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Theorem gsummpt2d 30615
Description: Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 19023. (Contributed by Thierry Arnoux, 27-Apr-2020.)
Hypotheses
Ref Expression
gsummpt2d.c 𝑧𝐶
gsummpt2d.0 𝑦𝜑
gsummpt2d.b 𝐵 = (Base‘𝑊)
gsummpt2d.1 (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
gsummpt2d.r (𝜑 → Rel 𝐴)
gsummpt2d.2 (𝜑𝐴 ∈ Fin)
gsummpt2d.m (𝜑𝑊 ∈ CMnd)
gsummpt2d.3 ((𝜑𝑥𝐴) → 𝐶𝐵)
Assertion
Ref Expression
gsummpt2d (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑦,𝐶   𝑥,𝐷   𝑥,𝑊,𝑦   𝜑,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑦)   𝐶(𝑥,𝑧)   𝐷(𝑦,𝑧)   𝑊(𝑧)

Proof of Theorem gsummpt2d
StepHypRef Expression
1 gsummpt2d.b . . 3 𝐵 = (Base‘𝑊)
2 eqid 2821 . . 3 (0g𝑊) = (0g𝑊)
3 gsummpt2d.m . . 3 (𝜑𝑊 ∈ CMnd)
4 gsummpt2d.2 . . 3 (𝜑𝐴 ∈ Fin)
54dmexd 7603 . . 3 (𝜑 → dom 𝐴 ∈ V)
6 gsummpt2d.3 . . 3 ((𝜑𝑥𝐴) → 𝐶𝐵)
7 gsummpt2d.r . . . 4 (𝜑 → Rel 𝐴)
8 1stdm 7730 . . . 4 ((Rel 𝐴𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
97, 8sylan 580 . . 3 ((𝜑𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
10 fo1st 7700 . . . . . 6 1st :V–onto→V
11 fofn 6586 . . . . . 6 (1st :V–onto→V → 1st Fn V)
12 dffn5 6718 . . . . . . 7 (1st Fn V ↔ 1st = (𝑥 ∈ V ↦ (1st𝑥)))
1312biimpi 217 . . . . . 6 (1st Fn V → 1st = (𝑥 ∈ V ↦ (1st𝑥)))
1410, 11, 13mp2b 10 . . . . 5 1st = (𝑥 ∈ V ↦ (1st𝑥))
1514reseq1i 5843 . . . 4 (1st𝐴) = ((𝑥 ∈ V ↦ (1st𝑥)) ↾ 𝐴)
16 ssv 3990 . . . . 5 𝐴 ⊆ V
17 resmpt 5899 . . . . 5 (𝐴 ⊆ V → ((𝑥 ∈ V ↦ (1st𝑥)) ↾ 𝐴) = (𝑥𝐴 ↦ (1st𝑥)))
1816, 17ax-mp 5 . . . 4 ((𝑥 ∈ V ↦ (1st𝑥)) ↾ 𝐴) = (𝑥𝐴 ↦ (1st𝑥))
1915, 18eqtri 2844 . . 3 (1st𝐴) = (𝑥𝐴 ↦ (1st𝑥))
201, 2, 3, 4, 5, 6, 9, 19gsummpt2co 30614 . 2 (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)))))
21 gsummpt2d.0 . . . 4 𝑦𝜑
223adantr 481 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → 𝑊 ∈ CMnd)
234adantr 481 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → 𝐴 ∈ Fin)
24 imaexg 7608 . . . . . . 7 (𝐴 ∈ Fin → (𝐴 “ {𝑦}) ∈ V)
2523, 24syl 17 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (𝐴 “ {𝑦}) ∈ V)
26 gsummpt2d.1 . . . . . . . . . 10 (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
2726adantl 482 . . . . . . . . 9 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐶 = 𝐷)
28 simp-4l 779 . . . . . . . . . 10 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝜑)
29 simplr 765 . . . . . . . . . 10 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝑥𝐴)
3028, 29, 6syl2anc 584 . . . . . . . . 9 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐶𝐵)
3127, 30eqeltrrd 2914 . . . . . . . 8 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐷𝐵)
32 vex 3498 . . . . . . . . . . . 12 𝑦 ∈ V
33 vex 3498 . . . . . . . . . . . 12 𝑧 ∈ V
3432, 33elimasn 5948 . . . . . . . . . . 11 (𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴)
3534biimpi 217 . . . . . . . . . 10 (𝑧 ∈ (𝐴 “ {𝑦}) → ⟨𝑦, 𝑧⟩ ∈ 𝐴)
3635adantl 482 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ 𝐴)
37 simpr 485 . . . . . . . . . 10 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝑥 = ⟨𝑦, 𝑧⟩)
3837eqeq1d 2823 . . . . . . . . 9 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩))
39 eqidd 2822 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
4036, 38, 39rspcedvd 3625 . . . . . . . 8 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ∃𝑥𝐴 𝑥 = ⟨𝑦, 𝑧⟩)
4131, 40r19.29a 3289 . . . . . . 7 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → 𝐷𝐵)
4241fmpttd 6872 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷):(𝐴 “ {𝑦})⟶𝐵)
43 eqid 2821 . . . . . . 7 (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) = (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)
44 imafi2 30374 . . . . . . . . 9 (𝐴 ∈ Fin → (𝐴 “ {𝑦}) ∈ Fin)
454, 44syl 17 . . . . . . . 8 (𝜑 → (𝐴 “ {𝑦}) ∈ Fin)
4645adantr 481 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (𝐴 “ {𝑦}) ∈ Fin)
47 fvex 6677 . . . . . . . 8 (0g𝑊) ∈ V
4847a1i 11 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (0g𝑊) ∈ V)
4943, 46, 41, 48fsuppmptdm 8833 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) finSupp (0g𝑊))
50 2ndconst 7787 . . . . . . . 8 (𝑦 ∈ dom 𝐴 → (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
5150adantl 482 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
52 1stpreimas 30368 . . . . . . . . . 10 ((Rel 𝐴𝑦 ∈ dom 𝐴) → ((1st𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
537, 52sylan 580 . . . . . . . . 9 ((𝜑𝑦 ∈ dom 𝐴) → ((1st𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
5453reseq2d 5847 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ((1st𝐴) “ {𝑦})) = (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))))
55 f1oeq1 6598 . . . . . . . 8 ((2nd ↾ ((1st𝐴) “ {𝑦})) = (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))) → ((2nd ↾ ((1st𝐴) “ {𝑦})):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}) ↔ (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦})))
5654, 55syl 17 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → ((2nd ↾ ((1st𝐴) “ {𝑦})):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}) ↔ (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦})))
5751, 56mpbird 258 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ((1st𝐴) “ {𝑦})):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
581, 2, 22, 25, 42, 49, 57gsumf1o 18967 . . . . 5 ((𝜑𝑦 ∈ dom 𝐴) → (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)) = (𝑊 Σg ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦})))))
59 simpr 485 . . . . . . . . . . 11 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → 𝑥 ∈ ((1st𝐴) “ {𝑦}))
6053adantr 481 . . . . . . . . . . 11 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → ((1st𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
6159, 60eleqtrd 2915 . . . . . . . . . 10 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → 𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})))
62 xp2nd 7713 . . . . . . . . . 10 (𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) → (2nd𝑥) ∈ (𝐴 “ {𝑦}))
6361, 62syl 17 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → (2nd𝑥) ∈ (𝐴 “ {𝑦}))
6463ralrimiva 3182 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → ∀𝑥 ∈ ((1st𝐴) “ {𝑦})(2nd𝑥) ∈ (𝐴 “ {𝑦}))
65 fo2nd 7701 . . . . . . . . . . . 12 2nd :V–onto→V
66 fofn 6586 . . . . . . . . . . . 12 (2nd :V–onto→V → 2nd Fn V)
67 dffn5 6718 . . . . . . . . . . . . 13 (2nd Fn V ↔ 2nd = (𝑥 ∈ V ↦ (2nd𝑥)))
6867biimpi 217 . . . . . . . . . . . 12 (2nd Fn V → 2nd = (𝑥 ∈ V ↦ (2nd𝑥)))
6965, 66, 68mp2b 10 . . . . . . . . . . 11 2nd = (𝑥 ∈ V ↦ (2nd𝑥))
7069reseq1i 5843 . . . . . . . . . 10 (2nd ↾ ((1st𝐴) “ {𝑦})) = ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ((1st𝐴) “ {𝑦}))
71 ssv 3990 . . . . . . . . . . 11 ((1st𝐴) “ {𝑦}) ⊆ V
72 resmpt 5899 . . . . . . . . . . 11 (((1st𝐴) “ {𝑦}) ⊆ V → ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ((1st𝐴) “ {𝑦})) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥)))
7371, 72ax-mp 5 . . . . . . . . . 10 ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ((1st𝐴) “ {𝑦})) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥))
7470, 73eqtri 2844 . . . . . . . . 9 (2nd ↾ ((1st𝐴) “ {𝑦})) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥))
7574a1i 11 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ((1st𝐴) “ {𝑦})) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥)))
76 eqidd 2822 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) = (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))
7764, 75, 76fmptcos 6886 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦}))) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥) / 𝑧𝐷))
78 nfv 1906 . . . . . . . . 9 𝑧((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦}))
79 gsummpt2d.c . . . . . . . . . 10 𝑧𝐶
8079a1i 11 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → 𝑧𝐶)
8161adantr 481 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})))
82 xp1st 7712 . . . . . . . . . . . . . 14 (𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) → (1st𝑥) ∈ {𝑦})
8381, 82syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → (1st𝑥) ∈ {𝑦})
84 fvex 6677 . . . . . . . . . . . . . 14 (1st𝑥) ∈ V
8584elsn 4574 . . . . . . . . . . . . 13 ((1st𝑥) ∈ {𝑦} ↔ (1st𝑥) = 𝑦)
8683, 85sylib 219 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → (1st𝑥) = 𝑦)
87 simpr 485 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝑧 = (2nd𝑥))
8887eqcomd 2827 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → (2nd𝑥) = 𝑧)
89 eqopi 7716 . . . . . . . . . . . 12 ((𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) ∧ ((1st𝑥) = 𝑦 ∧ (2nd𝑥) = 𝑧)) → 𝑥 = ⟨𝑦, 𝑧⟩)
9081, 86, 88, 89syl12anc 832 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝑥 = ⟨𝑦, 𝑧⟩)
9190, 26syl 17 . . . . . . . . . 10 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝐶 = 𝐷)
9291eqcomd 2827 . . . . . . . . 9 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝐷 = 𝐶)
9378, 80, 63, 92csbiedf 3912 . . . . . . . 8 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → (2nd𝑥) / 𝑧𝐷 = 𝐶)
9493mpteq2dva 5153 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥) / 𝑧𝐷) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶))
9577, 94eqtrd 2856 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦}))) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶))
9695oveq2d 7161 . . . . 5 ((𝜑𝑦 ∈ dom 𝐴) → (𝑊 Σg ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦})))) = (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)))
9758, 96eqtr2d 2857 . . . 4 ((𝜑𝑦 ∈ dom 𝐴) → (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)) = (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))
9821, 97mpteq2da 5152 . . 3 (𝜑 → (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶))) = (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))))
9998oveq2d 7161 . 2 (𝜑 → (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)))) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))))
10020, 99eqtrd 2856 1 (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wnf 1775  wcel 2105  wnfc 2961  Vcvv 3495  csb 3882  wss 3935  {csn 4559  cop 4565  cmpt 5138   × cxp 5547  ccnv 5548  dom cdm 5549  cres 5551  cima 5552  ccom 5553  Rel wrel 5554   Fn wfn 6344  ontowfo 6347  1-1-ontowf1o 6348  cfv 6349  (class class class)co 7145  1st c1st 7678  2nd c2nd 7679  Fincfn 8498  Basecbs 16473  0gc0g 16703   Σg cgsu 16704  CMndccmn 18837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7569  df-1st 7680  df-2nd 7681  df-supp 7822  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-fsupp 8823  df-oi 8963  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-n0 11887  df-z 11971  df-uz 12233  df-fz 12883  df-fzo 13024  df-seq 13360  df-hash 13681  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-0g 16705  df-gsum 16706  df-mre 16847  df-mrc 16848  df-acs 16850  df-mgm 17842  df-sgrp 17891  df-mnd 17902  df-submnd 17947  df-mulg 18165  df-cntz 18387  df-cmn 18839
This theorem is referenced by:  esum2d  31252
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