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Theorem gsummpt2d 30680
Description: Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 19084. (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 2819 . . 3 (0g𝑊) = (0g𝑊)
3 gsummpt2d.m . . 3 (𝜑𝑊 ∈ CMnd)
4 gsummpt2d.2 . . 3 (𝜑𝐴 ∈ Fin)
54dmexd 7607 . . 3 (𝜑 → dom 𝐴 ∈ V)
6 gsummpt2d.3 . . 3 ((𝜑𝑥𝐴) → 𝐶𝐵)
7 gsummpt2d.r . . . 4 (𝜑 → Rel 𝐴)
8 1stdm 7731 . . . 4 ((Rel 𝐴𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
97, 8sylan 582 . . 3 ((𝜑𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
10 fo1st 7701 . . . . . 6 1st :V–onto→V
11 fofn 6585 . . . . . 6 (1st :V–onto→V → 1st Fn V)
12 dffn5 6717 . . . . . . 7 (1st Fn V ↔ 1st = (𝑥 ∈ V ↦ (1st𝑥)))
1312biimpi 218 . . . . . 6 (1st Fn V → 1st = (𝑥 ∈ V ↦ (1st𝑥)))
1410, 11, 13mp2b 10 . . . . 5 1st = (𝑥 ∈ V ↦ (1st𝑥))
1514reseq1i 5842 . . . 4 (1st𝐴) = ((𝑥 ∈ V ↦ (1st𝑥)) ↾ 𝐴)
16 ssv 3989 . . . . 5 𝐴 ⊆ V
17 resmpt 5898 . . . . 5 (𝐴 ⊆ V → ((𝑥 ∈ V ↦ (1st𝑥)) ↾ 𝐴) = (𝑥𝐴 ↦ (1st𝑥)))
1816, 17ax-mp 5 . . . 4 ((𝑥 ∈ V ↦ (1st𝑥)) ↾ 𝐴) = (𝑥𝐴 ↦ (1st𝑥))
1915, 18eqtri 2842 . . 3 (1st𝐴) = (𝑥𝐴 ↦ (1st𝑥))
201, 2, 3, 4, 5, 6, 9, 19gsummpt2co 30679 . 2 (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)))))
21 gsummpt2d.0 . . . 4 𝑦𝜑
223adantr 483 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → 𝑊 ∈ CMnd)
234adantr 483 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → 𝐴 ∈ Fin)
24 imaexg 7612 . . . . . . 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 484 . . . . . . . . 9 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐶 = 𝐷)
28 simp-4l 781 . . . . . . . . . 10 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝜑)
29 simplr 767 . . . . . . . . . 10 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝑥𝐴)
3028, 29, 6syl2anc 586 . . . . . . . . 9 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐶𝐵)
3127, 30eqeltrrd 2912 . . . . . . . 8 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐷𝐵)
32 vex 3496 . . . . . . . . . . . 12 𝑦 ∈ V
33 vex 3496 . . . . . . . . . . . 12 𝑧 ∈ V
3432, 33elimasn 5947 . . . . . . . . . . 11 (𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴)
3534biimpi 218 . . . . . . . . . 10 (𝑧 ∈ (𝐴 “ {𝑦}) → ⟨𝑦, 𝑧⟩ ∈ 𝐴)
3635adantl 484 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ 𝐴)
37 simpr 487 . . . . . . . . . 10 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝑥 = ⟨𝑦, 𝑧⟩)
3837eqeq1d 2821 . . . . . . . . 9 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩))
39 eqidd 2820 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
4036, 38, 39rspcedvd 3624 . . . . . . . 8 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ∃𝑥𝐴 𝑥 = ⟨𝑦, 𝑧⟩)
4131, 40r19.29a 3287 . . . . . . 7 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → 𝐷𝐵)
4241fmpttd 6872 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷):(𝐴 “ {𝑦})⟶𝐵)
43 eqid 2819 . . . . . . 7 (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) = (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)
44 imafi2 30439 . . . . . . . . 9 (𝐴 ∈ Fin → (𝐴 “ {𝑦}) ∈ Fin)
454, 44syl 17 . . . . . . . 8 (𝜑 → (𝐴 “ {𝑦}) ∈ Fin)
4645adantr 483 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (𝐴 “ {𝑦}) ∈ Fin)
47 fvex 6676 . . . . . . . 8 (0g𝑊) ∈ V
4847a1i 11 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (0g𝑊) ∈ V)
4943, 46, 41, 48fsuppmptdm 8836 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) finSupp (0g𝑊))
50 2ndconst 7788 . . . . . . . 8 (𝑦 ∈ dom 𝐴 → (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
5150adantl 484 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
52 1stpreimas 30433 . . . . . . . . . 10 ((Rel 𝐴𝑦 ∈ dom 𝐴) → ((1st𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
537, 52sylan 582 . . . . . . . . 9 ((𝜑𝑦 ∈ dom 𝐴) → ((1st𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
5453reseq2d 5846 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ((1st𝐴) “ {𝑦})) = (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))))
55 f1oeq1 6597 . . . . . . . 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 259 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ((1st𝐴) “ {𝑦})):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
581, 2, 22, 25, 42, 49, 57gsumf1o 19028 . . . . 5 ((𝜑𝑦 ∈ dom 𝐴) → (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)) = (𝑊 Σg ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦})))))
59 simpr 487 . . . . . . . . . . 11 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → 𝑥 ∈ ((1st𝐴) “ {𝑦}))
6053adantr 483 . . . . . . . . . . 11 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → ((1st𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
6159, 60eleqtrd 2913 . . . . . . . . . 10 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → 𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})))
62 xp2nd 7714 . . . . . . . . . 10 (𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) → (2nd𝑥) ∈ (𝐴 “ {𝑦}))
6361, 62syl 17 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → (2nd𝑥) ∈ (𝐴 “ {𝑦}))
6463ralrimiva 3180 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → ∀𝑥 ∈ ((1st𝐴) “ {𝑦})(2nd𝑥) ∈ (𝐴 “ {𝑦}))
65 fo2nd 7702 . . . . . . . . . . . 12 2nd :V–onto→V
66 fofn 6585 . . . . . . . . . . . 12 (2nd :V–onto→V → 2nd Fn V)
67 dffn5 6717 . . . . . . . . . . . . 13 (2nd Fn V ↔ 2nd = (𝑥 ∈ V ↦ (2nd𝑥)))
6867biimpi 218 . . . . . . . . . . . 12 (2nd Fn V → 2nd = (𝑥 ∈ V ↦ (2nd𝑥)))
6965, 66, 68mp2b 10 . . . . . . . . . . 11 2nd = (𝑥 ∈ V ↦ (2nd𝑥))
7069reseq1i 5842 . . . . . . . . . 10 (2nd ↾ ((1st𝐴) “ {𝑦})) = ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ((1st𝐴) “ {𝑦}))
71 ssv 3989 . . . . . . . . . . 11 ((1st𝐴) “ {𝑦}) ⊆ V
72 resmpt 5898 . . . . . . . . . . 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 2842 . . . . . . . . 9 (2nd ↾ ((1st𝐴) “ {𝑦})) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥))
7574a1i 11 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ((1st𝐴) “ {𝑦})) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥)))
76 eqidd 2820 . . . . . . . 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 1908 . . . . . . . . 9 𝑧((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦}))
79 gsummpt2d.c . . . . . . . . . 10 𝑧𝐶
8079a1i 11 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → 𝑧𝐶)
8161adantr 483 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})))
82 xp1st 7713 . . . . . . . . . . . . . 14 (𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) → (1st𝑥) ∈ {𝑦})
8381, 82syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → (1st𝑥) ∈ {𝑦})
84 fvex 6676 . . . . . . . . . . . . . 14 (1st𝑥) ∈ V
8584elsn 4574 . . . . . . . . . . . . 13 ((1st𝑥) ∈ {𝑦} ↔ (1st𝑥) = 𝑦)
8683, 85sylib 220 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → (1st𝑥) = 𝑦)
87 simpr 487 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝑧 = (2nd𝑥))
8887eqcomd 2825 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → (2nd𝑥) = 𝑧)
89 eqopi 7717 . . . . . . . . . . . 12 ((𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) ∧ ((1st𝑥) = 𝑦 ∧ (2nd𝑥) = 𝑧)) → 𝑥 = ⟨𝑦, 𝑧⟩)
9081, 86, 88, 89syl12anc 834 . . . . . . . . . . 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 2825 . . . . . . . . 9 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝐷 = 𝐶)
9378, 80, 63, 92csbiedf 3911 . . . . . . . 8 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → (2nd𝑥) / 𝑧𝐷 = 𝐶)
9493mpteq2dva 5152 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥) / 𝑧𝐷) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶))
9577, 94eqtrd 2854 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦}))) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶))
9695oveq2d 7164 . . . . 5 ((𝜑𝑦 ∈ dom 𝐴) → (𝑊 Σg ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦})))) = (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)))
9758, 96eqtr2d 2855 . . . 4 ((𝜑𝑦 ∈ dom 𝐴) → (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)) = (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))
9821, 97mpteq2da 5151 . . 3 (𝜑 → (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶))) = (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))))
9998oveq2d 7164 . 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 2854 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 208  wa 398   = wceq 1530  wnf 1777  wcel 2107  wnfc 2959  Vcvv 3493  csb 3881  wss 3934  {csn 4559  cop 4565  cmpt 5137   × cxp 5546  ccnv 5547  dom cdm 5548  cres 5550  cima 5551  ccom 5552  Rel wrel 5553   Fn wfn 6343  ontowfo 6346  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7148  1st c1st 7679  2nd c2nd 7680  Fincfn 8501  Basecbs 16475  0gc0g 16705   Σg cgsu 16706  CMndccmn 18898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-om 7573  df-1st 7681  df-2nd 7682  df-supp 7823  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885  df-fzo 13026  df-seq 13362  df-hash 13683  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-0g 16707  df-gsum 16708  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-mulg 18217  df-cntz 18439  df-cmn 18900
This theorem is referenced by:  esum2d  31345
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