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Theorem mplmonmul 19512
 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 2651 . . 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 19511 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)
12 mplmonmul.x . . . 4 (𝜑𝑌𝐷)
131, 2, 6, 7, 5, 8, 9, 12mplmon 19511 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) ∈ 𝐵)
141, 2, 3, 4, 5, 11, 13mplmul 19491 . 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 2655 . . . . 5 (𝑦 = 𝑘 → (𝑦 = (𝑋𝑓 + 𝑌) ↔ 𝑘 = (𝑋𝑓 + 𝑌)))
1615ifbid 4141 . . . 4 (𝑦 = 𝑘 → if(𝑦 = (𝑋𝑓 + 𝑌), 1 , 0 ) = if(𝑘 = (𝑋𝑓 + 𝑌), 1 , 0 ))
1716cbvmptv 4783 . . 3 (𝑦𝐷 ↦ if(𝑦 = (𝑋𝑓 + 𝑌), 1 , 0 )) = (𝑘𝐷 ↦ if(𝑘 = (𝑋𝑓 + 𝑌), 1 , 0 ))
18 simpr 476 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘})
1918snssd 4372 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → {𝑋} ⊆ {𝑥𝐷𝑥𝑟𝑘})
2019resmptd 5487 . . . . . . . 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 6706 . . . . . . 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 762 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑅 ∈ Ring)
23 ringmnd 18602 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
2422, 23syl 17 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑅 ∈ Mnd)
2510ad2antrr 762 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑋𝐷)
26 iftrue 4125 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1 , 0 ) = 1 )
27 eqid 2651 . . . . . . . . . . . . 13 (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))
28 fvex 6239 . . . . . . . . . . . . . 14 (1r𝑅) ∈ V
297, 28eqeltri 2726 . . . . . . . . . . . . 13 1 ∈ V
3026, 27, 29fvmpt 6321 . . . . . . . . . . . 12 (𝑋𝐷 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋) = 1 )
3125, 30syl 17 . . . . . . . . . . 11 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋) = 1 )
32 ssrab2 3720 . . . . . . . . . . . . 13 {𝑥𝐷𝑥𝑟𝑘} ⊆ 𝐷
338ad2antrr 762 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝐼𝑊)
34 simplr 807 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑘𝐷)
35 eqid 2651 . . . . . . . . . . . . . . 15 {𝑥𝐷𝑥𝑟𝑘} = {𝑥𝐷𝑥𝑟𝑘}
365, 35psrbagconcl 19421 . . . . . . . . . . . . . 14 ((𝐼𝑊𝑘𝐷𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑋) ∈ {𝑥𝐷𝑥𝑟𝑘})
3733, 34, 18, 36syl3anc 1366 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑋) ∈ {𝑥𝐷𝑥𝑟𝑘})
3832, 37sseldi 3634 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑋) ∈ 𝐷)
39 eqeq1 2655 . . . . . . . . . . . . . 14 (𝑦 = (𝑘𝑓𝑋) → (𝑦 = 𝑌 ↔ (𝑘𝑓𝑋) = 𝑌))
4039ifbid 4141 . . . . . . . . . . . . 13 (𝑦 = (𝑘𝑓𝑋) → if(𝑦 = 𝑌, 1 , 0 ) = if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ))
41 eqid 2651 . . . . . . . . . . . . 13 (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))
42 fvex 6239 . . . . . . . . . . . . . . 15 (0g𝑅) ∈ V
436, 42eqeltri 2726 . . . . . . . . . . . . . 14 0 ∈ V
4429, 43ifex 4189 . . . . . . . . . . . . 13 if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ) ∈ V
4540, 41, 44fvmpt 6321 . . . . . . . . . . . 12 ((𝑘𝑓𝑋) ∈ 𝐷 → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑋)) = if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ))
4638, 45syl 17 . . . . . . . . . . 11 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑋)) = if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ))
4731, 46oveq12d 6708 . . . . . . . . . 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 )))
48 eqid 2651 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘𝑅)
4948, 7ringidcl 18614 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
5048, 6ring0cl 18615 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
5149, 50ifcld 4164 . . . . . . . . . . . 12 (𝑅 ∈ Ring → if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅))
5222, 51syl 17 . . . . . . . . . . 11 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅))
5348, 3, 7ringlidm 18617 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅)) → ( 1 (.r𝑅)if((𝑘𝑓𝑋) = 𝑌, 1 , 0 )) = if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ))
5422, 52, 53syl2anc 694 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ( 1 (.r𝑅)if((𝑘𝑓𝑋) = 𝑌, 1 , 0 )) = if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ))
555psrbagf 19413 . . . . . . . . . . . . . . . . . 18 ((𝐼𝑊𝑘𝐷) → 𝑘:𝐼⟶ℕ0)
5633, 34, 55syl2anc 694 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑘:𝐼⟶ℕ0)
5756ffvelrnda 6399 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
588adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐷) → 𝐼𝑊)
5910adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐷) → 𝑋𝐷)
605psrbagf 19413 . . . . . . . . . . . . . . . . . . 19 ((𝐼𝑊𝑋𝐷) → 𝑋:𝐼⟶ℕ0)
6158, 59, 60syl2anc 694 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑋:𝐼⟶ℕ0)
6261ffvelrnda 6399 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
6362adantlr 751 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
645psrbagf 19413 . . . . . . . . . . . . . . . . . . . 20 ((𝐼𝑊𝑌𝐷) → 𝑌:𝐼⟶ℕ0)
658, 12, 64syl2anc 694 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑌:𝐼⟶ℕ0)
6665adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑌:𝐼⟶ℕ0)
6766ffvelrnda 6399 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
6867adantlr 751 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
69 nn0cn 11340 . . . . . . . . . . . . . . . . 17 ((𝑘𝑧) ∈ ℕ0 → (𝑘𝑧) ∈ ℂ)
70 nn0cn 11340 . . . . . . . . . . . . . . . . 17 ((𝑋𝑧) ∈ ℕ0 → (𝑋𝑧) ∈ ℂ)
71 nn0cn 11340 . . . . . . . . . . . . . . . . 17 ((𝑌𝑧) ∈ ℕ0 → (𝑌𝑧) ∈ ℂ)
72 subadd 10322 . . . . . . . . . . . . . . . . 17 (((𝑘𝑧) ∈ ℂ ∧ (𝑋𝑧) ∈ ℂ ∧ (𝑌𝑧) ∈ ℂ) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
7369, 70, 71, 72syl3an 1408 . . . . . . . . . . . . . . . 16 (((𝑘𝑧) ∈ ℕ0 ∧ (𝑋𝑧) ∈ ℕ0 ∧ (𝑌𝑧) ∈ ℕ0) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
7457, 63, 68, 73syl3anc 1366 . . . . . . . . . . . . . . 15 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) ∧ 𝑧𝐼) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
75 eqcom 2658 . . . . . . . . . . . . . . 15 (((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧) ↔ (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧)))
7674, 75syl6bb 276 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) ∧ 𝑧𝐼) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
7776ralbidva 3014 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
78 mpteqb 6338 . . . . . . . . . . . . . 14 (∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) ∈ V → ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ ∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧)))
79 ovexd 6720 . . . . . . . . . . . . . 14 (𝑧𝐼 → ((𝑘𝑧) − (𝑋𝑧)) ∈ V)
8078, 79mprg 2955 . . . . . . . . . . . . 13 ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ ∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧))
81 mpteqb 6338 . . . . . . . . . . . . . 14 (∀𝑧𝐼 (𝑘𝑧) ∈ V → ((𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
82 fvex 6239 . . . . . . . . . . . . . . 15 (𝑘𝑧) ∈ V
8382a1i 11 . . . . . . . . . . . . . 14 (𝑧𝐼 → (𝑘𝑧) ∈ V)
8481, 83mprg 2955 . . . . . . . . . . . . 13 ((𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧)))
8577, 80, 843bitr4g 303 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ (𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧)))))
8656feqmptd 6288 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑘 = (𝑧𝐼 ↦ (𝑘𝑧)))
8761feqmptd 6288 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐷) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
8887adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
8933, 57, 63, 86, 88offval2 6956 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑋) = (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))))
9066feqmptd 6288 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐷) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
9190adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
9289, 91eqeq12d 2666 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑘𝑓𝑋) = 𝑌 ↔ (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧))))
9358, 62, 67, 87, 90offval2 6956 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐷) → (𝑋𝑓 + 𝑌) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))))
9493adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑋𝑓 + 𝑌) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))))
9586, 94eqeq12d 2666 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘 = (𝑋𝑓 + 𝑌) ↔ (𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧)))))
9685, 92, 953bitr4d 300 . . . . . . . . . . 11 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑘𝑓𝑋) = 𝑌𝑘 = (𝑋𝑓 + 𝑌)))
9796ifbid 4141 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ) = if(𝑘 = (𝑋𝑓 + 𝑌), 1 , 0 ))
9847, 54, 973eqtrd 2689 . . . . . . . . 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 ))
9997, 52eqeltrrd 2731 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → if(𝑘 = (𝑋𝑓 + 𝑌), 1 , 0 ) ∈ (Base‘𝑅))
10098, 99eqeltrd 2730 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑋))) ∈ (Base‘𝑅))
101 fveq2 6229 . . . . . . . . . 10 (𝑗 = 𝑋 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋))
102 oveq2 6698 . . . . . . . . . . 11 (𝑗 = 𝑋 → (𝑘𝑓𝑗) = (𝑘𝑓𝑋))
103102fveq2d 6233 . . . . . . . . . 10 (𝑗 = 𝑋 → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)) = ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑋)))
104101, 103oveq12d 6708 . . . . . . . . 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))))
10548, 104gsumsn 18400 . . . . . . . 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))))
10624, 25, 100, 105syl3anc 1366 . . . . . . 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))))
10721, 106, 983eqtrd 2689 . . . . . 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 ))
1086gsum0 17325 . . . . . . 7 (𝑅 Σg ∅) = 0
109 disjsn 4278 . . . . . . . . 9 (({𝑥𝐷𝑥𝑟𝑘} ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘})
1109ad2antrr 762 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑅 ∈ Ring)
1111, 48, 2, 5, 11mplelf 19481 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
112111ad2antrr 762 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
113 simpr 476 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘})
11432, 113sseldi 3634 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑗𝐷)
115112, 114ffvelrnd 6400 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) ∈ (Base‘𝑅))
1161, 48, 2, 5, 13mplelf 19481 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
117116ad2antrr 762 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
1188ad2antrr 762 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝐼𝑊)
119 simplr 807 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑘𝐷)
1205, 35psrbagconcl 19421 . . . . . . . . . . . . . . . 16 ((𝐼𝑊𝑘𝐷𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑗) ∈ {𝑥𝐷𝑥𝑟𝑘})
121118, 119, 113, 120syl3anc 1366 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑗) ∈ {𝑥𝐷𝑥𝑟𝑘})
12232, 121sseldi 3634 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑗) ∈ 𝐷)
123117, 122ffvelrnd 6400 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)) ∈ (Base‘𝑅))
12448, 3ringcl 18607 . . . . . . . . . . . . 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))
125110, 115, 123, 124syl3anc 1366 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))) ∈ (Base‘𝑅))
126 eqid 2651 . . . . . . . . . . . 12 (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) = (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))))
127125, 126fmptd 6425 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))):{𝑥𝐷𝑥𝑟𝑘}⟶(Base‘𝑅))
128 ffn 6083 . . . . . . . . . . 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})
129 fnresdisj 6039 . . . . . . . . . . 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}) = ∅))
130127, 128, 1293syl 18 . . . . . . . . . 10 ((𝜑𝑘𝐷) → (({𝑥𝐷𝑥𝑟𝑘} ∩ {𝑋}) = ∅ ↔ ((𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) ↾ {𝑋}) = ∅))
131130biimpa 500 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ ({𝑥𝐷𝑥𝑟𝑘} ∩ {𝑋}) = ∅) → ((𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) ↾ {𝑋}) = ∅)
132109, 131sylan2br 492 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) ↾ {𝑋}) = ∅)
133132oveq2d 6706 . . . . . . 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 ∅))
13462nn0red 11390 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℝ)
135 nn0addge1 11377 . . . . . . . . . . . . . 14 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℕ0) → (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
136134, 67, 135syl2anc 694 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
137136ralrimiva 2995 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
138 ovexd 6720 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → ((𝑋𝑧) + (𝑌𝑧)) ∈ V)
13958, 62, 138, 87, 93ofrfval2 6957 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → (𝑋𝑟 ≤ (𝑋𝑓 + 𝑌) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧))))
140137, 139mpbird 247 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 𝑋𝑟 ≤ (𝑋𝑓 + 𝑌))
141 breq1 4688 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥𝑟 ≤ (𝑋𝑓 + 𝑌) ↔ 𝑋𝑟 ≤ (𝑋𝑓 + 𝑌)))
142141elrab 3396 . . . . . . . . . . 11 (𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝑋𝑓 + 𝑌)} ↔ (𝑋𝐷𝑋𝑟 ≤ (𝑋𝑓 + 𝑌)))
14359, 140, 142sylanbrc 699 . . . . . . . . . 10 ((𝜑𝑘𝐷) → 𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝑋𝑓 + 𝑌)})
144 breq2 4689 . . . . . . . . . . . 12 (𝑘 = (𝑋𝑓 + 𝑌) → (𝑥𝑟𝑘𝑥𝑟 ≤ (𝑋𝑓 + 𝑌)))
145144rabbidv 3220 . . . . . . . . . . 11 (𝑘 = (𝑋𝑓 + 𝑌) → {𝑥𝐷𝑥𝑟𝑘} = {𝑥𝐷𝑥𝑟 ≤ (𝑋𝑓 + 𝑌)})
146145eleq2d 2716 . . . . . . . . . 10 (𝑘 = (𝑋𝑓 + 𝑌) → (𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘} ↔ 𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝑋𝑓 + 𝑌)}))
147143, 146syl5ibrcom 237 . . . . . . . . 9 ((𝜑𝑘𝐷) → (𝑘 = (𝑋𝑓 + 𝑌) → 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}))
148147con3dimp 456 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ¬ 𝑘 = (𝑋𝑓 + 𝑌))
149148iffalsed 4130 . . . . . . 7 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → if(𝑘 = (𝑋𝑓 + 𝑌), 1 , 0 ) = 0 )
150108, 133, 1493eqtr4a 2711 . . . . . 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 ))
151107, 150pm2.61dan 849 . . . . 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 ))
1529adantr 480 . . . . . . 7 ((𝜑𝑘𝐷) → 𝑅 ∈ Ring)
153 ringcmn 18627 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
154152, 153syl 17 . . . . . 6 ((𝜑𝑘𝐷) → 𝑅 ∈ CMnd)
1555psrbaglefi 19420 . . . . . . 7 ((𝐼𝑊𝑘𝐷) → {𝑥𝐷𝑥𝑟𝑘} ∈ Fin)
1568, 155sylan 487 . . . . . 6 ((𝜑𝑘𝐷) → {𝑥𝐷𝑥𝑟𝑘} ∈ Fin)
157 ssdif 3778 . . . . . . . . . . . 12 ({𝑥𝐷𝑥𝑟𝑘} ⊆ 𝐷 → ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋}))
15832, 157ax-mp 5 . . . . . . . . . . 11 ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋})
159158sseli 3632 . . . . . . . . . 10 (𝑗 ∈ ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋}) → 𝑗 ∈ (𝐷 ∖ {𝑋}))
160111adantr 480 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
161 eldifsni 4353 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦𝑋)
162161adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦𝑋)
163162neneqd 2828 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋)
164163iffalsed 4130 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 )
165 ovex 6718 . . . . . . . . . . . . . 14 (ℕ0𝑚 𝐼) ∈ V
1665, 165rabex2 4847 . . . . . . . . . . . . 13 𝐷 ∈ V
167166a1i 11 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → 𝐷 ∈ V)
168164, 167suppss2 7374 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋})
16943a1i 11 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 0 ∈ V)
170160, 168, 167, 169suppssr 7371 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ (𝐷 ∖ {𝑋})) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
171159, 170sylan2 490 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋})) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
172171oveq1d 6705 . . . . . . . 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))))
173 eldifi 3765 . . . . . . . . 9 (𝑗 ∈ ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋}) → 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘})
17448, 3, 6ringlz 18633 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)) ∈ (Base‘𝑅)) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))) = 0 )
175110, 123, 174syl2anc 694 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))) = 0 )
176173, 175sylan2 490 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋})) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))) = 0 )
177172, 176eqtrd 2685 . . . . . . 7 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋})) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))) = 0 )
178166rabex 4845 . . . . . . . 8 {𝑥𝐷𝑥𝑟𝑘} ∈ V
179178a1i 11 . . . . . . 7 ((𝜑𝑘𝐷) → {𝑥𝐷𝑥𝑟𝑘} ∈ V)
180177, 179suppss2 7374 . . . . . 6 ((𝜑𝑘𝐷) → ((𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) supp 0 ) ⊆ {𝑋})
181166mptrabex 6529 . . . . . . . . 9 (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) ∈ V
182 funmpt 5964 . . . . . . . . 9 Fun (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))))
183181, 182, 433pm3.2i 1259 . . . . . . . 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)
184183a1i 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))
185 snfi 8079 . . . . . . . 8 {𝑋} ∈ Fin
186185a1i 11 . . . . . . 7 ((𝜑𝑘𝐷) → {𝑋} ∈ Fin)
187 suppssfifsupp 8331 . . . . . . 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 )
188184, 186, 180, 187syl12anc 1364 . . . . . 6 ((𝜑𝑘𝐷) → (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) finSupp 0 )
18948, 6, 154, 156, 127, 180, 188gsumres 18360 . . . . 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))))))
190151, 189eqtr3d 2687 . . . 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))))))
191190mpteq2dva 4777 . . 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)))))))
19217, 191syl5eq 2697 . 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)))))))
19314, 192eqtr4d 2688 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 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  {crab 2945  Vcvv 3231   ∖ cdif 3604   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  ifcif 4119  {csn 4210   class class class wbr 4685   ↦ cmpt 4762  ◡ccnv 5142   ↾ cres 5145   “ cima 5146  Fun wfun 5920   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ∘𝑓 cof 6937   ∘𝑟 cofr 6938   supp csupp 7340   ↑𝑚 cmap 7899  Fincfn 7997   finSupp cfsupp 8316  ℂcc 9972  ℝcr 9973   + caddc 9977   ≤ cle 10113   − cmin 10304  ℕcn 11058  ℕ0cn0 11330  Basecbs 15904  .rcmulr 15989  0gc0g 16147   Σg cgsu 16148  Mndcmnd 17341  CMndccmn 18239  1rcur 18547  Ringcrg 18593   mPoly cmpl 19401 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-tset 16007  df-0g 16149  df-gsum 16150  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-psr 19404  df-mpl 19406 This theorem is referenced by:  mplcoe3  19514  mplcoe5  19516  mplmon2mul  19549
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