MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mplmonmul Structured version   Visualization version   GIF version

Theorem mplmonmul 19587
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 2724 . . 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 19586 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)
12 mplmonmul.x . . . 4 (𝜑𝑌𝐷)
131, 2, 6, 7, 5, 8, 9, 12mplmon 19586 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) ∈ 𝐵)
141, 2, 3, 4, 5, 11, 13mplmul 19566 . 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 2728 . . . . 5 (𝑦 = 𝑘 → (𝑦 = (𝑋𝑓 + 𝑌) ↔ 𝑘 = (𝑋𝑓 + 𝑌)))
1615ifbid 4216 . . . 4 (𝑦 = 𝑘 → if(𝑦 = (𝑋𝑓 + 𝑌), 1 , 0 ) = if(𝑘 = (𝑋𝑓 + 𝑌), 1 , 0 ))
1716cbvmptv 4858 . . 3 (𝑦𝐷 ↦ if(𝑦 = (𝑋𝑓 + 𝑌), 1 , 0 )) = (𝑘𝐷 ↦ if(𝑘 = (𝑋𝑓 + 𝑌), 1 , 0 ))
18 simpr 479 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘})
1918snssd 4448 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → {𝑋} ⊆ {𝑥𝐷𝑥𝑟𝑘})
2019resmptd 5562 . . . . . . . 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 6781 . . . . . . 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 764 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑅 ∈ Ring)
23 ringmnd 18677 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
2422, 23syl 17 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑅 ∈ Mnd)
2510ad2antrr 764 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑋𝐷)
26 iftrue 4200 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1 , 0 ) = 1 )
27 eqid 2724 . . . . . . . . . . . . 13 (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))
28 fvex 6314 . . . . . . . . . . . . . 14 (1r𝑅) ∈ V
297, 28eqeltri 2799 . . . . . . . . . . . . 13 1 ∈ V
3026, 27, 29fvmpt 6396 . . . . . . . . . . . 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 3793 . . . . . . . . . . . . 13 {𝑥𝐷𝑥𝑟𝑘} ⊆ 𝐷
338ad2antrr 764 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝐼𝑊)
34 simplr 809 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑘𝐷)
35 eqid 2724 . . . . . . . . . . . . . . 15 {𝑥𝐷𝑥𝑟𝑘} = {𝑥𝐷𝑥𝑟𝑘}
365, 35psrbagconcl 19496 . . . . . . . . . . . . . 14 ((𝐼𝑊𝑘𝐷𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑋) ∈ {𝑥𝐷𝑥𝑟𝑘})
3733, 34, 18, 36syl3anc 1439 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑋) ∈ {𝑥𝐷𝑥𝑟𝑘})
3832, 37sseldi 3707 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑋) ∈ 𝐷)
39 eqeq1 2728 . . . . . . . . . . . . . 14 (𝑦 = (𝑘𝑓𝑋) → (𝑦 = 𝑌 ↔ (𝑘𝑓𝑋) = 𝑌))
4039ifbid 4216 . . . . . . . . . . . . 13 (𝑦 = (𝑘𝑓𝑋) → if(𝑦 = 𝑌, 1 , 0 ) = if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ))
41 eqid 2724 . . . . . . . . . . . . 13 (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))
42 fvex 6314 . . . . . . . . . . . . . . 15 (0g𝑅) ∈ V
436, 42eqeltri 2799 . . . . . . . . . . . . . 14 0 ∈ V
4429, 43ifex 4264 . . . . . . . . . . . . 13 if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ) ∈ V
4540, 41, 44fvmpt 6396 . . . . . . . . . . . 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 6783 . . . . . . . . . 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 2724 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘𝑅)
4948, 7ringidcl 18689 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
5048, 6ring0cl 18690 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
5149, 50ifcld 4239 . . . . . . . . . . . 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 18692 . . . . . . . . . . 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 696 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ( 1 (.r𝑅)if((𝑘𝑓𝑋) = 𝑌, 1 , 0 )) = if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ))
555psrbagf 19488 . . . . . . . . . . . . . . . . . 18 ((𝐼𝑊𝑘𝐷) → 𝑘:𝐼⟶ℕ0)
5633, 34, 55syl2anc 696 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑘:𝐼⟶ℕ0)
5756ffvelrnda 6474 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
588adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐷) → 𝐼𝑊)
5910adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐷) → 𝑋𝐷)
605psrbagf 19488 . . . . . . . . . . . . . . . . . . 19 ((𝐼𝑊𝑋𝐷) → 𝑋:𝐼⟶ℕ0)
6158, 59, 60syl2anc 696 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑋:𝐼⟶ℕ0)
6261ffvelrnda 6474 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
6362adantlr 753 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
645psrbagf 19488 . . . . . . . . . . . . . . . . . . . 20 ((𝐼𝑊𝑌𝐷) → 𝑌:𝐼⟶ℕ0)
658, 12, 64syl2anc 696 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑌:𝐼⟶ℕ0)
6665adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑌:𝐼⟶ℕ0)
6766ffvelrnda 6474 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
6867adantlr 753 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
69 nn0cn 11415 . . . . . . . . . . . . . . . . 17 ((𝑘𝑧) ∈ ℕ0 → (𝑘𝑧) ∈ ℂ)
70 nn0cn 11415 . . . . . . . . . . . . . . . . 17 ((𝑋𝑧) ∈ ℕ0 → (𝑋𝑧) ∈ ℂ)
71 nn0cn 11415 . . . . . . . . . . . . . . . . 17 ((𝑌𝑧) ∈ ℕ0 → (𝑌𝑧) ∈ ℂ)
72 subadd 10397 . . . . . . . . . . . . . . . . 17 (((𝑘𝑧) ∈ ℂ ∧ (𝑋𝑧) ∈ ℂ ∧ (𝑌𝑧) ∈ ℂ) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
7369, 70, 71, 72syl3an 1481 . . . . . . . . . . . . . . . 16 (((𝑘𝑧) ∈ ℕ0 ∧ (𝑋𝑧) ∈ ℕ0 ∧ (𝑌𝑧) ∈ ℕ0) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
7457, 63, 68, 73syl3anc 1439 . . . . . . . . . . . . . . 15 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) ∧ 𝑧𝐼) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
75 eqcom 2731 . . . . . . . . . . . . . . 15 (((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧) ↔ (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧)))
7674, 75syl6bb 276 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) ∧ 𝑧𝐼) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
7776ralbidva 3087 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
78 mpteqb 6413 . . . . . . . . . . . . . 14 (∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) ∈ V → ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ ∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧)))
79 ovexd 6795 . . . . . . . . . . . . . 14 (𝑧𝐼 → ((𝑘𝑧) − (𝑋𝑧)) ∈ V)
8078, 79mprg 3028 . . . . . . . . . . . . 13 ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ ∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧))
81 mpteqb 6413 . . . . . . . . . . . . . 14 (∀𝑧𝐼 (𝑘𝑧) ∈ V → ((𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
82 fvex 6314 . . . . . . . . . . . . . . 15 (𝑘𝑧) ∈ V
8382a1i 11 . . . . . . . . . . . . . 14 (𝑧𝐼 → (𝑘𝑧) ∈ V)
8481, 83mprg 3028 . . . . . . . . . . . . 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 6363 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑘 = (𝑧𝐼 ↦ (𝑘𝑧)))
8761feqmptd 6363 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐷) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
8887adantr 472 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
8933, 57, 63, 86, 88offval2 7031 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑋) = (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))))
9066feqmptd 6363 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐷) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
9190adantr 472 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
9289, 91eqeq12d 2739 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑘𝑓𝑋) = 𝑌 ↔ (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧))))
9358, 62, 67, 87, 90offval2 7031 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐷) → (𝑋𝑓 + 𝑌) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))))
9493adantr 472 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑋𝑓 + 𝑌) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))))
9586, 94eqeq12d 2739 . . . . . . . . . . . 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 4216 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ) = if(𝑘 = (𝑋𝑓 + 𝑌), 1 , 0 ))
9847, 54, 973eqtrd 2762 . . . . . . . . 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 2804 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → if(𝑘 = (𝑋𝑓 + 𝑌), 1 , 0 ) ∈ (Base‘𝑅))
10098, 99eqeltrd 2803 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑋))) ∈ (Base‘𝑅))
101 fveq2 6304 . . . . . . . . . 10 (𝑗 = 𝑋 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋))
102 oveq2 6773 . . . . . . . . . . 11 (𝑗 = 𝑋 → (𝑘𝑓𝑗) = (𝑘𝑓𝑋))
103102fveq2d 6308 . . . . . . . . . 10 (𝑗 = 𝑋 → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)) = ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑋)))
104101, 103oveq12d 6783 . . . . . . . . 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 18475 . . . . . . . 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 1439 . . . . . . 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 2762 . . . . . 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 17400 . . . . . . 7 (𝑅 Σg ∅) = 0
109 disjsn 4353 . . . . . . . . 9 (({𝑥𝐷𝑥𝑟𝑘} ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘})
1109ad2antrr 764 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑅 ∈ Ring)
1111, 48, 2, 5, 11mplelf 19556 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
112111ad2antrr 764 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
113 simpr 479 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘})
11432, 113sseldi 3707 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑗𝐷)
115112, 114ffvelrnd 6475 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) ∈ (Base‘𝑅))
1161, 48, 2, 5, 13mplelf 19556 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
117116ad2antrr 764 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
1188ad2antrr 764 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝐼𝑊)
119 simplr 809 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑘𝐷)
1205, 35psrbagconcl 19496 . . . . . . . . . . . . . . . 16 ((𝐼𝑊𝑘𝐷𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑗) ∈ {𝑥𝐷𝑥𝑟𝑘})
121118, 119, 113, 120syl3anc 1439 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑗) ∈ {𝑥𝐷𝑥𝑟𝑘})
12232, 121sseldi 3707 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑗) ∈ 𝐷)
123117, 122ffvelrnd 6475 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)) ∈ (Base‘𝑅))
12448, 3ringcl 18682 . . . . . . . . . . . . 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 1439 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))) ∈ (Base‘𝑅))
126 eqid 2724 . . . . . . . . . . . 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 6500 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))):{𝑥𝐷𝑥𝑟𝑘}⟶(Base‘𝑅))
128 ffn 6158 . . . . . . . . . . 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 6114 . . . . . . . . . . 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 502 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ ({𝑥𝐷𝑥𝑟𝑘} ∩ {𝑋}) = ∅) → ((𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) ↾ {𝑋}) = ∅)
132109, 131sylan2br 494 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) ↾ {𝑋}) = ∅)
133132oveq2d 6781 . . . . . . 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 11465 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℝ)
135 nn0addge1 11452 . . . . . . . . . . . . . 14 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℕ0) → (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
136134, 67, 135syl2anc 696 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
137136ralrimiva 3068 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
138 ovexd 6795 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → ((𝑋𝑧) + (𝑌𝑧)) ∈ V)
13958, 62, 138, 87, 93ofrfval2 7032 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → (𝑋𝑟 ≤ (𝑋𝑓 + 𝑌) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧))))
140137, 139mpbird 247 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 𝑋𝑟 ≤ (𝑋𝑓 + 𝑌))
141 breq1 4763 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥𝑟 ≤ (𝑋𝑓 + 𝑌) ↔ 𝑋𝑟 ≤ (𝑋𝑓 + 𝑌)))
142141elrab 3469 . . . . . . . . . . 11 (𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝑋𝑓 + 𝑌)} ↔ (𝑋𝐷𝑋𝑟 ≤ (𝑋𝑓 + 𝑌)))
14359, 140, 142sylanbrc 701 . . . . . . . . . 10 ((𝜑𝑘𝐷) → 𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝑋𝑓 + 𝑌)})
144 breq2 4764 . . . . . . . . . . . 12 (𝑘 = (𝑋𝑓 + 𝑌) → (𝑥𝑟𝑘𝑥𝑟 ≤ (𝑋𝑓 + 𝑌)))
145144rabbidv 3293 . . . . . . . . . . 11 (𝑘 = (𝑋𝑓 + 𝑌) → {𝑥𝐷𝑥𝑟𝑘} = {𝑥𝐷𝑥𝑟 ≤ (𝑋𝑓 + 𝑌)})
146145eleq2d 2789 . . . . . . . . . 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 4205 . . . . . . 7 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → if(𝑘 = (𝑋𝑓 + 𝑌), 1 , 0 ) = 0 )
150108, 133, 1493eqtr4a 2784 . . . . . 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 867 . . . . 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 472 . . . . . . 7 ((𝜑𝑘𝐷) → 𝑅 ∈ Ring)
153 ringcmn 18702 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
154152, 153syl 17 . . . . . 6 ((𝜑𝑘𝐷) → 𝑅 ∈ CMnd)
1555psrbaglefi 19495 . . . . . . 7 ((𝐼𝑊𝑘𝐷) → {𝑥𝐷𝑥𝑟𝑘} ∈ Fin)
1568, 155sylan 489 . . . . . 6 ((𝜑𝑘𝐷) → {𝑥𝐷𝑥𝑟𝑘} ∈ Fin)
157 ssdif 3853 . . . . . . . . . . . 12 ({𝑥𝐷𝑥𝑟𝑘} ⊆ 𝐷 → ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋}))
15832, 157ax-mp 5 . . . . . . . . . . 11 ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋})
159158sseli 3705 . . . . . . . . . 10 (𝑗 ∈ ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋}) → 𝑗 ∈ (𝐷 ∖ {𝑋}))
160111adantr 472 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
161 eldifsni 4429 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦𝑋)
162161adantl 473 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦𝑋)
163162neneqd 2901 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋)
164163iffalsed 4205 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 )
165 ovex 6793 . . . . . . . . . . . . . 14 (ℕ0𝑚 𝐼) ∈ V
1665, 165rabex2 4922 . . . . . . . . . . . . 13 𝐷 ∈ V
167166a1i 11 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → 𝐷 ∈ V)
168164, 167suppss2 7449 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋})
16943a1i 11 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 0 ∈ V)
170160, 168, 167, 169suppssr 7446 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ (𝐷 ∖ {𝑋})) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
171159, 170sylan2 492 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋})) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
172171oveq1d 6780 . . . . . . . 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 3840 . . . . . . . . 9 (𝑗 ∈ ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋}) → 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘})
17448, 3, 6ringlz 18708 . . . . . . . . . 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 696 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))) = 0 )
176173, 175sylan2 492 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋})) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))) = 0 )
177172, 176eqtrd 2758 . . . . . . 7 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋})) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))) = 0 )
178166rabex 4920 . . . . . . . 8 {𝑥𝐷𝑥𝑟𝑘} ∈ V
179178a1i 11 . . . . . . 7 ((𝜑𝑘𝐷) → {𝑥𝐷𝑥𝑟𝑘} ∈ V)
180177, 179suppss2 7449 . . . . . 6 ((𝜑𝑘𝐷) → ((𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) supp 0 ) ⊆ {𝑋})
181166mptrabex 6604 . . . . . . . . 9 (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) ∈ V
182 funmpt 6039 . . . . . . . . 9 Fun (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))))
183181, 182, 433pm3.2i 1376 . . . . . . . 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 8154 . . . . . . . 8 {𝑋} ∈ Fin
186185a1i 11 . . . . . . 7 ((𝜑𝑘𝐷) → {𝑋} ∈ Fin)
187 suppssfifsupp 8406 . . . . . . 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 1437 . . . . . 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 18435 . . . . 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 2760 . . . 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 4852 . . 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 2770 . 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 2761 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 1072   = wceq 1596  wcel 2103  wne 2896  wral 3014  {crab 3018  Vcvv 3304  cdif 3677  cin 3679  wss 3680  c0 4023  ifcif 4194  {csn 4285   class class class wbr 4760  cmpt 4837  ccnv 5217  cres 5220  cima 5221  Fun wfun 5995   Fn wfn 5996  wf 5997  cfv 6001  (class class class)co 6765  𝑓 cof 7012  𝑟 cofr 7013   supp csupp 7415  𝑚 cmap 7974  Fincfn 8072   finSupp cfsupp 8391  cc 10047  cr 10048   + caddc 10052  cle 10188  cmin 10379  cn 11133  0cn0 11405  Basecbs 15980  .rcmulr 16065  0gc0g 16223   Σg cgsu 16224  Mndcmnd 17416  CMndccmn 18314  1rcur 18622  Ringcrg 18668   mPoly cmpl 19476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-of 7014  df-ofr 7015  df-om 7183  df-1st 7285  df-2nd 7286  df-supp 7416  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-2o 7681  df-oadd 7684  df-er 7862  df-map 7976  df-pm 7977  df-ixp 8026  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-fsupp 8392  df-oi 8531  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-2 11192  df-3 11193  df-4 11194  df-5 11195  df-6 11196  df-7 11197  df-8 11198  df-9 11199  df-n0 11406  df-z 11491  df-uz 11801  df-fz 12441  df-fzo 12581  df-seq 12917  df-hash 13233  df-struct 15982  df-ndx 15983  df-slot 15984  df-base 15986  df-sets 15987  df-ress 15988  df-plusg 16077  df-mulr 16078  df-sca 16080  df-vsca 16081  df-tset 16083  df-0g 16225  df-gsum 16226  df-mgm 17364  df-sgrp 17406  df-mnd 17417  df-grp 17547  df-minusg 17548  df-mulg 17663  df-cntz 17871  df-cmn 18316  df-abl 18317  df-mgp 18611  df-ur 18623  df-ring 18670  df-psr 19479  df-mpl 19481
This theorem is referenced by:  mplcoe3  19589  mplcoe5  19591  mplmon2mul  19624
  Copyright terms: Public domain W3C validator