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Theorem mamufval 20414
 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 20413 . . . 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 6364 . . . . 5 (1st ‘(1st𝑜)) ∈ V
5 fvex 6364 . . . . 5 (2nd ‘(1st𝑜)) ∈ V
6 fvex 6364 . . . . . . 7 (2nd𝑜) ∈ V
7 eqidd 2762 . . . . . . . 8 (𝑝 = (2nd𝑜) → ((Base‘𝑟) ↑𝑚 (𝑚 × 𝑛)) = ((Base‘𝑟) ↑𝑚 (𝑚 × 𝑛)))
8 xpeq2 5287 . . . . . . . . 9 (𝑝 = (2nd𝑜) → (𝑛 × 𝑝) = (𝑛 × (2nd𝑜)))
98oveq2d 6831 . . . . . . . 8 (𝑝 = (2nd𝑜) → ((Base‘𝑟) ↑𝑚 (𝑛 × 𝑝)) = ((Base‘𝑟) ↑𝑚 (𝑛 × (2nd𝑜))))
10 eqidd 2762 . . . . . . . . 9 (𝑝 = (2nd𝑜) → 𝑚 = 𝑚)
11 id 22 . . . . . . . . 9 (𝑝 = (2nd𝑜) → 𝑝 = (2nd𝑜))
12 eqidd 2762 . . . . . . . . 9 (𝑝 = (2nd𝑜) → (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))) = (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))
1310, 11, 12mpt2eq123dv 6884 . . . . . . . 8 (𝑝 = (2nd𝑜) → (𝑖𝑚, 𝑘𝑝 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))))) = (𝑖𝑚, 𝑘 ∈ (2nd𝑜) ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))))))
147, 9, 13mpt2eq123dv 6884 . . . . . . 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 3701 . . . . . 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 5292 . . . . . . . 8 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → (𝑚 × 𝑛) = ((1st ‘(1st𝑜)) × (2nd ‘(1st𝑜))))
1716oveq2d 6831 . . . . . . 7 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → ((Base‘𝑟) ↑𝑚 (𝑚 × 𝑛)) = ((Base‘𝑟) ↑𝑚 ((1st ‘(1st𝑜)) × (2nd ‘(1st𝑜)))))
18 simpr 479 . . . . . . . . 9 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → 𝑛 = (2nd ‘(1st𝑜)))
1918xpeq1d 5296 . . . . . . . 8 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → (𝑛 × (2nd𝑜)) = ((2nd ‘(1st𝑜)) × (2nd𝑜)))
2019oveq2d 6831 . . . . . . 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 472 . . . . . . . 8 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → 𝑚 = (1st ‘(1st𝑜)))
23 eqidd 2762 . . . . . . . 8 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → (2nd𝑜) = (2nd𝑜))
24 eqidd 2762 . . . . . . . . . 10 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)) = ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))
2518, 24mpteq12dv 4886 . . . . . . . . 9 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))) = (𝑗 ∈ (2nd ‘(1st𝑜)) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))))
2625oveq2d 6831 . . . . . . . 8 ((𝑚 = (1st ‘(1st𝑜)) ∧ 𝑛 = (2nd ‘(1st𝑜))) → (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))) = (𝑟 Σg (𝑗 ∈ (2nd ‘(1st𝑜)) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))
2722, 23, 26mpt2eq123dv 6884 . . . . . . 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, 27mpt2eq123dv 6884 . . . . . 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, 28syl5eq 2807 . . . . 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 3705 . . . 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 811 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → 𝑟 = 𝑅)
3231fveq2d 6358 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (Base‘𝑟) = (Base‘𝑅))
33 mamufval.b . . . . . . 7 𝐵 = (Base‘𝑅)
3432, 33syl6eqr 2813 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (Base‘𝑟) = 𝐵)
35 fveq2 6354 . . . . . . . . . 10 (𝑜 = ⟨𝑀, 𝑁, 𝑃⟩ → (1st𝑜) = (1st ‘⟨𝑀, 𝑁, 𝑃⟩))
3635fveq2d 6358 . . . . . . . . 9 (𝑜 = ⟨𝑀, 𝑁, 𝑃⟩ → (1st ‘(1st𝑜)) = (1st ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)))
3736ad2antll 767 . . . . . . . 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 7349 . . . . . . . . . 10 ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin) → (1st ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)) = 𝑀)
4238, 39, 40, 41syl3anc 1477 . . . . . . . . 9 (𝜑 → (1st ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)) = 𝑀)
4342adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (1st ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)) = 𝑀)
4437, 43eqtrd 2795 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (1st ‘(1st𝑜)) = 𝑀)
4535fveq2d 6358 . . . . . . . . 9 (𝑜 = ⟨𝑀, 𝑁, 𝑃⟩ → (2nd ‘(1st𝑜)) = (2nd ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)))
4645ad2antll 767 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (2nd ‘(1st𝑜)) = (2nd ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)))
47 ot2ndg 7350 . . . . . . . . . 10 ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin) → (2nd ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)) = 𝑁)
4838, 39, 40, 47syl3anc 1477 . . . . . . . . 9 (𝜑 → (2nd ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)) = 𝑁)
4948adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (2nd ‘(1st ‘⟨𝑀, 𝑁, 𝑃⟩)) = 𝑁)
5046, 49eqtrd 2795 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (2nd ‘(1st𝑜)) = 𝑁)
5144, 50xpeq12d 5298 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → ((1st ‘(1st𝑜)) × (2nd ‘(1st𝑜))) = (𝑀 × 𝑁))
5234, 51oveq12d 6833 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → ((Base‘𝑟) ↑𝑚 ((1st ‘(1st𝑜)) × (2nd ‘(1st𝑜)))) = (𝐵𝑚 (𝑀 × 𝑁)))
53 fveq2 6354 . . . . . . . . 9 (𝑜 = ⟨𝑀, 𝑁, 𝑃⟩ → (2nd𝑜) = (2nd ‘⟨𝑀, 𝑁, 𝑃⟩))
5453ad2antll 767 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (2nd𝑜) = (2nd ‘⟨𝑀, 𝑁, 𝑃⟩))
55 ot3rdg 7351 . . . . . . . . . 10 (𝑃 ∈ Fin → (2nd ‘⟨𝑀, 𝑁, 𝑃⟩) = 𝑃)
5640, 55syl 17 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝑀, 𝑁, 𝑃⟩) = 𝑃)
5756adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (2nd ‘⟨𝑀, 𝑁, 𝑃⟩) = 𝑃)
5854, 57eqtrd 2795 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (2nd𝑜) = 𝑃)
5950, 58xpeq12d 5298 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → ((2nd ‘(1st𝑜)) × (2nd𝑜)) = (𝑁 × 𝑃))
6034, 59oveq12d 6833 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → ((Base‘𝑟) ↑𝑚 ((2nd ‘(1st𝑜)) × (2nd𝑜))) = (𝐵𝑚 (𝑁 × 𝑃)))
6131fveq2d 6358 . . . . . . . . . 10 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (.r𝑟) = (.r𝑅))
62 mamufval.t . . . . . . . . . 10 · = (.r𝑅)
6361, 62syl6eqr 2813 . . . . . . . . 9 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (.r𝑟) = · )
6463oveqd 6832 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)) = ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))
6550, 64mpteq12dv 4886 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (𝑗 ∈ (2nd ‘(1st𝑜)) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘))) = (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘))))
6631, 65oveq12d 6833 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁, 𝑃⟩)) → (𝑟 Σg (𝑗 ∈ (2nd ‘(1st𝑜)) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))
6744, 58, 66mpt2eq123dv 6884 . . . . 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, 67mpt2eq123dv 6884 . . . 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, 68syl5eq 2807 . . 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)
71 elex 3353 . . . 4 (𝑅𝑉𝑅 ∈ V)
7270, 71syl 17 . . 3 (𝜑𝑅 ∈ V)
73 otex 5083 . . . 4 𝑀, 𝑁, 𝑃⟩ ∈ V
7473a1i 11 . . 3 (𝜑 → ⟨𝑀, 𝑁, 𝑃⟩ ∈ V)
75 ovex 6843 . . . . 5 (𝐵𝑚 (𝑀 × 𝑁)) ∈ V
76 ovex 6843 . . . . 5 (𝐵𝑚 (𝑁 × 𝑃)) ∈ V
7775, 76mpt2ex 7417 . . . 4 (𝑥 ∈ (𝐵𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵𝑚 (𝑁 × 𝑃)) ↦ (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))) ∈ V
7877a1i 11 . . 3 (𝜑 → (𝑥 ∈ (𝐵𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵𝑚 (𝑁 × 𝑃)) ↦ (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))) ∈ V)
793, 69, 72, 74, 78ovmpt2d 6955 . 2 (𝜑 → (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩) = (𝑥 ∈ (𝐵𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵𝑚 (𝑁 × 𝑃)) ↦ (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))))
801, 79syl5eq 2807 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 383   = wceq 1632   ∈ wcel 2140  Vcvv 3341  ⦋csb 3675  ⟨cotp 4330   ↦ cmpt 4882   × cxp 5265  ‘cfv 6050  (class class class)co 6815   ↦ cmpt2 6817  1st c1st 7333  2nd c2nd 7334   ↑𝑚 cmap 8026  Fincfn 8124  Basecbs 16080  .rcmulr 16165   Σg cgsu 16324   maMul cmmul 20412 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-ot 4331  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-mamu 20413 This theorem is referenced by:  mamuval  20415  mamudm  20417
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