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Theorem mplmonmul 20245
 Description: The product of two monomials adds the exponent vectors together. For example, the product of (𝑥↑2)(𝑦↑2) with (𝑦↑1)(𝑧↑3) is (𝑥↑2)(𝑦↑3)(𝑧↑3), where the exponent vectors ⟨2, 2, 0⟩ and ⟨0, 1, 3⟩ are added to give ⟨2, 3, 3⟩. (Contributed by Mario Carneiro, 9-Jan-2015.)
Hypotheses
Ref Expression
mplmon.s 𝑃 = (𝐼 mPoly 𝑅)
mplmon.b 𝐵 = (Base‘𝑃)
mplmon.z 0 = (0g𝑅)
mplmon.o 1 = (1r𝑅)
mplmon.d 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
mplmon.i (𝜑𝐼𝑊)
mplmon.r (𝜑𝑅 ∈ Ring)
mplmon.x (𝜑𝑋𝐷)
mplmonmul.t · = (.r𝑃)
mplmonmul.x (𝜑𝑌𝐷)
Assertion
Ref Expression
mplmonmul (𝜑 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) · (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))) = (𝑦𝐷 ↦ if(𝑦 = (𝑋f + 𝑌), 1 , 0 )))
Distinct variable groups:   𝑦,𝐷   𝑓,𝐼   𝜑,𝑦   𝑦,𝑓,𝑋   𝑦, 0   𝑦, 1   𝑦,𝑅   𝑓,𝑌,𝑦
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑦,𝑓)   𝐷(𝑓)   𝑃(𝑦,𝑓)   𝑅(𝑓)   · (𝑦,𝑓)   1 (𝑓)   𝐼(𝑦)   𝑊(𝑦,𝑓)   0 (𝑓)

Proof of Theorem mplmonmul
Dummy variables 𝑗 𝑘 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mplmon.s . . 3 𝑃 = (𝐼 mPoly 𝑅)
2 mplmon.b . . 3 𝐵 = (Base‘𝑃)
3 eqid 2824 . . 3 (.r𝑅) = (.r𝑅)
4 mplmonmul.t . . 3 · = (.r𝑃)
5 mplmon.d . . 3 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
6 mplmon.z . . . 4 0 = (0g𝑅)
7 mplmon.o . . . 4 1 = (1r𝑅)
8 mplmon.i . . . 4 (𝜑𝐼𝑊)
9 mplmon.r . . . 4 (𝜑𝑅 ∈ Ring)
10 mplmon.x . . . 4 (𝜑𝑋𝐷)
111, 2, 6, 7, 5, 8, 9, 10mplmon 20244 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)
12 mplmonmul.x . . . 4 (𝜑𝑌𝐷)
131, 2, 6, 7, 5, 8, 9, 12mplmon 20244 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) ∈ 𝐵)
141, 2, 3, 4, 5, 11, 13mplmul 20223 . 2 (𝜑 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) · (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))) = (𝑘𝐷 ↦ (𝑅 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))))))
15 eqeq1 2828 . . . . 5 (𝑦 = 𝑘 → (𝑦 = (𝑋f + 𝑌) ↔ 𝑘 = (𝑋f + 𝑌)))
1615ifbid 4472 . . . 4 (𝑦 = 𝑘 → if(𝑦 = (𝑋f + 𝑌), 1 , 0 ) = if(𝑘 = (𝑋f + 𝑌), 1 , 0 ))
1716cbvmptv 5155 . . 3 (𝑦𝐷 ↦ if(𝑦 = (𝑋f + 𝑌), 1 , 0 )) = (𝑘𝐷 ↦ if(𝑘 = (𝑋f + 𝑌), 1 , 0 ))
18 simpr 488 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑋 ∈ {𝑥𝐷𝑥r𝑘})
1918snssd 4726 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → {𝑋} ⊆ {𝑥𝐷𝑥r𝑘})
2019resmptd 5895 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋}) = (𝑗 ∈ {𝑋} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))))
2120oveq2d 7165 . . . . . . 7 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑅 Σg ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋})) = (𝑅 Σg (𝑗 ∈ {𝑋} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))))))
229ad2antrr 725 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑅 ∈ Ring)
23 ringmnd 19307 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
2422, 23syl 17 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑅 ∈ Mnd)
2510ad2antrr 725 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑋𝐷)
26 iftrue 4456 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1 , 0 ) = 1 )
27 eqid 2824 . . . . . . . . . . . . 13 (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))
287fvexi 6675 . . . . . . . . . . . . 13 1 ∈ V
2926, 27, 28fvmpt 6759 . . . . . . . . . . . 12 (𝑋𝐷 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋) = 1 )
3025, 29syl 17 . . . . . . . . . . 11 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋) = 1 )
31 ssrab2 4042 . . . . . . . . . . . . 13 {𝑥𝐷𝑥r𝑘} ⊆ 𝐷
328ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝐼𝑊)
33 simplr 768 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑘𝐷)
34 eqid 2824 . . . . . . . . . . . . . . 15 {𝑥𝐷𝑥r𝑘} = {𝑥𝐷𝑥r𝑘}
355, 34psrbagconcl 20153 . . . . . . . . . . . . . 14 ((𝐼𝑊𝑘𝐷𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑋) ∈ {𝑥𝐷𝑥r𝑘})
3632, 33, 18, 35syl3anc 1368 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑋) ∈ {𝑥𝐷𝑥r𝑘})
3731, 36sseldi 3951 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑋) ∈ 𝐷)
38 eqeq1 2828 . . . . . . . . . . . . . 14 (𝑦 = (𝑘f𝑋) → (𝑦 = 𝑌 ↔ (𝑘f𝑋) = 𝑌))
3938ifbid 4472 . . . . . . . . . . . . 13 (𝑦 = (𝑘f𝑋) → if(𝑦 = 𝑌, 1 , 0 ) = if((𝑘f𝑋) = 𝑌, 1 , 0 ))
40 eqid 2824 . . . . . . . . . . . . 13 (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))
416fvexi 6675 . . . . . . . . . . . . . 14 0 ∈ V
4228, 41ifex 4498 . . . . . . . . . . . . 13 if((𝑘f𝑋) = 𝑌, 1 , 0 ) ∈ V
4339, 40, 42fvmpt 6759 . . . . . . . . . . . 12 ((𝑘f𝑋) ∈ 𝐷 → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋)) = if((𝑘f𝑋) = 𝑌, 1 , 0 ))
4437, 43syl 17 . . . . . . . . . . 11 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋)) = if((𝑘f𝑋) = 𝑌, 1 , 0 ))
4530, 44oveq12d 7167 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋))) = ( 1 (.r𝑅)if((𝑘f𝑋) = 𝑌, 1 , 0 )))
46 eqid 2824 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘𝑅)
4746, 7ringidcl 19321 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
4846, 6ring0cl 19322 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
4947, 48ifcld 4495 . . . . . . . . . . . 12 (𝑅 ∈ Ring → if((𝑘f𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅))
5022, 49syl 17 . . . . . . . . . . 11 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → if((𝑘f𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅))
5146, 3, 7ringlidm 19324 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ if((𝑘f𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅)) → ( 1 (.r𝑅)if((𝑘f𝑋) = 𝑌, 1 , 0 )) = if((𝑘f𝑋) = 𝑌, 1 , 0 ))
5222, 50, 51syl2anc 587 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ( 1 (.r𝑅)if((𝑘f𝑋) = 𝑌, 1 , 0 )) = if((𝑘f𝑋) = 𝑌, 1 , 0 ))
535psrbagf 20145 . . . . . . . . . . . . . . . . . 18 ((𝐼𝑊𝑘𝐷) → 𝑘:𝐼⟶ℕ0)
5432, 33, 53syl2anc 587 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑘:𝐼⟶ℕ0)
5554ffvelrnda 6842 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
568adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐷) → 𝐼𝑊)
5710adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐷) → 𝑋𝐷)
585psrbagf 20145 . . . . . . . . . . . . . . . . . . 19 ((𝐼𝑊𝑋𝐷) → 𝑋:𝐼⟶ℕ0)
5956, 57, 58syl2anc 587 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑋:𝐼⟶ℕ0)
6059ffvelrnda 6842 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
6160adantlr 714 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
625psrbagf 20145 . . . . . . . . . . . . . . . . . . . 20 ((𝐼𝑊𝑌𝐷) → 𝑌:𝐼⟶ℕ0)
638, 12, 62syl2anc 587 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑌:𝐼⟶ℕ0)
6463adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑌:𝐼⟶ℕ0)
6564ffvelrnda 6842 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
6665adantlr 714 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
67 nn0cn 11904 . . . . . . . . . . . . . . . . 17 ((𝑘𝑧) ∈ ℕ0 → (𝑘𝑧) ∈ ℂ)
68 nn0cn 11904 . . . . . . . . . . . . . . . . 17 ((𝑋𝑧) ∈ ℕ0 → (𝑋𝑧) ∈ ℂ)
69 nn0cn 11904 . . . . . . . . . . . . . . . . 17 ((𝑌𝑧) ∈ ℕ0 → (𝑌𝑧) ∈ ℂ)
70 subadd 10887 . . . . . . . . . . . . . . . . 17 (((𝑘𝑧) ∈ ℂ ∧ (𝑋𝑧) ∈ ℂ ∧ (𝑌𝑧) ∈ ℂ) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
7167, 68, 69, 70syl3an 1157 . . . . . . . . . . . . . . . 16 (((𝑘𝑧) ∈ ℕ0 ∧ (𝑋𝑧) ∈ ℕ0 ∧ (𝑌𝑧) ∈ ℕ0) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
7255, 61, 66, 71syl3anc 1368 . . . . . . . . . . . . . . 15 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) ∧ 𝑧𝐼) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
73 eqcom 2831 . . . . . . . . . . . . . . 15 (((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧) ↔ (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧)))
7472, 73syl6bb 290 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) ∧ 𝑧𝐼) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
7574ralbidva 3191 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
76 mpteqb 6778 . . . . . . . . . . . . . 14 (∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) ∈ V → ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ ∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧)))
77 ovexd 7184 . . . . . . . . . . . . . 14 (𝑧𝐼 → ((𝑘𝑧) − (𝑋𝑧)) ∈ V)
7876, 77mprg 3147 . . . . . . . . . . . . 13 ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ ∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧))
79 mpteqb 6778 . . . . . . . . . . . . . 14 (∀𝑧𝐼 (𝑘𝑧) ∈ V → ((𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
80 fvexd 6676 . . . . . . . . . . . . . 14 (𝑧𝐼 → (𝑘𝑧) ∈ V)
8179, 80mprg 3147 . . . . . . . . . . . . 13 ((𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧)))
8275, 78, 813bitr4g 317 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ (𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧)))))
8354feqmptd 6724 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑘 = (𝑧𝐼 ↦ (𝑘𝑧)))
8459feqmptd 6724 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐷) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
8584adantr 484 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
8632, 55, 61, 83, 85offval2 7420 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑋) = (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))))
8764feqmptd 6724 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐷) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
8887adantr 484 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
8986, 88eqeq12d 2840 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑘f𝑋) = 𝑌 ↔ (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧))))
9056, 60, 65, 84, 87offval2 7420 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐷) → (𝑋f + 𝑌) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))))
9190adantr 484 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑋f + 𝑌) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))))
9283, 91eqeq12d 2840 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘 = (𝑋f + 𝑌) ↔ (𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧)))))
9382, 89, 923bitr4d 314 . . . . . . . . . . 11 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑘f𝑋) = 𝑌𝑘 = (𝑋f + 𝑌)))
9493ifbid 4472 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → if((𝑘f𝑋) = 𝑌, 1 , 0 ) = if(𝑘 = (𝑋f + 𝑌), 1 , 0 ))
9545, 52, 943eqtrd 2863 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋))) = if(𝑘 = (𝑋f + 𝑌), 1 , 0 ))
9694, 50eqeltrrd 2917 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → if(𝑘 = (𝑋f + 𝑌), 1 , 0 ) ∈ (Base‘𝑅))
9795, 96eqeltrd 2916 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋))) ∈ (Base‘𝑅))
98 fveq2 6661 . . . . . . . . . 10 (𝑗 = 𝑋 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋))
99 oveq2 7157 . . . . . . . . . . 11 (𝑗 = 𝑋 → (𝑘f𝑗) = (𝑘f𝑋))
10099fveq2d 6665 . . . . . . . . . 10 (𝑗 = 𝑋 → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)) = ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋)))
10198, 100oveq12d 7167 . . . . . . . . 9 (𝑗 = 𝑋 → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) = (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋))))
10246, 101gsumsn 19074 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ 𝑋𝐷 ∧ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋))) ∈ (Base‘𝑅)) → (𝑅 Σg (𝑗 ∈ {𝑋} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))))) = (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋))))
10324, 25, 97, 102syl3anc 1368 . . . . . . 7 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑅 Σg (𝑗 ∈ {𝑋} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))))) = (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋))))
10421, 103, 953eqtrd 2863 . . . . . 6 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑅 Σg ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋})) = if(𝑘 = (𝑋f + 𝑌), 1 , 0 ))
1056gsum0 17894 . . . . . . 7 (𝑅 Σg ∅) = 0
106 disjsn 4632 . . . . . . . . 9 (({𝑥𝐷𝑥r𝑘} ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ {𝑥𝐷𝑥r𝑘})
1079ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑅 ∈ Ring)
1081, 46, 2, 5, 11mplelf 20213 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
109108ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
110 simpr 488 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑗 ∈ {𝑥𝐷𝑥r𝑘})
11131, 110sseldi 3951 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑗𝐷)
112109, 111ffvelrnd 6843 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) ∈ (Base‘𝑅))
1131, 46, 2, 5, 13mplelf 20213 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
114113ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
1158ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → 𝐼𝑊)
116 simplr 768 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑘𝐷)
1175, 34psrbagconcl 20153 . . . . . . . . . . . . . . . 16 ((𝐼𝑊𝑘𝐷𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑗) ∈ {𝑥𝐷𝑥r𝑘})
118115, 116, 110, 117syl3anc 1368 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑗) ∈ {𝑥𝐷𝑥r𝑘})
11931, 118sseldi 3951 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑗) ∈ 𝐷)
120114, 119ffvelrnd 6843 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)) ∈ (Base‘𝑅))
12146, 3ringcl 19314 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) ∈ (Base‘𝑅) ∧ ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)) ∈ (Base‘𝑅)) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) ∈ (Base‘𝑅))
122107, 112, 120, 121syl3anc 1368 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) ∈ (Base‘𝑅))
123122fmpttd 6870 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))):{𝑥𝐷𝑥r𝑘}⟶(Base‘𝑅))
124 ffn 6503 . . . . . . . . . . 11 ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))):{𝑥𝐷𝑥r𝑘}⟶(Base‘𝑅) → (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) Fn {𝑥𝐷𝑥r𝑘})
125 fnresdisj 6456 . . . . . . . . . . 11 ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) Fn {𝑥𝐷𝑥r𝑘} → (({𝑥𝐷𝑥r𝑘} ∩ {𝑋}) = ∅ ↔ ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋}) = ∅))
126123, 124, 1253syl 18 . . . . . . . . . 10 ((𝜑𝑘𝐷) → (({𝑥𝐷𝑥r𝑘} ∩ {𝑋}) = ∅ ↔ ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋}) = ∅))
127126biimpa 480 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ ({𝑥𝐷𝑥r𝑘} ∩ {𝑋}) = ∅) → ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋}) = ∅)
128106, 127sylan2br 597 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋}) = ∅)
129128oveq2d 7165 . . . . . . 7 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑅 Σg ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋})) = (𝑅 Σg ∅))
130 breq1 5055 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥r ≤ (𝑋f + 𝑌) ↔ 𝑋r ≤ (𝑋f + 𝑌)))
13160nn0red 11953 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℝ)
132 nn0addge1 11940 . . . . . . . . . . . . . 14 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℕ0) → (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
133131, 65, 132syl2anc 587 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
134133ralrimiva 3177 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
135 ovexd 7184 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → ((𝑋𝑧) + (𝑌𝑧)) ∈ V)
13656, 60, 135, 84, 90ofrfval2 7421 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → (𝑋r ≤ (𝑋f + 𝑌) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧))))
137134, 136mpbird 260 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 𝑋r ≤ (𝑋f + 𝑌))
138130, 57, 137elrabd 3668 . . . . . . . . . 10 ((𝜑𝑘𝐷) → 𝑋 ∈ {𝑥𝐷𝑥r ≤ (𝑋f + 𝑌)})
139 breq2 5056 . . . . . . . . . . . 12 (𝑘 = (𝑋f + 𝑌) → (𝑥r𝑘𝑥r ≤ (𝑋f + 𝑌)))
140139rabbidv 3465 . . . . . . . . . . 11 (𝑘 = (𝑋f + 𝑌) → {𝑥𝐷𝑥r𝑘} = {𝑥𝐷𝑥r ≤ (𝑋f + 𝑌)})
141140eleq2d 2901 . . . . . . . . . 10 (𝑘 = (𝑋f + 𝑌) → (𝑋 ∈ {𝑥𝐷𝑥r𝑘} ↔ 𝑋 ∈ {𝑥𝐷𝑥r ≤ (𝑋f + 𝑌)}))
142138, 141syl5ibrcom 250 . . . . . . . . 9 ((𝜑𝑘𝐷) → (𝑘 = (𝑋f + 𝑌) → 𝑋 ∈ {𝑥𝐷𝑥r𝑘}))
143142con3dimp 412 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ¬ 𝑘 = (𝑋f + 𝑌))
144143iffalsed 4461 . . . . . . 7 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → if(𝑘 = (𝑋f + 𝑌), 1 , 0 ) = 0 )
145105, 129, 1443eqtr4a 2885 . . . . . 6 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑅 Σg ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋})) = if(𝑘 = (𝑋f + 𝑌), 1 , 0 ))
146104, 145pm2.61dan 812 . . . . 5 ((𝜑𝑘𝐷) → (𝑅 Σg ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋})) = if(𝑘 = (𝑋f + 𝑌), 1 , 0 ))
1479adantr 484 . . . . . . 7 ((𝜑𝑘𝐷) → 𝑅 ∈ Ring)
148 ringcmn 19334 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
149147, 148syl 17 . . . . . 6 ((𝜑𝑘𝐷) → 𝑅 ∈ CMnd)
1505psrbaglefi 20152 . . . . . . 7 ((𝐼𝑊𝑘𝐷) → {𝑥𝐷𝑥r𝑘} ∈ Fin)
1518, 150sylan 583 . . . . . 6 ((𝜑𝑘𝐷) → {𝑥𝐷𝑥r𝑘} ∈ Fin)
152 ssdif 4102 . . . . . . . . . . . 12 ({𝑥𝐷𝑥r𝑘} ⊆ 𝐷 → ({𝑥𝐷𝑥r𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋}))
15331, 152ax-mp 5 . . . . . . . . . . 11 ({𝑥𝐷𝑥r𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋})
154153sseli 3949 . . . . . . . . . 10 (𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋}) → 𝑗 ∈ (𝐷 ∖ {𝑋}))
155108adantr 484 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
156 eldifsni 4707 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦𝑋)
157156adantl 485 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦𝑋)
158157neneqd 3019 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋)
159158iffalsed 4461 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 )
160 ovex 7182 . . . . . . . . . . . . . 14 (ℕ0m 𝐼) ∈ V
1615, 160rabex2 5223 . . . . . . . . . . . . 13 𝐷 ∈ V
162161a1i 11 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → 𝐷 ∈ V)
163159, 162suppss2 7860 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋})
16441a1i 11 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 0 ∈ V)
165155, 163, 162, 164suppssr 7857 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ (𝐷 ∖ {𝑋})) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
166154, 165sylan2 595 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋})) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
167166oveq1d 7164 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋})) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) = ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))))
168 eldifi 4089 . . . . . . . . 9 (𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋}) → 𝑗 ∈ {𝑥𝐷𝑥r𝑘})
16946, 3, 6ringlz 19340 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)) ∈ (Base‘𝑅)) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) = 0 )
170107, 120, 169syl2anc 587 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) = 0 )
171168, 170sylan2 595 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋})) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) = 0 )
172167, 171eqtrd 2859 . . . . . . 7 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋})) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) = 0 )
173161rabex 5221 . . . . . . . 8 {𝑥𝐷𝑥r𝑘} ∈ V
174173a1i 11 . . . . . . 7 ((𝜑𝑘𝐷) → {𝑥𝐷𝑥r𝑘} ∈ V)
175172, 174suppss2 7860 . . . . . 6 ((𝜑𝑘𝐷) → ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) supp 0 ) ⊆ {𝑋})
176161mptrabex 6979 . . . . . . . . 9 (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ∈ V
177 funmpt 6381 . . . . . . . . 9 Fun (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))))
178176, 177, 413pm3.2i 1336 . . . . . . . 8 ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ∈ V ∧ Fun (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ∧ 0 ∈ V)
179178a1i 11 . . . . . . 7 ((𝜑𝑘𝐷) → ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ∈ V ∧ Fun (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ∧ 0 ∈ V))
180 snfi 8590 . . . . . . . 8 {𝑋} ∈ Fin
181180a1i 11 . . . . . . 7 ((𝜑𝑘𝐷) → {𝑋} ∈ Fin)
182 suppssfifsupp 8845 . . . . . . 7 ((((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ∈ V ∧ Fun (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ∧ 0 ∈ V) ∧ ({𝑋} ∈ Fin ∧ ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) supp 0 ) ⊆ {𝑋})) → (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) finSupp 0 )
183179, 181, 175, 182syl12anc 835 . . . . . 6 ((𝜑𝑘𝐷) → (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) finSupp 0 )
18446, 6, 149, 151, 123, 175, 183gsumres 19033 . . . . 5 ((𝜑𝑘𝐷) → (𝑅 Σg ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋})) = (𝑅 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))))))
185146, 184eqtr3d 2861 . . . 4 ((𝜑𝑘𝐷) → if(𝑘 = (𝑋f + 𝑌), 1 , 0 ) = (𝑅 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))))))
186185mpteq2dva 5147 . . 3 (𝜑 → (𝑘𝐷 ↦ if(𝑘 = (𝑋f + 𝑌), 1 , 0 )) = (𝑘𝐷 ↦ (𝑅 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))))))
18717, 186syl5eq 2871 . 2 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑋f + 𝑌), 1 , 0 )) = (𝑘𝐷 ↦ (𝑅 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))))))
18814, 187eqtr4d 2862 1 (𝜑 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) · (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))) = (𝑦𝐷 ↦ if(𝑦 = (𝑋f + 𝑌), 1 , 0 )))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  ∀wral 3133  {crab 3137  Vcvv 3480   ∖ cdif 3916   ∩ cin 3918   ⊆ wss 3919  ∅c0 4276  ifcif 4450  {csn 4550   class class class wbr 5052   ↦ cmpt 5132  ◡ccnv 5541   ↾ cres 5544   “ cima 5545  Fun wfun 6337   Fn wfn 6338  ⟶wf 6339  ‘cfv 6343  (class class class)co 7149   ∘f cof 7401   ∘r cofr 7402   supp csupp 7826   ↑m cmap 8402  Fincfn 8505   finSupp cfsupp 8830  ℂcc 10533  ℝcr 10534   + caddc 10538   ≤ cle 10674   − cmin 10868  ℕcn 11634  ℕ0cn0 11894  Basecbs 16483  .rcmulr 16566  0gc0g 16713   Σg cgsu 16714  Mndcmnd 17911  CMndccmn 18906  1rcur 19251  Ringcrg 19297   mPoly cmpl 20133 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-ofr 7404  df-om 7575  df-1st 7684  df-2nd 7685  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fsupp 8831  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-z 11979  df-uz 12241  df-fz 12895  df-fzo 13038  df-seq 13374  df-hash 13696  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-tset 16584  df-0g 16715  df-gsum 16716  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-mulg 18225  df-cntz 18447  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-ring 19299  df-psr 20136  df-mpl 20138 This theorem is referenced by:  mplcoe3  20247  mplcoe5  20249  mplmon2mul  20281
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