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Theorem esumrnmpt2 30258
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 3152 . . . . 5 𝑘{𝑘𝐴 ∣ ¬ 𝐵 = ∅}
2 esumrnmpt2.1 . . . . 5 (𝑦 = 𝐵𝐶 = 𝐷)
3 esumrnmpt2.2 . . . . . 6 (𝜑𝐴𝑉)
4 ssrab2 3720 . . . . . . 7 {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴
54a1i 11 . . . . . 6 (𝜑 → {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴)
63, 5ssexd 4838 . . . . 5 (𝜑 → {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ∈ V)
75sselda 3636 . . . . . 6 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → 𝑘𝐴)
8 esumrnmpt2.3 . . . . . 6 ((𝜑𝑘𝐴) → 𝐷 ∈ (0[,]+∞))
97, 8syldan 486 . . . . 5 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
10 esumrnmpt2.4 . . . . . . 7 ((𝜑𝑘𝐴) → 𝐵𝑊)
117, 10syldan 486 . . . . . 6 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵𝑊)
12 rabid 3145 . . . . . . . . 9 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↔ (𝑘𝐴 ∧ ¬ 𝐵 = ∅))
1312simprbi 479 . . . . . . . 8 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} → ¬ 𝐵 = ∅)
1413adantl 481 . . . . . . 7 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → ¬ 𝐵 = ∅)
15 elsng 4224 . . . . . . . 8 (𝐵𝑊 → (𝐵 ∈ {∅} ↔ 𝐵 = ∅))
1611, 15syl 17 . . . . . . 7 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → (𝐵 ∈ {∅} ↔ 𝐵 = ∅))
1714, 16mtbird 314 . . . . . 6 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → ¬ 𝐵 ∈ {∅})
1811, 17eldifd 3618 . . . . 5 ((𝜑𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵 ∈ (𝑊 ∖ {∅}))
19 esumrnmpt2.6 . . . . . 6 (𝜑Disj 𝑘𝐴 𝐵)
20 nfcv 2793 . . . . . . 7 𝑘𝐴
211, 20disjss1f 29512 . . . . . 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 30242 . . . 4 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 = Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
24 nfv 1883 . . . . . . . . . . 11 𝑦(𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅)
25 snex 4938 . . . . . . . . . . . 12 {∅} ∈ V
2625a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → {∅} ∈ V)
27 velsn 4226 . . . . . . . . . . . . . . 15 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
2827biimpi 206 . . . . . . . . . . . . . 14 (𝑦 ∈ {∅} → 𝑦 = ∅)
2928adantl 481 . . . . . . . . . . . . 13 (((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝑦 = ∅)
30 nfv 1883 . . . . . . . . . . . . . . . 16 𝑘𝜑
31 nfre1 3034 . . . . . . . . . . . . . . . 16 𝑘𝑘𝐴 𝐵 = ∅
3230, 31nfan 1868 . . . . . . . . . . . . . . 15 𝑘(𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅)
33 nfv 1883 . . . . . . . . . . . . . . 15 𝑘 𝑦 = ∅
3432, 33nfan 1868 . . . . . . . . . . . . . 14 𝑘((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅)
35 nfv 1883 . . . . . . . . . . . . . 14 𝑘 𝐶 = 0
36 simpllr 815 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝑦 = ∅)
37 simpr 476 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝐵 = ∅)
3836, 37eqtr4d 2688 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝑦 = 𝐵)
3938, 2syl 17 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝐶 = 𝐷)
40 simp-4l 823 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝜑)
41 simplr 807 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝑘𝐴)
42 esumrnmpt2.5 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐴) ∧ 𝐵 = ∅) → 𝐷 = 0)
4340, 41, 37, 42syl21anc 1365 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝐷 = 0)
4439, 43eqtrd 2685 . . . . . . . . . . . . . 14 (((((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘𝐴) ∧ 𝐵 = ∅) → 𝐶 = 0)
45 simplr 807 . . . . . . . . . . . . . 14 (((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → ∃𝑘𝐴 𝐵 = ∅)
4634, 35, 44, 45r19.29af2 3104 . . . . . . . . . . . . 13 (((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → 𝐶 = 0)
4729, 46syldan 486 . . . . . . . . . . . 12 (((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 = 0)
48 0e0iccpnf 12321 . . . . . . . . . . . 12 0 ∈ (0[,]+∞)
4947, 48syl6eqel 2738 . . . . . . . . . . 11 (((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 ∈ (0[,]+∞))
50 nfcv 2793 . . . . . . . . . . . . . . . . 17 𝑘𝑦
51 nfmpt1 4780 . . . . . . . . . . . . . . . . . 18 𝑘(𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)
5251nfrn 5400 . . . . . . . . . . . . . . . . 17 𝑘ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)
5350, 52nfel 2806 . . . . . . . . . . . . . . . 16 𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)
5430, 53nfan 1868 . . . . . . . . . . . . . . 15 𝑘(𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵))
55 simpr 476 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
56 rabid 3145 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↔ (𝑘𝐴𝐵 = ∅))
5756simprbi 479 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ {𝑘𝐴𝐵 = ∅} → 𝐵 = ∅)
5857ad2antlr 763 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐵 = ∅)
5955, 58eqtrd 2685 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = ∅)
6059, 27sylibr 224 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 ∈ {∅})
61 vex 3234 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
62 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)
6362elrnmpt 5404 . . . . . . . . . . . . . . . . . 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 206 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) → ∃𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝑦 = 𝐵)
6665adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) → ∃𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝑦 = 𝐵)
6754, 60, 66r19.29af 3105 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) → 𝑦 ∈ {∅})
6867ex 449 . . . . . . . . . . . . 13 (𝜑 → (𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) → 𝑦 ∈ {∅}))
6968ssrdv 3642 . . . . . . . . . . . 12 (𝜑 → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ⊆ {∅})
7069adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ⊆ {∅})
7124, 26, 49, 70esummono 30244 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ Σ*𝑦 ∈ {∅}𝐶)
72 0ex 4823 . . . . . . . . . . . 12 ∅ ∈ V
7372a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → ∅ ∈ V)
7448a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → 0 ∈ (0[,]+∞))
7546, 73, 74esumsn 30255 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → Σ*𝑦 ∈ {∅}𝐶 = 0)
7671, 75breqtrd 4711 . . . . . . . . 9 ((𝜑 ∧ ∃𝑘𝐴 𝐵 = ∅) → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
77 simpr 476 . . . . . . . . . 10 ((𝜑 ∧ ¬ ∃𝑘𝐴 𝐵 = ∅) → ¬ ∃𝑘𝐴 𝐵 = ∅)
78 nfv 1883 . . . . . . . . . . . . 13 𝑦 ¬ ∃𝑘𝐴 𝐵 = ∅
7931nfn 1824 . . . . . . . . . . . . . . . . 17 𝑘 ¬ ∃𝑘𝐴 𝐵 = ∅
80 nfrab1 3152 . . . . . . . . . . . . . . . . 17 𝑘{𝑘𝐴𝐵 = ∅}
81 nfcv 2793 . . . . . . . . . . . . . . . . 17 𝑘
82 rabn0 3991 . . . . . . . . . . . . . . . . . . 19 ({𝑘𝐴𝐵 = ∅} ≠ ∅ ↔ ∃𝑘𝐴 𝐵 = ∅)
8382biimpi 206 . . . . . . . . . . . . . . . . . 18 ({𝑘𝐴𝐵 = ∅} ≠ ∅ → ∃𝑘𝐴 𝐵 = ∅)
8483necon1bi 2851 . . . . . . . . . . . . . . . . 17 (¬ ∃𝑘𝐴 𝐵 = ∅ → {𝑘𝐴𝐵 = ∅} = ∅)
85 eqid 2651 . . . . . . . . . . . . . . . . . 18 𝐵 = 𝐵
8685a1i 11 . . . . . . . . . . . . . . . . 17 (¬ ∃𝑘𝐴 𝐵 = ∅ → 𝐵 = 𝐵)
8779, 80, 81, 84, 86mpteq12df 4768 . . . . . . . . . . . . . . . 16 (¬ ∃𝑘𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ ∅ ↦ 𝐵))
88 mpt0 6059 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ∅ ↦ 𝐵) = ∅
8987, 88syl6eq 2701 . . . . . . . . . . . . . . 15 (¬ ∃𝑘𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) = ∅)
9089rneqd 5385 . . . . . . . . . . . . . 14 (¬ ∃𝑘𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) = ran ∅)
91 rn0 5409 . . . . . . . . . . . . . 14 ran ∅ = ∅
9290, 91syl6eq 2701 . . . . . . . . . . . . 13 (¬ ∃𝑘𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) = ∅)
9378, 92esumeq1d 30225 . . . . . . . . . . . 12 (¬ ∃𝑘𝐴 𝐵 = ∅ → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 = Σ*𝑦 ∈ ∅𝐶)
94 esumnul 30238 . . . . . . . . . . . 12 Σ*𝑦 ∈ ∅𝐶 = 0
9593, 94syl6eq 2701 . . . . . . . . . . 11 (¬ ∃𝑘𝐴 𝐵 = ∅ → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
96 0le0 11148 . . . . . . . . . . 11 0 ≤ 0
9795, 96syl6eqbr 4724 . . . . . . . . . 10 (¬ ∃𝑘𝐴 𝐵 = ∅ → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
9877, 97syl 17 . . . . . . . . 9 ((𝜑 ∧ ¬ ∃𝑘𝐴 𝐵 = ∅) → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
9976, 98pm2.61dan 849 . . . . . . . 8 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
100 ssrab2 3720 . . . . . . . . . . . . 13 {𝑘𝐴𝐵 = ∅} ⊆ 𝐴
101100a1i 11 . . . . . . . . . . . 12 (𝜑 → {𝑘𝐴𝐵 = ∅} ⊆ 𝐴)
1023, 101ssexd 4838 . . . . . . . . . . 11 (𝜑 → {𝑘𝐴𝐵 = ∅} ∈ V)
10380mptexgf 6526 . . . . . . . . . . 11 ({𝑘𝐴𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∈ V)
104 rnexg 7140 . . . . . . . . . . 11 ((𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∈ V → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∈ V)
105102, 103, 1043syl 18 . . . . . . . . . 10 (𝜑 → ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∈ V)
1062adantl 481 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
107 simplll 813 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝜑)
108101sselda 3636 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝑘𝐴)
109108adantlr 751 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝑘𝐴)
110109adantr 480 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘𝐴)
111107, 110, 8syl2anc 694 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
112106, 111eqeltrd 2730 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
11354, 112, 66r19.29af 3105 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
114113ralrimiva 2995 . . . . . . . . . 10 (𝜑 → ∀𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
115 nfcv 2793 . . . . . . . . . . 11 𝑦ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)
116115esumcl 30220 . . . . . . . . . 10 ((ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∈ V ∧ ∀𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞)) → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
117105, 114, 116syl2anc 694 . . . . . . . . 9 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
118 elxrge0 12319 . . . . . . . . . 10 *𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞) ↔ (Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ* ∧ 0 ≤ Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶))
119118simprbi 479 . . . . . . . . 9 *𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞) → 0 ≤ Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶)
120117, 119syl 17 . . . . . . . 8 (𝜑 → 0 ≤ Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶)
12199, 120jca 553 . . . . . . 7 (𝜑 → (Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0 ∧ 0 ≤ Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶))
122 iccssxr 12294 . . . . . . . . 9 (0[,]+∞) ⊆ ℝ*
123122, 117sseldi 3634 . . . . . . . 8 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ*)
124122, 48sselii 3633 . . . . . . . . 9 0 ∈ ℝ*
125124a1i 11 . . . . . . . 8 (𝜑 → 0 ∈ ℝ*)
126 xrletri3 12023 . . . . . . . 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)))
127123, 125, 126syl2anc 694 . . . . . . 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)))
128121, 127mpbird 247 . . . . . 6 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
129128oveq1d 6705 . . . . 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))
1309ralrimiva 2995 . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
1311esumcl 30220 . . . . . . . . 9 (({𝑘𝐴 ∣ ¬ 𝐵 = ∅} ∈ V ∧ ∀𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞)) → Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
1326, 130, 131syl2anc 694 . . . . . . . 8 (𝜑 → Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
133122, 132sseldi 3634 . . . . . . 7 (𝜑 → Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ ℝ*)
13423, 133eqeltrd 2730 . . . . . 6 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ*)
135 xaddid2 12111 . . . . . 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)
136134, 135syl 17 . . . . 5 (𝜑 → (0 +𝑒 Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶) = Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶)
137129, 136eqtrd 2685 . . . 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)
138 simpl 472 . . . . . . . . . 10 ((𝜑𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝜑)
13957adantl 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝐵 = ∅)
140138, 108, 139, 42syl21anc 1365 . . . . . . . . 9 ((𝜑𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝐷 = 0)
141140ralrimiva 2995 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 = 0)
14230, 141esumeq2d 30227 . . . . . . 7 (𝜑 → Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 = Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}0)
14380esum0 30239 . . . . . . . 8 ({𝑘𝐴𝐵 = ∅} ∈ V → Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}0 = 0)
144102, 143syl 17 . . . . . . 7 (𝜑 → Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}0 = 0)
145142, 144eqtrd 2685 . . . . . 6 (𝜑 → Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 = 0)
146145oveq1d 6705 . . . . 5 (𝜑 → (Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = (0 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷))
147 xaddid2 12111 . . . . . 6 *𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ ℝ* → (0 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
148133, 147syl 17 . . . . 5 (𝜑 → (0 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
149146, 148eqtrd 2685 . . . 4 (𝜑 → (Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
15023, 137, 1493eqtr4d 2695 . . 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))
151 nfv 1883 . . . 4 𝑦𝜑
152 nfcv 2793 . . . 4 𝑦ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
1531mptexgf 6526 . . . . 5 ({𝑘𝐴 ∣ ¬ 𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
154 rnexg 7140 . . . . 5 ((𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V → ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
1556, 153, 1543syl 18 . . . 4 (𝜑 → ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
156 ssrin 3871 . . . . . . 7 (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ⊆ {∅} → (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ({∅} ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)))
15769, 156syl 17 . . . . . 6 (𝜑 → (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ({∅} ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)))
158 incom 3838 . . . . . . 7 (ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ({∅} ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
15913neqned 2830 . . . . . . . . . . . 12 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} → 𝐵 ≠ ∅)
160159necomd 2878 . . . . . . . . . . 11 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} → ∅ ≠ 𝐵)
161160neneqd 2828 . . . . . . . . . 10 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} → ¬ ∅ = 𝐵)
162161nrex 3029 . . . . . . . . 9 ¬ ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵
163 eqid 2651 . . . . . . . . . . 11 (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
164163elrnmpt 5404 . . . . . . . . . 10 (∅ ∈ V → (∅ ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵))
16572, 164ax-mp 5 . . . . . . . . 9 (∅ ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵)
166162, 165mtbir 312 . . . . . . . 8 ¬ ∅ ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
167 disjsn 4278 . . . . . . . 8 ((ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
168166, 167mpbir 221 . . . . . . 7 (ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅
169158, 168eqtr3i 2675 . . . . . 6 ({∅} ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅
170157, 169syl6sseq 3684 . . . . 5 (𝜑 → (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ∅)
171 ss0 4007 . . . . 5 ((ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ∅ → (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅)
172170, 171syl 17 . . . 4 (𝜑 → (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅)
173 nfmpt1 4780 . . . . . . . 8 𝑘(𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
174173nfrn 5400 . . . . . . 7 𝑘ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
17550, 174nfel 2806 . . . . . 6 𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
17630, 175nfan 1868 . . . . 5 𝑘(𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
1772adantl 481 . . . . . 6 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
178 simplll 813 . . . . . . 7 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝜑)
1797adantlr 751 . . . . . . . 8 (((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) → 𝑘𝐴)
180179adantr 480 . . . . . . 7 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘𝐴)
181178, 180, 8syl2anc 694 . . . . . 6 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
182177, 181eqeltrd 2730 . . . . 5 ((((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
183163elrnmpt 5404 . . . . . . . 8 (𝑦 ∈ V → (𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵))
18461, 183ax-mp 5 . . . . . . 7 (𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
185184biimpi 206 . . . . . 6 (𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) → ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
186185adantl 481 . . . . 5 ((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → ∃𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
187176, 182, 186r19.29af 3105 . . . 4 ((𝜑𝑦 ∈ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
188151, 115, 152, 105, 155, 172, 113, 187esumsplit 30243 . . 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))
189 rabnc 3995 . . . . 5 ({𝑘𝐴𝐵 = ∅} ∩ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) = ∅
190189a1i 11 . . . 4 (𝜑 → ({𝑘𝐴𝐵 = ∅} ∩ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) = ∅)
191108, 8syldan 486 . . . 4 ((𝜑𝑘 ∈ {𝑘𝐴𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
19230, 80, 1, 102, 6, 190, 191, 9esumsplit 30243 . . 3 (𝜑 → Σ*𝑘 ∈ ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅})𝐷 = (Σ*𝑘 ∈ {𝑘𝐴𝐵 = ∅}𝐷 +𝑒 Σ*𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}𝐷))
193150, 188, 1923eqtr4d 2695 . 2 (𝜑 → Σ*𝑦 ∈ (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))𝐶 = Σ*𝑘 ∈ ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅})𝐷)
194 rabxm 3994 . . . . . . . 8 𝐴 = ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅})
195194, 85mpteq12i 4775 . . . . . . 7 (𝑘𝐴𝐵) = (𝑘 ∈ ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵)
196 mptun 6063 . . . . . . 7 (𝑘 ∈ ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵) = ((𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
197195, 196eqtri 2673 . . . . . 6 (𝑘𝐴𝐵) = ((𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
198197rneqi 5384 . . . . 5 ran (𝑘𝐴𝐵) = ran ((𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
199 rnun 5576 . . . . 5 ran ((𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
200198, 199eqtri 2673 . . . 4 ran (𝑘𝐴𝐵) = (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
201200a1i 11 . . 3 (𝜑 → ran (𝑘𝐴𝐵) = (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)))
202151, 201esumeq1d 30225 . 2 (𝜑 → Σ*𝑦 ∈ ran (𝑘𝐴𝐵)𝐶 = Σ*𝑦 ∈ (ran (𝑘 ∈ {𝑘𝐴𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))𝐶)
203194a1i 11 . . 3 (𝜑𝐴 = ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅}))
20430, 203esumeq1d 30225 . 2 (𝜑 → Σ*𝑘𝐴𝐷 = Σ*𝑘 ∈ ({𝑘𝐴𝐵 = ∅} ∪ {𝑘𝐴 ∣ ¬ 𝐵 = ∅})𝐷)
205193, 202, 2043eqtr4d 2695 1 (𝜑 → Σ*𝑦 ∈ ran (𝑘𝐴𝐵)𝐶 = Σ*𝑘𝐴𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210  Disj wdisj 4652   class class class wbr 4685  cmpt 4762  ran crn 5144  (class class class)co 6690  0cc0 9974  +∞cpnf 10109  *cxr 10111  cle 10113   +𝑒 cxad 11982  [,]cicc 12216  Σ*cesum 30217
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-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-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-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
This theorem is referenced by:  carsggect  30508  carsgclctunlem2  30509  pmeasadd  30515
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