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Theorem mamufval 21879
Description: Functional value of the matrix multiplication operator. (Contributed by Stefan O'Rear, 2-Sep-2015.)
Hypotheses
Ref Expression
mamufval.f 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)
mamufval.b 𝐵 = (Base‘𝑅)
mamufval.t · = (.r𝑅)
mamufval.r (𝜑𝑅𝑉)
mamufval.m (𝜑𝑀 ∈ Fin)
mamufval.n (𝜑𝑁 ∈ Fin)
mamufval.p (𝜑𝑃 ∈ Fin)
Assertion
Ref Expression
mamufval (𝜑𝐹 = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m (𝑁 × 𝑃)) ↦ (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))))
Distinct variable groups:   𝑖,𝑗,𝑘,𝑥,𝑦,𝑀   𝑖,𝑁,𝑗,𝑘,𝑥,𝑦   𝑃,𝑖,𝑗,𝑘,𝑥,𝑦   𝑅,𝑖,𝑗,𝑘,𝑥,𝑦   𝜑,𝑖,𝑗,𝑘,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥, · ,𝑦,𝑖,𝑘
Allowed substitution hints:   𝐵(𝑖,𝑗,𝑘)   · (𝑗)   𝐹(𝑥,𝑦,𝑖,𝑗,𝑘)   𝑉(𝑥,𝑦,𝑖,𝑗,𝑘)

Proof of Theorem mamufval
Dummy variables 𝑚 𝑛 𝑜 𝑝 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mamufval.f . 2 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)
2 df-mamu 21878 . . . 4 maMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ (1st ‘(1st𝑜)) / 𝑚(2nd ‘(1st𝑜)) / 𝑛(2nd𝑜) / 𝑝(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m (𝑛 × 𝑝)) ↦ (𝑖𝑚, 𝑘𝑝 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))))
32a1i 11 . . 3 (𝜑 → maMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ (1st ‘(1st𝑜)) / 𝑚(2nd ‘(1st𝑜)) / 𝑛(2nd𝑜) / 𝑝(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m (𝑛 × 𝑝)) ↦ (𝑖𝑚, 𝑘𝑝 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))))))))
4 fvex 6902 . . . . 5 (1st ‘(1st𝑜)) ∈ V
5 fvex 6902 . . . . 5 (2nd ‘(1st𝑜)) ∈ V
6 fvex 6902 . . . . . . 7 (2nd𝑜) ∈ V
7 eqidd 2734 . . . . . . . 8 (𝑝 = (2nd𝑜) → ((Base‘𝑟) ↑m (𝑚 × 𝑛)) = ((Base‘𝑟) ↑m (𝑚 × 𝑛)))
8 xpeq2 5697 . . . . . . . . 9 (𝑝 = (2nd𝑜) → (𝑛 × 𝑝) = (𝑛 × (2nd𝑜)))
98oveq2d 7422 . . . . . . . 8 (𝑝 = (2nd𝑜) → ((Base‘𝑟) ↑m (𝑛 × 𝑝)) = ((Base‘𝑟) ↑m (𝑛 × (2nd𝑜))))
10 eqidd 2734 . . . . . . . . 9 (𝑝 = (2nd𝑜) → 𝑚 = 𝑚)
11 id 22 . . . . . . . . 9 (𝑝 = (2nd𝑜) → 𝑝 = (2nd𝑜))
12 eqidd 2734 . . . . . . . . 9 (𝑝 = (2nd𝑜) → (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))) = (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))
1310, 11, 12mpoeq123dv 7481 . . . . . . . 8 (𝑝 = (2nd𝑜) → (𝑖𝑚, 𝑘𝑝 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))))) = (𝑖𝑚, 𝑘 ∈ (2nd𝑜) ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))))))
147, 9, 13mpoeq123dv 7481 . . . . . . 7 (𝑝 = (2nd𝑜) → (𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m (𝑛 × 𝑝)) ↦ (𝑖𝑚, 𝑘𝑝 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))) = (𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m (𝑛 × (2nd𝑜))) ↦ (𝑖𝑚, 𝑘 ∈ (2nd𝑜) ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))))
156, 14csbie 3929 . . . . . 6 (2nd𝑜) / 𝑝(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m (𝑛 × 𝑝)) ↦ (𝑖𝑚, 𝑘𝑝 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))) = (𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m (𝑛 × (2nd𝑜))) ↦ (𝑖𝑚, 𝑘 ∈ (2nd𝑜) ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))))))
16 xpeq12 5701 . . . . . . . 8 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → (𝑚 × 𝑛) = ((1st ‘(1st𝑜)) × (2nd ‘(1st𝑜))))
1716oveq2d 7422 . . . . . . 7 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → ((Base‘𝑟) ↑m (𝑚 × 𝑛)) = ((Base‘𝑟) ↑m ((1st ‘(1st𝑜)) × (2nd ‘(1st𝑜)))))
18 simpr 486 . . . . . . . . 9 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → 𝑛 = (2nd ‘(1st𝑜)))
1918xpeq1d 5705 . . . . . . . 8 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → (𝑛 × (2nd𝑜)) = ((2nd ‘(1st𝑜)) × (2nd𝑜)))
2019oveq2d 7422 . . . . . . 7 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → ((Base‘𝑟) ↑m (𝑛 × (2nd𝑜))) = ((Base‘𝑟) ↑m ((2nd ‘(1st𝑜)) × (2nd𝑜))))
21 id 22 . . . . . . . . 9 (𝑚 = (1st ‘(1st𝑜)) → 𝑚 = (1st ‘(1st𝑜)))
2221adantr 482 . . . . . . . 8 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → 𝑚 = (1st ‘(1st𝑜)))
23 eqidd 2734 . . . . . . . 8 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → (2nd𝑜) = (2nd𝑜))
24 eqidd 2734 . . . . . . . . . 10 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)) = ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))
2518, 24mpteq12dv 5239 . . . . . . . . 9 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))) = (𝑗 ∈ (2nd ‘(1st𝑜)) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))))
2625oveq2d 7422 . . . . . . . 8 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))) = (𝑟 Σg (𝑗 ∈ (2nd ‘(1st𝑜)) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))
2722, 23, 26mpoeq123dv 7481 . . . . . . 7 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → (𝑖𝑚, 𝑘 ∈ (2nd𝑜) ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))))) = (𝑖 ∈ (1st ‘(1st𝑜)), 𝑘 ∈ (2nd𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd ‘(1st𝑜)) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))))))
2817, 20, 27mpoeq123dv 7481 . . . . . 6 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → (𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m (𝑛 × (2nd𝑜))) ↦ (𝑖𝑚, 𝑘 ∈ (2nd𝑜) ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))) = (𝑥 ∈ ((Base‘𝑟) ↑m ((1st ‘(1st𝑜)) × (2nd ‘(1st𝑜)))), 𝑦 ∈ ((Base‘𝑟) ↑m ((2nd ‘(1st𝑜)) × (2nd𝑜))) ↦ (𝑖 ∈ (1st ‘(1st𝑜)), 𝑘 ∈ (2nd𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd ‘(1st𝑜)) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))))
2915, 28eqtrid 2785 . . . . 5 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → (2nd𝑜) / 𝑝(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m (𝑛 × 𝑝)) ↦ (𝑖𝑚, 𝑘𝑝 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))) = (𝑥 ∈ ((Base‘𝑟) ↑m ((1st ‘(1st𝑜)) × (2nd ‘(1st𝑜)))), 𝑦 ∈ ((Base‘𝑟) ↑m ((2nd ‘(1st𝑜)) × (2nd𝑜))) ↦ (𝑖 ∈ (1st ‘(1st𝑜)), 𝑘 ∈ (2nd𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd ‘(1st𝑜)) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))))
304, 5, 29csbie2 3935 . . . 4 (1st ‘(1st𝑜)) / 𝑚(2nd ‘(1st𝑜)) / 𝑛(2nd𝑜) / 𝑝(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m (𝑛 × 𝑝)) ↦ (𝑖𝑚, 𝑘𝑝 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))) = (𝑥 ∈ ((Base‘𝑟) ↑m ((1st ‘(1st𝑜)) × (2nd ‘(1st𝑜)))), 𝑦 ∈ ((Base‘𝑟) ↑m ((2nd ‘(1st𝑜)) × (2nd𝑜))) ↦ (𝑖 ∈ (1st ‘(1st𝑜)), 𝑘 ∈ (2nd𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd ‘(1st𝑜)) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))))))
31 simprl 770 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → 𝑟 = 𝑅)
3231fveq2d 6893 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (Base‘𝑟) = (Base‘𝑅))
33 mamufval.b . . . . . . 7 𝐵 = (Base‘𝑅)
3432, 33eqtr4di 2791 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (Base‘𝑟) = 𝐵)
35 fveq2 6889 . . . . . . . . . 10 (𝑜 = ⟨𝑀, 𝑁, 𝑃⟩ → (1st𝑜) = (1st ‘⟨𝑀, 𝑁, 𝑃⟩))
3635fveq2d 6893 . . . . . . . . 9 (𝑜 = ⟨𝑀, 𝑁, 𝑃⟩ → (1st ‘(1st𝑜)) = (1st ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)))
3736ad2antll 728 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (1st ‘(1st𝑜)) = (1st ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)))
38 mamufval.m . . . . . . . . . 10 (𝜑𝑀 ∈ Fin)
39 mamufval.n . . . . . . . . . 10 (𝜑𝑁 ∈ Fin)
40 mamufval.p . . . . . . . . . 10 (𝜑𝑃 ∈ Fin)
41 ot1stg 7986 . . . . . . . . . 10 ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin) → (1st ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)) = 𝑀)
4238, 39, 40, 41syl3anc 1372 . . . . . . . . 9 (𝜑 → (1st ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)) = 𝑀)
4342adantr 482 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (1st ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)) = 𝑀)
4437, 43eqtrd 2773 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (1st ‘(1st𝑜)) = 𝑀)
4535fveq2d 6893 . . . . . . . . 9 (𝑜 = ⟨𝑀, 𝑁, 𝑃⟩ → (2nd ‘(1st𝑜)) = (2nd ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)))
4645ad2antll 728 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (2nd ‘(1st𝑜)) = (2nd ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)))
47 ot2ndg 7987 . . . . . . . . . 10 ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin) → (2nd ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)) = 𝑁)
4838, 39, 40, 47syl3anc 1372 . . . . . . . . 9 (𝜑 → (2nd ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)) = 𝑁)
4948adantr 482 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (2nd ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)) = 𝑁)
5046, 49eqtrd 2773 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (2nd ‘(1st𝑜)) = 𝑁)
5144, 50xpeq12d 5707 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → ((1st ‘(1st𝑜)) × (2nd ‘(1st𝑜))) = (𝑀 × 𝑁))
5234, 51oveq12d 7424 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → ((Base‘𝑟) ↑m ((1st ‘(1st𝑜)) × (2nd ‘(1st𝑜)))) = (𝐵m (𝑀 × 𝑁)))
53 fveq2 6889 . . . . . . . . 9 (𝑜 = ⟨𝑀, 𝑁, 𝑃⟩ → (2nd𝑜) = (2nd ‘⟨𝑀, 𝑁, 𝑃⟩))
5453ad2antll 728 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (2nd𝑜) = (2nd ‘⟨𝑀, 𝑁, 𝑃⟩))
55 ot3rdg 7988 . . . . . . . . . 10 (𝑃 ∈ Fin → (2nd ‘⟨𝑀, 𝑁, 𝑃⟩) = 𝑃)
5640, 55syl 17 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝑀, 𝑁, 𝑃⟩) = 𝑃)
5756adantr 482 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (2nd ‘⟨𝑀, 𝑁, 𝑃⟩) = 𝑃)
5854, 57eqtrd 2773 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (2nd𝑜) = 𝑃)
5950, 58xpeq12d 5707 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → ((2nd ‘(1st𝑜)) × (2nd𝑜)) = (𝑁 × 𝑃))
6034, 59oveq12d 7424 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → ((Base‘𝑟) ↑m ((2nd ‘(1st𝑜)) × (2nd𝑜))) = (𝐵m (𝑁 × 𝑃)))
6131fveq2d 6893 . . . . . . . . . 10 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (.r𝑟) = (.r𝑅))
62 mamufval.t . . . . . . . . . 10 · = (.r𝑅)
6361, 62eqtr4di 2791 . . . . . . . . 9 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (.r𝑟) = · )
6463oveqd 7423 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)) = ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))
6550, 64mpteq12dv 5239 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (𝑗 ∈ (2nd ‘(1st𝑜)) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))) = (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘))))
6631, 65oveq12d 7424 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (𝑟 Σg (𝑗 ∈ (2nd ‘(1st𝑜)) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))
6744, 58, 66mpoeq123dv 7481 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (𝑖 ∈ (1st ‘(1st𝑜)), 𝑘 ∈ (2nd𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd ‘(1st𝑜)) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))))) = (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘))))))
6852, 60, 67mpoeq123dv 7481 . . . 4 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (𝑥 ∈ ((Base‘𝑟) ↑m ((1st ‘(1st𝑜)) × (2nd ‘(1st𝑜)))), 𝑦 ∈ ((Base‘𝑟) ↑m ((2nd ‘(1st𝑜)) × (2nd𝑜))) ↦ (𝑖 ∈ (1st ‘(1st𝑜)), 𝑘 ∈ (2nd𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd ‘(1st𝑜)) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))) = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m (𝑁 × 𝑃)) ↦ (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))))
6930, 68eqtrid 2785 . . 3 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (1st ‘(1st𝑜)) / 𝑚(2nd ‘(1st𝑜)) / 𝑛(2nd𝑜) / 𝑝(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m (𝑛 × 𝑝)) ↦ (𝑖𝑚, 𝑘𝑝 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))) = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m (𝑁 × 𝑃)) ↦ (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))))
70 mamufval.r . . . 4 (𝜑𝑅𝑉)
7170elexd 3495 . . 3 (𝜑𝑅 ∈ V)
72 otex 5465 . . . 4 𝑀, 𝑁, 𝑃⟩ ∈ V
7372a1i 11 . . 3 (𝜑 → ⟨𝑀, 𝑁, 𝑃⟩ ∈ V)
74 ovex 7439 . . . . 5 (𝐵m (𝑀 × 𝑁)) ∈ V
75 ovex 7439 . . . . 5 (𝐵m (𝑁 × 𝑃)) ∈ V
7674, 75mpoex 8063 . . . 4 (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m (𝑁 × 𝑃)) ↦ (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))) ∈ V
7776a1i 11 . . 3 (𝜑 → (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m (𝑁 × 𝑃)) ↦ (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))) ∈ V)
783, 69, 71, 73, 77ovmpod 7557 . 2 (𝜑 → (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩) = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m (𝑁 × 𝑃)) ↦ (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))))
791, 78eqtrid 2785 1 (𝜑𝐹 = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m (𝑁 × 𝑃)) ↦ (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  csb 3893  cotp 4636  cmpt 5231   × cxp 5674  cfv 6541  (class class class)co 7406  cmpo 7408  1st c1st 7970  2nd c2nd 7971  m cmap 8817  Fincfn 8936  Basecbs 17141  .rcmulr 17195   Σg cgsu 17383   maMul cmmul 21877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-ot 4637  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-mamu 21878
This theorem is referenced by:  mamuval  21880  mamudm  21882
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