af 3251 . . . 4 ((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
186150, 114, 151, 104, 154, 170, 112, 185esumsplit 32652 . . 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 4347 . . . . 5 ({𝑘𝐴𝐵 = ∅} ∩ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) = ∅
188187a1i 11 . . . 4 (𝜑 → ({𝑘𝐴𝐵 = ∅} ∩ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) = ∅)
189107, 8syldan 591 . . . 4 ((𝜑𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
19030, 101, 1, 100, 6, 188, 189, 9esumsplit 32652 . . 3 (𝜑 → Σ*𝑘 ∈ ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅})𝐷 = (Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷))
191149, 186, 1903eqtr4d 2786 . 2 (𝜑 → Σ*𝑦 ∈ (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))𝐶 = Σ*𝑘 ∈ ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅})𝐷)
192 rabxm 4346 . . . . . . . 8 𝐴 = ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅})
193192, 83mpteq12i 5211 . . . . . . 7 (𝑘𝐴𝐵) = (𝑘 ∈ ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵)
194 mptun 6647 . . . . . . 7 (𝑘 ∈ ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵) = ((𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
195193, 194eqtri 2764 . . . . . 6 (𝑘𝐴𝐵) = ((𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
196195rneqi 5892 . . . . 5 ran (𝑘𝐴𝐵) = ran ((𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
197 rnun 6098 . . . . 5 ran ((𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
198196, 197eqtri 2764 . . . 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 32634 . 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 32634 . 2 (𝜑 → Σ*𝑘𝐴𝐷 = Σ*𝑘 ∈ ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅})𝐷)
203191, 200, 2023eqtr4d 2786 1 (𝜑 → Σ*𝑦 ∈ ran (𝑘𝐴𝐵)𝐶 = Σ*𝑘𝐴𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cun 3908  cin 3909  wss 3910  c0 4282  {csn 4586  Disj wdisj 5070   class class class wbr 5105  cmpt 5188  ran crn 5634  (class class class)co 7357  0cc0 11051  +∞cpnf 11186  *cxr 11188  cle 11190   +𝑒 cxad 13031  [,]cicc 13267  Σ*cesum 32626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950  df-sin 15952  df-cos 15953  df-pi 15955  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-ordt 17383  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-ps 18455  df-tsr 18456  df-plusf 18496  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-subrg 20220  df-abv 20276  df-lmod 20324  df-scaf 20325  df-sra 20633  df-rgmod 20634  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-tmd 23423  df-tgp 23424  df-tsms 23478  df-trg 23511  df-xms 23673  df-ms 23674  df-tms 23675  df-nm 23938  df-ngp 23939  df-nrg 23941  df-nlm 23942  df-ii 24240  df-cncf 24241  df-limc 25230  df-dv 25231  df-log 25912  df-esum 32627
This theorem is referenced by:  carsggect  32918  carsgclctunlem2  32919  pmeasadd  32925
  Copyright terms: Public domain W3C validator