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Theorem gsumzaddlem 19919
Description: The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)
Hypotheses
Ref Expression
gsumzadd.b 𝐵 = (Base‘𝐺)
gsumzadd.0 0 = (0g𝐺)
gsumzadd.p + = (+g𝐺)
gsumzadd.z 𝑍 = (Cntz‘𝐺)
gsumzadd.g (𝜑𝐺 ∈ Mnd)
gsumzadd.a (𝜑𝐴𝑉)
gsumzadd.fn (𝜑𝐹 finSupp 0 )
gsumzadd.hn (𝜑𝐻 finSupp 0 )
gsumzaddlem.w 𝑊 = ((𝐹𝐻) supp 0 )
gsumzaddlem.f (𝜑𝐹:𝐴𝐵)
gsumzaddlem.h (𝜑𝐻:𝐴𝐵)
gsumzaddlem.1 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumzaddlem.2 (𝜑 → ran 𝐻 ⊆ (𝑍‘ran 𝐻))
gsumzaddlem.3 (𝜑 → ran (𝐹f + 𝐻) ⊆ (𝑍‘ran (𝐹f + 𝐻)))
gsumzaddlem.4 ((𝜑 ∧ (𝑥𝐴𝑘 ∈ (𝐴𝑥))) → (𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))}))
Assertion
Ref Expression
gsumzaddlem (𝜑 → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
Distinct variable groups:   𝑥,𝑘, +   0 ,𝑘,𝑥   𝑘,𝐹,𝑥   𝑘,𝐺,𝑥   𝐴,𝑘,𝑥   𝐵,𝑘,𝑥   𝑘,𝐻,𝑥   𝜑,𝑘,𝑥   𝑥,𝑉   𝑘,𝑊,𝑥   𝑘,𝑍,𝑥
Allowed substitution hint:   𝑉(𝑘)

Proof of Theorem gsumzaddlem
Dummy variables 𝑓 𝑛 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzadd.g . . . . . 6 (𝜑𝐺 ∈ Mnd)
2 gsumzadd.b . . . . . . . 8 𝐵 = (Base‘𝐺)
3 gsumzadd.0 . . . . . . . 8 0 = (0g𝐺)
42, 3mndidcl 18742 . . . . . . 7 (𝐺 ∈ Mnd → 0𝐵)
51, 4syl 17 . . . . . 6 (𝜑0𝐵)
6 gsumzadd.p . . . . . . 7 + = (+g𝐺)
72, 6, 3mndlid 18747 . . . . . 6 ((𝐺 ∈ Mnd ∧ 0𝐵) → ( 0 + 0 ) = 0 )
81, 5, 7syl2anc 582 . . . . 5 (𝜑 → ( 0 + 0 ) = 0 )
98adantr 479 . . . 4 ((𝜑𝑊 = ∅) → ( 0 + 0 ) = 0 )
10 gsumzaddlem.f . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
11 gsumzadd.a . . . . . . . 8 (𝜑𝐴𝑉)
123fvexi 6915 . . . . . . . . 9 0 ∈ V
1312a1i 11 . . . . . . . 8 (𝜑0 ∈ V)
14 gsumzaddlem.h . . . . . . . . . . 11 (𝜑𝐻:𝐴𝐵)
1514, 11fexd 7244 . . . . . . . . . 10 (𝜑𝐻 ∈ V)
1615suppun 8198 . . . . . . . . 9 (𝜑 → (𝐹 supp 0 ) ⊆ ((𝐹𝐻) supp 0 ))
17 gsumzaddlem.w . . . . . . . . 9 𝑊 = ((𝐹𝐻) supp 0 )
1816, 17sseqtrrdi 4031 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)
1910, 11, 13, 18gsumcllem 19906 . . . . . . 7 ((𝜑𝑊 = ∅) → 𝐹 = (𝑥𝐴0 ))
2019oveq2d 7440 . . . . . 6 ((𝜑𝑊 = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥𝐴0 )))
213gsumz 18826 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑥𝐴0 )) = 0 )
221, 11, 21syl2anc 582 . . . . . . 7 (𝜑 → (𝐺 Σg (𝑥𝐴0 )) = 0 )
2322adantr 479 . . . . . 6 ((𝜑𝑊 = ∅) → (𝐺 Σg (𝑥𝐴0 )) = 0 )
2420, 23eqtrd 2766 . . . . 5 ((𝜑𝑊 = ∅) → (𝐺 Σg 𝐹) = 0 )
2510, 11fexd 7244 . . . . . . . . . . 11 (𝜑𝐹 ∈ V)
2625suppun 8198 . . . . . . . . . 10 (𝜑 → (𝐻 supp 0 ) ⊆ ((𝐻𝐹) supp 0 ))
27 uncom 4153 . . . . . . . . . . 11 (𝐹𝐻) = (𝐻𝐹)
2827oveq1i 7434 . . . . . . . . . 10 ((𝐹𝐻) supp 0 ) = ((𝐻𝐹) supp 0 )
2926, 28sseqtrrdi 4031 . . . . . . . . 9 (𝜑 → (𝐻 supp 0 ) ⊆ ((𝐹𝐻) supp 0 ))
3029, 17sseqtrrdi 4031 . . . . . . . 8 (𝜑 → (𝐻 supp 0 ) ⊆ 𝑊)
3114, 11, 13, 30gsumcllem 19906 . . . . . . 7 ((𝜑𝑊 = ∅) → 𝐻 = (𝑥𝐴0 ))
3231oveq2d 7440 . . . . . 6 ((𝜑𝑊 = ∅) → (𝐺 Σg 𝐻) = (𝐺 Σg (𝑥𝐴0 )))
3332, 23eqtrd 2766 . . . . 5 ((𝜑𝑊 = ∅) → (𝐺 Σg 𝐻) = 0 )
3424, 33oveq12d 7442 . . . 4 ((𝜑𝑊 = ∅) → ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)) = ( 0 + 0 ))
3511adantr 479 . . . . . . . 8 ((𝜑𝑊 = ∅) → 𝐴𝑉)
365ad2antrr 724 . . . . . . . 8 (((𝜑𝑊 = ∅) ∧ 𝑥𝐴) → 0𝐵)
3735, 36, 36, 19, 31offval2 7710 . . . . . . 7 ((𝜑𝑊 = ∅) → (𝐹f + 𝐻) = (𝑥𝐴 ↦ ( 0 + 0 )))
389mpteq2dv 5255 . . . . . . 7 ((𝜑𝑊 = ∅) → (𝑥𝐴 ↦ ( 0 + 0 )) = (𝑥𝐴0 ))
3937, 38eqtrd 2766 . . . . . 6 ((𝜑𝑊 = ∅) → (𝐹f + 𝐻) = (𝑥𝐴0 ))
4039oveq2d 7440 . . . . 5 ((𝜑𝑊 = ∅) → (𝐺 Σg (𝐹f + 𝐻)) = (𝐺 Σg (𝑥𝐴0 )))
4140, 23eqtrd 2766 . . . 4 ((𝜑𝑊 = ∅) → (𝐺 Σg (𝐹f + 𝐻)) = 0 )
429, 34, 413eqtr4rd 2777 . . 3 ((𝜑𝑊 = ∅) → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
4342ex 411 . 2 (𝜑 → (𝑊 = ∅ → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))))
441adantr 479 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐺 ∈ Mnd)
452, 6mndcl 18735 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ 𝑧𝐵𝑤𝐵) → (𝑧 + 𝑤) ∈ 𝐵)
46453expb 1117 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ (𝑧𝐵𝑤𝐵)) → (𝑧 + 𝑤) ∈ 𝐵)
4744, 46sylan 578 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑧𝐵𝑤𝐵)) → (𝑧 + 𝑤) ∈ 𝐵)
4847caovclg 7618 . . . . . . 7 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
49 simprl 769 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (♯‘𝑊) ∈ ℕ)
50 nnuz 12917 . . . . . . . 8 ℕ = (ℤ‘1)
5149, 50eleqtrdi 2836 . . . . . . 7 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (♯‘𝑊) ∈ (ℤ‘1))
5210adantr 479 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐹:𝐴𝐵)
53 f1of1 6842 . . . . . . . . . . . 12 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑓:(1...(♯‘𝑊))–1-1𝑊)
5453ad2antll 727 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑓:(1...(♯‘𝑊))–1-1𝑊)
55 suppssdm 8191 . . . . . . . . . . . . . 14 ((𝐹𝐻) supp 0 ) ⊆ dom (𝐹𝐻)
5655a1i 11 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝐻) supp 0 ) ⊆ dom (𝐹𝐻))
5717a1i 11 . . . . . . . . . . . . 13 (𝜑𝑊 = ((𝐹𝐻) supp 0 ))
58 dmun 5917 . . . . . . . . . . . . . 14 dom (𝐹𝐻) = (dom 𝐹 ∪ dom 𝐻)
5910fdmd 6738 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝐹 = 𝐴)
6014fdmd 6738 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝐻 = 𝐴)
6159, 60uneq12d 4164 . . . . . . . . . . . . . . 15 (𝜑 → (dom 𝐹 ∪ dom 𝐻) = (𝐴𝐴))
62 unidm 4152 . . . . . . . . . . . . . . 15 (𝐴𝐴) = 𝐴
6361, 62eqtrdi 2782 . . . . . . . . . . . . . 14 (𝜑 → (dom 𝐹 ∪ dom 𝐻) = 𝐴)
6458, 63eqtr2id 2779 . . . . . . . . . . . . 13 (𝜑𝐴 = dom (𝐹𝐻))
6556, 57, 643sstr4d 4027 . . . . . . . . . . . 12 (𝜑𝑊𝐴)
6665adantr 479 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑊𝐴)
67 f1ss 6803 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝑊))–1-1𝑊𝑊𝐴) → 𝑓:(1...(♯‘𝑊))–1-1𝐴)
6854, 66, 67syl2anc 582 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑓:(1...(♯‘𝑊))–1-1𝐴)
69 f1f 6798 . . . . . . . . . 10 (𝑓:(1...(♯‘𝑊))–1-1𝐴𝑓:(1...(♯‘𝑊))⟶𝐴)
7068, 69syl 17 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑓:(1...(♯‘𝑊))⟶𝐴)
71 fco 6752 . . . . . . . . 9 ((𝐹:𝐴𝐵𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐹𝑓):(1...(♯‘𝑊))⟶𝐵)
7252, 70, 71syl2anc 582 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐹𝑓):(1...(♯‘𝑊))⟶𝐵)
7372ffvelcdmda 7098 . . . . . . 7 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) ∈ 𝐵)
7414adantr 479 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐻:𝐴𝐵)
75 fco 6752 . . . . . . . . 9 ((𝐻:𝐴𝐵𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐻𝑓):(1...(♯‘𝑊))⟶𝐵)
7674, 70, 75syl2anc 582 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐻𝑓):(1...(♯‘𝑊))⟶𝐵)
7776ffvelcdmda 7098 . . . . . . 7 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐻𝑓)‘𝑘) ∈ 𝐵)
7852ffnd 6729 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐹 Fn 𝐴)
7974ffnd 6729 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐻 Fn 𝐴)
8011adantr 479 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐴𝑉)
81 ovexd 7459 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (1...(♯‘𝑊)) ∈ V)
82 inidm 4220 . . . . . . . . . . 11 (𝐴𝐴) = 𝐴
8378, 79, 70, 80, 80, 81, 82ofco 7714 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ((𝐹f + 𝐻) ∘ 𝑓) = ((𝐹𝑓) ∘f + (𝐻𝑓)))
8483fveq1d 6903 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (((𝐹f + 𝐻) ∘ 𝑓)‘𝑘) = (((𝐹𝑓) ∘f + (𝐻𝑓))‘𝑘))
8584adantr 479 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (((𝐹f + 𝐻) ∘ 𝑓)‘𝑘) = (((𝐹𝑓) ∘f + (𝐻𝑓))‘𝑘))
86 fnfco 6767 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐹𝑓) Fn (1...(♯‘𝑊)))
8778, 70, 86syl2anc 582 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐹𝑓) Fn (1...(♯‘𝑊)))
88 fnfco 6767 . . . . . . . . . 10 ((𝐻 Fn 𝐴𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐻𝑓) Fn (1...(♯‘𝑊)))
8979, 70, 88syl2anc 582 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐻𝑓) Fn (1...(♯‘𝑊)))
90 inidm 4220 . . . . . . . . 9 ((1...(♯‘𝑊)) ∩ (1...(♯‘𝑊))) = (1...(♯‘𝑊))
91 eqidd 2727 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) = ((𝐹𝑓)‘𝑘))
92 eqidd 2727 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐻𝑓)‘𝑘) = ((𝐻𝑓)‘𝑘))
9387, 89, 81, 81, 90, 91, 92ofval 7701 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (((𝐹𝑓) ∘f + (𝐻𝑓))‘𝑘) = (((𝐹𝑓)‘𝑘) + ((𝐻𝑓)‘𝑘)))
9485, 93eqtrd 2766 . . . . . . 7 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (((𝐹f + 𝐻) ∘ 𝑓)‘𝑘) = (((𝐹𝑓)‘𝑘) + ((𝐻𝑓)‘𝑘)))
951ad2antrr 724 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝐺 ∈ Mnd)
96 elfzouz 13690 . . . . . . . . . 10 (𝑛 ∈ (1..^(♯‘𝑊)) → 𝑛 ∈ (ℤ‘1))
9796adantl 480 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑛 ∈ (ℤ‘1))
98 elfzouz2 13701 . . . . . . . . . . . . 13 (𝑛 ∈ (1..^(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ𝑛))
9998adantl 480 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (♯‘𝑊) ∈ (ℤ𝑛))
100 fzss2 13595 . . . . . . . . . . . 12 ((♯‘𝑊) ∈ (ℤ𝑛) → (1...𝑛) ⊆ (1...(♯‘𝑊)))
10199, 100syl 17 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (1...𝑛) ⊆ (1...(♯‘𝑊)))
102101sselda 3979 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ (1...(♯‘𝑊)))
10373adantlr 713 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) ∈ 𝐵)
104102, 103syldan 589 . . . . . . . . 9 ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → ((𝐹𝑓)‘𝑘) ∈ 𝐵)
1052, 6mndcl 18735 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑘𝐵𝑥𝐵) → (𝑘 + 𝑥) ∈ 𝐵)
1061053expb 1117 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ (𝑘𝐵𝑥𝐵)) → (𝑘 + 𝑥) ∈ 𝐵)
10795, 106sylan 578 . . . . . . . . 9 ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ (𝑘𝐵𝑥𝐵)) → (𝑘 + 𝑥) ∈ 𝐵)
10897, 104, 107seqcl 14042 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐹𝑓))‘𝑛) ∈ 𝐵)
10977adantlr 713 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐻𝑓)‘𝑘) ∈ 𝐵)
110102, 109syldan 589 . . . . . . . . 9 ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → ((𝐻𝑓)‘𝑘) ∈ 𝐵)
11197, 110, 107seqcl 14042 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐻𝑓))‘𝑛) ∈ 𝐵)
112 fzofzp1 13784 . . . . . . . . 9 (𝑛 ∈ (1..^(♯‘𝑊)) → (𝑛 + 1) ∈ (1...(♯‘𝑊)))
113 ffvelcdm 7095 . . . . . . . . 9 (((𝐹𝑓):(1...(♯‘𝑊))⟶𝐵 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘(𝑛 + 1)) ∈ 𝐵)
11472, 112, 113syl2an 594 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐹𝑓)‘(𝑛 + 1)) ∈ 𝐵)
115 ffvelcdm 7095 . . . . . . . . 9 (((𝐻𝑓):(1...(♯‘𝑊))⟶𝐵 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → ((𝐻𝑓)‘(𝑛 + 1)) ∈ 𝐵)
11676, 112, 115syl2an 594 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐻𝑓)‘(𝑛 + 1)) ∈ 𝐵)
117 fvco3 7001 . . . . . . . . . . . 12 ((𝑓:(1...(♯‘𝑊))⟶𝐴 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘(𝑛 + 1)) = (𝐹‘(𝑓‘(𝑛 + 1))))
11870, 112, 117syl2an 594 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐹𝑓)‘(𝑛 + 1)) = (𝐹‘(𝑓‘(𝑛 + 1))))
119 fveq2 6901 . . . . . . . . . . . . 13 (𝑘 = (𝑓‘(𝑛 + 1)) → (𝐹𝑘) = (𝐹‘(𝑓‘(𝑛 + 1))))
120119eleq1d 2811 . . . . . . . . . . . 12 (𝑘 = (𝑓‘(𝑛 + 1)) → ((𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}) ↔ (𝐹‘(𝑓‘(𝑛 + 1))) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})))
121 gsumzaddlem.4 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑥𝐴𝑘 ∈ (𝐴𝑥))) → (𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))}))
122121expr 455 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (𝑘 ∈ (𝐴𝑥) → (𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))})))
123122ralrimiv 3135 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → ∀𝑘 ∈ (𝐴𝑥)(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))}))
124123ex 411 . . . . . . . . . . . . . . 15 (𝜑 → (𝑥𝐴 → ∀𝑘 ∈ (𝐴𝑥)(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))})))
125124alrimiv 1923 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥(𝑥𝐴 → ∀𝑘 ∈ (𝐴𝑥)(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))})))
126125ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ∀𝑥(𝑥𝐴 → ∀𝑘 ∈ (𝐴𝑥)(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))})))
127 imassrn 6080 . . . . . . . . . . . . . 14 (𝑓 “ (1...𝑛)) ⊆ ran 𝑓
12870adantr 479 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑓:(1...(♯‘𝑊))⟶𝐴)
129128frnd 6736 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ran 𝑓𝐴)
130127, 129sstrid 3991 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 “ (1...𝑛)) ⊆ 𝐴)
131 vex 3466 . . . . . . . . . . . . . . 15 𝑓 ∈ V
132131imaex 7927 . . . . . . . . . . . . . 14 (𝑓 “ (1...𝑛)) ∈ V
133 sseq1 4005 . . . . . . . . . . . . . . 15 (𝑥 = (𝑓 “ (1...𝑛)) → (𝑥𝐴 ↔ (𝑓 “ (1...𝑛)) ⊆ 𝐴))
134 difeq2 4115 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑓 “ (1...𝑛)) → (𝐴𝑥) = (𝐴 ∖ (𝑓 “ (1...𝑛))))
135 reseq2 5984 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (𝑓 “ (1...𝑛)) → (𝐻𝑥) = (𝐻 ↾ (𝑓 “ (1...𝑛))))
136135oveq2d 7440 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑓 “ (1...𝑛)) → (𝐺 Σg (𝐻𝑥)) = (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛)))))
137136sneqd 4645 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑓 “ (1...𝑛)) → {(𝐺 Σg (𝐻𝑥))} = {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})
138137fveq2d 6905 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑓 “ (1...𝑛)) → (𝑍‘{(𝐺 Σg (𝐻𝑥))}) = (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))
139138eleq2d 2812 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑓 “ (1...𝑛)) → ((𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))}) ↔ (𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})))
140134, 139raleqbidv 3330 . . . . . . . . . . . . . . 15 (𝑥 = (𝑓 “ (1...𝑛)) → (∀𝑘 ∈ (𝐴𝑥)(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))}) ↔ ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})))
141133, 140imbi12d 343 . . . . . . . . . . . . . 14 (𝑥 = (𝑓 “ (1...𝑛)) → ((𝑥𝐴 → ∀𝑘 ∈ (𝐴𝑥)(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))})) ↔ ((𝑓 “ (1...𝑛)) ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))))
142132, 141spcv 3591 . . . . . . . . . . . . 13 (∀𝑥(𝑥𝐴 → ∀𝑘 ∈ (𝐴𝑥)(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))})) → ((𝑓 “ (1...𝑛)) ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})))
143126, 130, 142sylc 65 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))
144 ffvelcdm 7095 . . . . . . . . . . . . . 14 ((𝑓:(1...(♯‘𝑊))⟶𝐴 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → (𝑓‘(𝑛 + 1)) ∈ 𝐴)
14570, 112, 144syl2an 594 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓‘(𝑛 + 1)) ∈ 𝐴)
146 fzp1nel 13639 . . . . . . . . . . . . . 14 ¬ (𝑛 + 1) ∈ (1...𝑛)
14768adantr 479 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑓:(1...(♯‘𝑊))–1-1𝐴)
148112adantl 480 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑛 + 1) ∈ (1...(♯‘𝑊)))
149 f1elima 7278 . . . . . . . . . . . . . . 15 ((𝑓:(1...(♯‘𝑊))–1-1𝐴 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊)) ∧ (1...𝑛) ⊆ (1...(♯‘𝑊))) → ((𝑓‘(𝑛 + 1)) ∈ (𝑓 “ (1...𝑛)) ↔ (𝑛 + 1) ∈ (1...𝑛)))
150147, 148, 101, 149syl3anc 1368 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝑓‘(𝑛 + 1)) ∈ (𝑓 “ (1...𝑛)) ↔ (𝑛 + 1) ∈ (1...𝑛)))
151146, 150mtbiri 326 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ¬ (𝑓‘(𝑛 + 1)) ∈ (𝑓 “ (1...𝑛)))
152145, 151eldifd 3958 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓‘(𝑛 + 1)) ∈ (𝐴 ∖ (𝑓 “ (1...𝑛))))
153120, 143, 152rspcdva 3609 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝐹‘(𝑓‘(𝑛 + 1))) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))
154118, 153eqeltrd 2826 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐹𝑓)‘(𝑛 + 1)) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))
155 gsumzadd.z . . . . . . . . . . . . 13 𝑍 = (Cntz‘𝐺)
156132a1i 11 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 “ (1...𝑛)) ∈ V)
15714ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝐻:𝐴𝐵)
158157, 130fssresd 6769 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝐻 ↾ (𝑓 “ (1...𝑛))):(𝑓 “ (1...𝑛))⟶𝐵)
159 gsumzaddlem.2 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐻 ⊆ (𝑍‘ran 𝐻))
160159ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ran 𝐻 ⊆ (𝑍‘ran 𝐻))
161 resss 6011 . . . . . . . . . . . . . . 15 (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ 𝐻
162161rnssi 5946 . . . . . . . . . . . . . 14 ran (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ ran 𝐻
163155cntzidss 19334 . . . . . . . . . . . . . 14 ((ran 𝐻 ⊆ (𝑍‘ran 𝐻) ∧ ran (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ ran 𝐻) → ran (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ (𝑍‘ran (𝐻 ↾ (𝑓 “ (1...𝑛)))))
164160, 162, 163sylancl 584 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ran (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ (𝑍‘ran (𝐻 ↾ (𝑓 “ (1...𝑛)))))
16597, 50eleqtrrdi 2837 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑛 ∈ ℕ)
166 f1ores 6857 . . . . . . . . . . . . . . 15 ((𝑓:(1...(♯‘𝑊))–1-1𝐴 ∧ (1...𝑛) ⊆ (1...(♯‘𝑊))) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1-onto→(𝑓 “ (1...𝑛)))
167147, 101, 166syl2anc 582 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1-onto→(𝑓 “ (1...𝑛)))
168 f1of1 6842 . . . . . . . . . . . . . 14 ((𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1-onto→(𝑓 “ (1...𝑛)) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1→(𝑓 “ (1...𝑛)))
169167, 168syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1→(𝑓 “ (1...𝑛)))
170 suppssdm 8191 . . . . . . . . . . . . . . 15 ((𝐻 ↾ (𝑓 “ (1...𝑛))) supp 0 ) ⊆ dom (𝐻 ↾ (𝑓 “ (1...𝑛)))
171 dmres 6021 . . . . . . . . . . . . . . . 16 dom (𝐻 ↾ (𝑓 “ (1...𝑛))) = ((𝑓 “ (1...𝑛)) ∩ dom 𝐻)
172171a1i 11 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → dom (𝐻 ↾ (𝑓 “ (1...𝑛))) = ((𝑓 “ (1...𝑛)) ∩ dom 𝐻))
173170, 172sseqtrid 4032 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐻 ↾ (𝑓 “ (1...𝑛))) supp 0 ) ⊆ ((𝑓 “ (1...𝑛)) ∩ dom 𝐻))
174 inss1 4230 . . . . . . . . . . . . . . 15 ((𝑓 “ (1...𝑛)) ∩ dom 𝐻) ⊆ (𝑓 “ (1...𝑛))
175 df-ima 5695 . . . . . . . . . . . . . . . 16 (𝑓 “ (1...𝑛)) = ran (𝑓 ↾ (1...𝑛))
176175a1i 11 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 “ (1...𝑛)) = ran (𝑓 ↾ (1...𝑛)))
177174, 176sseqtrid 4032 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝑓 “ (1...𝑛)) ∩ dom 𝐻) ⊆ ran (𝑓 ↾ (1...𝑛)))
178173, 177sstrd 3990 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐻 ↾ (𝑓 “ (1...𝑛))) supp 0 ) ⊆ ran (𝑓 ↾ (1...𝑛)))
179 eqid 2726 . . . . . . . . . . . . 13 (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) supp 0 ) = (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) supp 0 )
1802, 3, 6, 155, 95, 156, 158, 164, 165, 169, 178, 179gsumval3 19905 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛)))) = (seq1( + , ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))))‘𝑛))
181175eqimss2i 4041 . . . . . . . . . . . . . . . . . 18 ran (𝑓 ↾ (1...𝑛)) ⊆ (𝑓 “ (1...𝑛))
182 cores 6260 . . . . . . . . . . . . . . . . . 18 (ran (𝑓 ↾ (1...𝑛)) ⊆ (𝑓 “ (1...𝑛)) → ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) = (𝐻 ∘ (𝑓 ↾ (1...𝑛))))
183181, 182ax-mp 5 . . . . . . . . . . . . . . . . 17 ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) = (𝐻 ∘ (𝑓 ↾ (1...𝑛)))
184 resco 6261 . . . . . . . . . . . . . . . . 17 ((𝐻𝑓) ↾ (1...𝑛)) = (𝐻 ∘ (𝑓 ↾ (1...𝑛)))
185183, 184eqtr4i 2757 . . . . . . . . . . . . . . . 16 ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) = ((𝐻𝑓) ↾ (1...𝑛))
186185fveq1i 6902 . . . . . . . . . . . . . . 15 (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛)))‘𝑘) = (((𝐻𝑓) ↾ (1...𝑛))‘𝑘)
187 fvres 6920 . . . . . . . . . . . . . . 15 (𝑘 ∈ (1...𝑛) → (((𝐻𝑓) ↾ (1...𝑛))‘𝑘) = ((𝐻𝑓)‘𝑘))
188186, 187eqtrid 2778 . . . . . . . . . . . . . 14 (𝑘 ∈ (1...𝑛) → (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛)))‘𝑘) = ((𝐻𝑓)‘𝑘))
189188adantl 480 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛)))‘𝑘) = ((𝐻𝑓)‘𝑘))
19097, 189seqfveq 14046 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))))‘𝑛) = (seq1( + , (𝐻𝑓))‘𝑛))
191180, 190eqtr2d 2767 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐻𝑓))‘𝑛) = (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛)))))
192 fvex 6914 . . . . . . . . . . . 12 (seq1( + , (𝐻𝑓))‘𝑛) ∈ V
193192elsn 4648 . . . . . . . . . . 11 ((seq1( + , (𝐻𝑓))‘𝑛) ∈ {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))} ↔ (seq1( + , (𝐻𝑓))‘𝑛) = (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛)))))
194191, 193sylibr 233 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐻𝑓))‘𝑛) ∈ {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})
1956, 155cntzi 19323 . . . . . . . . . 10 ((((𝐹𝑓)‘(𝑛 + 1)) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}) ∧ (seq1( + , (𝐻𝑓))‘𝑛) ∈ {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}) → (((𝐹𝑓)‘(𝑛 + 1)) + (seq1( + , (𝐻𝑓))‘𝑛)) = ((seq1( + , (𝐻𝑓))‘𝑛) + ((𝐹𝑓)‘(𝑛 + 1))))
196154, 194, 195syl2anc 582 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (((𝐹𝑓)‘(𝑛 + 1)) + (seq1( + , (𝐻𝑓))‘𝑛)) = ((seq1( + , (𝐻𝑓))‘𝑛) + ((𝐹𝑓)‘(𝑛 + 1))))
197196eqcomd 2732 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((seq1( + , (𝐻𝑓))‘𝑛) + ((𝐹𝑓)‘(𝑛 + 1))) = (((𝐹𝑓)‘(𝑛 + 1)) + (seq1( + , (𝐻𝑓))‘𝑛)))
1982, 6, 95, 108, 111, 114, 116, 197mnd4g 18741 . . . . . . 7 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (((seq1( + , (𝐹𝑓))‘𝑛) + (seq1( + , (𝐻𝑓))‘𝑛)) + (((𝐹𝑓)‘(𝑛 + 1)) + ((𝐻𝑓)‘(𝑛 + 1)))) = (((seq1( + , (𝐹𝑓))‘𝑛) + ((𝐹𝑓)‘(𝑛 + 1))) + ((seq1( + , (𝐻𝑓))‘𝑛) + ((𝐻𝑓)‘(𝑛 + 1)))))
19948, 48, 51, 73, 77, 94, 198seqcaopr3 14057 . . . . . 6 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (seq1( + , ((𝐹f + 𝐻) ∘ 𝑓))‘(♯‘𝑊)) = ((seq1( + , (𝐹𝑓))‘(♯‘𝑊)) + (seq1( + , (𝐻𝑓))‘(♯‘𝑊))))
20047, 52, 74, 80, 80, 82off 7708 . . . . . . 7 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐹f + 𝐻):𝐴𝐵)
201 gsumzaddlem.3 . . . . . . . 8 (𝜑 → ran (𝐹f + 𝐻) ⊆ (𝑍‘ran (𝐹f + 𝐻)))
202201adantr 479 . . . . . . 7 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ran (𝐹f + 𝐻) ⊆ (𝑍‘ran (𝐹f + 𝐻)))
20344, 106sylan 578 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑘𝐵𝑥𝐵)) → (𝑘 + 𝑥) ∈ 𝐵)
204203, 52, 74, 80, 80, 82off 7708 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐹f + 𝐻):𝐴𝐵)
205 eldifi 4126 . . . . . . . . . 10 (𝑥 ∈ (𝐴 ∖ ran 𝑓) → 𝑥𝐴)
206 eqidd 2727 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
207 eqidd 2727 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥𝐴) → (𝐻𝑥) = (𝐻𝑥))
20878, 79, 80, 80, 82, 206, 207ofval 7701 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥𝐴) → ((𝐹f + 𝐻)‘𝑥) = ((𝐹𝑥) + (𝐻𝑥)))
209205, 208sylan2 591 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ((𝐹f + 𝐻)‘𝑥) = ((𝐹𝑥) + (𝐻𝑥)))
21016adantr 479 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐹 supp 0 ) ⊆ ((𝐹𝐻) supp 0 ))
211 f1ofo 6850 . . . . . . . . . . . . . . . 16 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑓:(1...(♯‘𝑊))–onto𝑊)
212 forn 6818 . . . . . . . . . . . . . . . 16 (𝑓:(1...(♯‘𝑊))–onto𝑊 → ran 𝑓 = 𝑊)
213211, 212syl 17 . . . . . . . . . . . . . . 15 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊 → ran 𝑓 = 𝑊)
214213, 17eqtrdi 2782 . . . . . . . . . . . . . 14 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊 → ran 𝑓 = ((𝐹𝐻) supp 0 ))
215214sseq2d 4012 . . . . . . . . . . . . 13 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊 → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ ((𝐹𝐻) supp 0 )))
216215ad2antll 727 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ ((𝐹𝐻) supp 0 )))
217210, 216mpbird 256 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐹 supp 0 ) ⊆ ran 𝑓)
21812a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 0 ∈ V)
21952, 217, 80, 218suppssr 8210 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → (𝐹𝑥) = 0 )
22026adantr 479 . . . . . . . . . . . . 13 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐻 supp 0 ) ⊆ ((𝐻𝐹) supp 0 ))
221220, 28sseqtrrdi 4031 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐻 supp 0 ) ⊆ ((𝐹𝐻) supp 0 ))
222214sseq2d 4012 . . . . . . . . . . . . 13 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊 → ((𝐻 supp 0 ) ⊆ ran 𝑓 ↔ (𝐻 supp 0 ) ⊆ ((𝐹𝐻) supp 0 )))
223222ad2antll 727 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ((𝐻 supp 0 ) ⊆ ran 𝑓 ↔ (𝐻 supp 0 ) ⊆ ((𝐹𝐻) supp 0 )))
224221, 223mpbird 256 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐻 supp 0 ) ⊆ ran 𝑓)
22574, 224, 80, 218suppssr 8210 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → (𝐻𝑥) = 0 )
226219, 225oveq12d 7442 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ((𝐹𝑥) + (𝐻𝑥)) = ( 0 + 0 ))
2278ad2antrr 724 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ( 0 + 0 ) = 0 )
228209, 226, 2273eqtrd 2770 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ((𝐹f + 𝐻)‘𝑥) = 0 )
229204, 228suppss 8208 . . . . . . 7 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ((𝐹f + 𝐻) supp 0 ) ⊆ ran 𝑓)
230 ovex 7457 . . . . . . . . 9 (𝐹f + 𝐻) ∈ V
231230, 131coex 7943 . . . . . . . 8 ((𝐹f + 𝐻) ∘ 𝑓) ∈ V
232 suppimacnv 8188 . . . . . . . . 9 ((((𝐹f + 𝐻) ∘ 𝑓) ∈ V ∧ 0 ∈ V) → (((𝐹f + 𝐻) ∘ 𝑓) supp 0 ) = (((𝐹f + 𝐻) ∘ 𝑓) “ (V ∖ { 0 })))
233232eqcomd 2732 . . . . . . . 8 ((((𝐹f + 𝐻) ∘ 𝑓) ∈ V ∧ 0 ∈ V) → (((𝐹f + 𝐻) ∘ 𝑓) “ (V ∖ { 0 })) = (((𝐹f + 𝐻) ∘ 𝑓) supp 0 ))
234231, 12, 233mp2an 690 . . . . . . 7 (((𝐹f + 𝐻) ∘ 𝑓) “ (V ∖ { 0 })) = (((𝐹f + 𝐻) ∘ 𝑓) supp 0 )
2352, 3, 6, 155, 44, 80, 200, 202, 49, 68, 229, 234gsumval3 19905 . . . . . 6 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐺 Σg (𝐹f + 𝐻)) = (seq1( + , ((𝐹f + 𝐻) ∘ 𝑓))‘(♯‘𝑊)))
236 gsumzaddlem.1 . . . . . . . . 9 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
237236adantr 479 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
238 eqid 2726 . . . . . . . 8 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
2392, 3, 6, 155, 44, 80, 52, 237, 49, 68, 217, 238gsumval3 19905 . . . . . . 7 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))
240159adantr 479 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ran 𝐻 ⊆ (𝑍‘ran 𝐻))
241 eqid 2726 . . . . . . . 8 ((𝐻𝑓) supp 0 ) = ((𝐻𝑓) supp 0 )
2422, 3, 6, 155, 44, 80, 74, 240, 49, 68, 224, 241gsumval3 19905 . . . . . . 7 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐺 Σg 𝐻) = (seq1( + , (𝐻𝑓))‘(♯‘𝑊)))
243239, 242oveq12d 7442 . . . . . 6 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)) = ((seq1( + , (𝐹𝑓))‘(♯‘𝑊)) + (seq1( + , (𝐻𝑓))‘(♯‘𝑊))))
244199, 235, 2433eqtr4d 2776 . . . . 5 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
245244expr 455 . . . 4 ((𝜑 ∧ (♯‘𝑊) ∈ ℕ) → (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊 → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))))
246245exlimdv 1929 . . 3 ((𝜑 ∧ (♯‘𝑊) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊 → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))))
247246expimpd 452 . 2 (𝜑 → (((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊) → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))))
248 gsumzadd.fn . . . . 5 (𝜑𝐹 finSupp 0 )
249 gsumzadd.hn . . . . 5 (𝜑𝐻 finSupp 0 )
250248, 249fsuppun 9430 . . . 4 (𝜑 → ((𝐹𝐻) supp 0 ) ∈ Fin)
25117, 250eqeltrid 2830 . . 3 (𝜑𝑊 ∈ Fin)
252 fz1f1o 15714 . . 3 (𝑊 ∈ Fin → (𝑊 = ∅ ∨ ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)))
253251, 252syl 17 . 2 (𝜑 → (𝑊 = ∅ ∨ ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)))
25443, 247, 253mpjaod 858 1 (𝜑 → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845  wal 1532   = wceq 1534  wex 1774  wcel 2099  wral 3051  Vcvv 3462  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4325  {csn 4633   class class class wbr 5153  cmpt 5236  ccnv 5681  dom cdm 5682  ran crn 5683  cres 5684  cima 5685  ccom 5686   Fn wfn 6549  wf 6550  1-1wf1 6551  ontowfo 6552  1-1-ontowf1o 6553  cfv 6554  (class class class)co 7424  f cof 7688   supp csupp 8174  Fincfn 8974   finSupp cfsupp 9405  1c1 11159   + caddc 11161  cn 12264  cuz 12874  ...cfz 13538  ..^cfzo 13681  seqcseq 14021  chash 14347  Basecbs 17213  +gcplusg 17266  0gc0g 17454   Σg cgsu 17455  Mndcmnd 18727  Cntzccntz 19309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7690  df-om 7877  df-1st 8003  df-2nd 8004  df-supp 8175  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fsupp 9406  df-oi 9553  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-n0 12525  df-z 12611  df-uz 12875  df-fz 13539  df-fzo 13682  df-seq 14022  df-hash 14348  df-0g 17456  df-gsum 17457  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-cntz 19311
This theorem is referenced by:  gsumzadd  19920  dprdfadd  20020
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