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Theorem mplmonmul 21902
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 2731 . . 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 21901 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)
12 mplmonmul.x . . . 4 (𝜑𝑌𝐷)
131, 2, 6, 7, 5, 8, 9, 12mplmon 21901 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) ∈ 𝐵)
141, 2, 3, 4, 5, 11, 13mplmul 21881 . 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 2735 . . . . 5 (𝑦 = 𝑘 → (𝑦 = (𝑋f + 𝑌) ↔ 𝑘 = (𝑋f + 𝑌)))
1615ifbid 4551 . . . 4 (𝑦 = 𝑘 → if(𝑦 = (𝑋f + 𝑌), 1 , 0 ) = if(𝑘 = (𝑋f + 𝑌), 1 , 0 ))
1716cbvmptv 5261 . . 3 (𝑦𝐷 ↦ if(𝑦 = (𝑋f + 𝑌), 1 , 0 )) = (𝑘𝐷 ↦ if(𝑘 = (𝑋f + 𝑌), 1 , 0 ))
18 simpr 484 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑋 ∈ {𝑥𝐷𝑥r𝑘})
1918snssd 4812 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → {𝑋} ⊆ {𝑥𝐷𝑥r𝑘})
2019resmptd 6040 . . . . . . . 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 7428 . . . . . . 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 723 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑅 ∈ Ring)
23 ringmnd 20144 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
2422, 23syl 17 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑅 ∈ Mnd)
2510ad2antrr 723 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑋𝐷)
26 iftrue 4534 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1 , 0 ) = 1 )
27 eqid 2731 . . . . . . . . . . . . 13 (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))
287fvexi 6905 . . . . . . . . . . . . 13 1 ∈ V
2926, 27, 28fvmpt 6998 . . . . . . . . . . . 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 4077 . . . . . . . . . . . . 13 {𝑥𝐷𝑥r𝑘} ⊆ 𝐷
32 simplr 766 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑘𝐷)
33 eqid 2731 . . . . . . . . . . . . . . 15 {𝑥𝐷𝑥r𝑘} = {𝑥𝐷𝑥r𝑘}
345, 33psrbagconcl 21797 . . . . . . . . . . . . . 14 ((𝑘𝐷𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑋) ∈ {𝑥𝐷𝑥r𝑘})
3532, 18, 34syl2anc 583 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑋) ∈ {𝑥𝐷𝑥r𝑘})
3631, 35sselid 3980 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑋) ∈ 𝐷)
37 eqeq1 2735 . . . . . . . . . . . . . 14 (𝑦 = (𝑘f𝑋) → (𝑦 = 𝑌 ↔ (𝑘f𝑋) = 𝑌))
3837ifbid 4551 . . . . . . . . . . . . 13 (𝑦 = (𝑘f𝑋) → if(𝑦 = 𝑌, 1 , 0 ) = if((𝑘f𝑋) = 𝑌, 1 , 0 ))
39 eqid 2731 . . . . . . . . . . . . 13 (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))
406fvexi 6905 . . . . . . . . . . . . . 14 0 ∈ V
4128, 40ifex 4578 . . . . . . . . . . . . 13 if((𝑘f𝑋) = 𝑌, 1 , 0 ) ∈ V
4238, 39, 41fvmpt 6998 . . . . . . . . . . . 12 ((𝑘f𝑋) ∈ 𝐷 → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋)) = if((𝑘f𝑋) = 𝑌, 1 , 0 ))
4336, 42syl 17 . . . . . . . . . . 11 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋)) = if((𝑘f𝑋) = 𝑌, 1 , 0 ))
4430, 43oveq12d 7430 . . . . . . . . . 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 )))
45 eqid 2731 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘𝑅)
4645, 7ringidcl 20161 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
4745, 6ring0cl 20162 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
4846, 47ifcld 4574 . . . . . . . . . . . 12 (𝑅 ∈ Ring → if((𝑘f𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅))
4922, 48syl 17 . . . . . . . . . . 11 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → if((𝑘f𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅))
5045, 3, 7ringlidm 20164 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ if((𝑘f𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅)) → ( 1 (.r𝑅)if((𝑘f𝑋) = 𝑌, 1 , 0 )) = if((𝑘f𝑋) = 𝑌, 1 , 0 ))
5122, 49, 50syl2anc 583 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ( 1 (.r𝑅)if((𝑘f𝑋) = 𝑌, 1 , 0 )) = if((𝑘f𝑋) = 𝑌, 1 , 0 ))
525psrbagf 21781 . . . . . . . . . . . . . . . . . 18 (𝑘𝐷𝑘:𝐼⟶ℕ0)
5332, 52syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑘:𝐼⟶ℕ0)
5453ffvelcdmda 7086 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
5510adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐷) → 𝑋𝐷)
565psrbagf 21781 . . . . . . . . . . . . . . . . . . 19 (𝑋𝐷𝑋:𝐼⟶ℕ0)
5755, 56syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑋:𝐼⟶ℕ0)
5857ffvelcdmda 7086 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
5958adantlr 712 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
605psrbagf 21781 . . . . . . . . . . . . . . . . . . . 20 (𝑌𝐷𝑌:𝐼⟶ℕ0)
6112, 60syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑌:𝐼⟶ℕ0)
6261adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑌:𝐼⟶ℕ0)
6362ffvelcdmda 7086 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
6463adantlr 712 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
65 nn0cn 12489 . . . . . . . . . . . . . . . . 17 ((𝑘𝑧) ∈ ℕ0 → (𝑘𝑧) ∈ ℂ)
66 nn0cn 12489 . . . . . . . . . . . . . . . . 17 ((𝑋𝑧) ∈ ℕ0 → (𝑋𝑧) ∈ ℂ)
67 nn0cn 12489 . . . . . . . . . . . . . . . . 17 ((𝑌𝑧) ∈ ℕ0 → (𝑌𝑧) ∈ ℂ)
68 subadd 11470 . . . . . . . . . . . . . . . . 17 (((𝑘𝑧) ∈ ℂ ∧ (𝑋𝑧) ∈ ℂ ∧ (𝑌𝑧) ∈ ℂ) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
6965, 66, 67, 68syl3an 1159 . . . . . . . . . . . . . . . 16 (((𝑘𝑧) ∈ ℕ0 ∧ (𝑋𝑧) ∈ ℕ0 ∧ (𝑌𝑧) ∈ ℕ0) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
7054, 59, 64, 69syl3anc 1370 . . . . . . . . . . . . . . 15 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) ∧ 𝑧𝐼) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
71 eqcom 2738 . . . . . . . . . . . . . . 15 (((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧) ↔ (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧)))
7270, 71bitrdi 287 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) ∧ 𝑧𝐼) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
7372ralbidva 3174 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
74 mpteqb 7017 . . . . . . . . . . . . . 14 (∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) ∈ V → ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ ∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧)))
75 ovexd 7447 . . . . . . . . . . . . . 14 (𝑧𝐼 → ((𝑘𝑧) − (𝑋𝑧)) ∈ V)
7674, 75mprg 3066 . . . . . . . . . . . . 13 ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ ∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧))
77 mpteqb 7017 . . . . . . . . . . . . . 14 (∀𝑧𝐼 (𝑘𝑧) ∈ V → ((𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
78 fvexd 6906 . . . . . . . . . . . . . 14 (𝑧𝐼 → (𝑘𝑧) ∈ V)
7977, 78mprg 3066 . . . . . . . . . . . . 13 ((𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧)))
8073, 76, 793bitr4g 314 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ (𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧)))))
818ad2antrr 723 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝐼𝑊)
8253feqmptd 6960 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑘 = (𝑧𝐼 ↦ (𝑘𝑧)))
8357feqmptd 6960 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐷) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
8483adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
8581, 54, 59, 82, 84offval2 7694 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑋) = (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))))
8662feqmptd 6960 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐷) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
8786adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
8885, 87eqeq12d 2747 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑘f𝑋) = 𝑌 ↔ (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧))))
898adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐷) → 𝐼𝑊)
9089, 58, 63, 83, 86offval2 7694 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐷) → (𝑋f + 𝑌) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))))
9190adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑋f + 𝑌) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))))
9282, 91eqeq12d 2747 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘 = (𝑋f + 𝑌) ↔ (𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧)))))
9380, 88, 923bitr4d 311 . . . . . . . . . . 11 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑘f𝑋) = 𝑌𝑘 = (𝑋f + 𝑌)))
9493ifbid 4551 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → if((𝑘f𝑋) = 𝑌, 1 , 0 ) = if(𝑘 = (𝑋f + 𝑌), 1 , 0 ))
9544, 51, 943eqtrd 2775 . . . . . . . . 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, 49eqeltrrd 2833 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → if(𝑘 = (𝑋f + 𝑌), 1 , 0 ) ∈ (Base‘𝑅))
9795, 96eqeltrd 2832 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋))) ∈ (Base‘𝑅))
98 fveq2 6891 . . . . . . . . . 10 (𝑗 = 𝑋 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋))
99 oveq2 7420 . . . . . . . . . . 11 (𝑗 = 𝑋 → (𝑘f𝑗) = (𝑘f𝑋))
10099fveq2d 6895 . . . . . . . . . 10 (𝑗 = 𝑋 → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)) = ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋)))
10198, 100oveq12d 7430 . . . . . . . . 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))))
10245, 101gsumsn 19870 . . . . . . . 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 1370 . . . . . . 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 2775 . . . . . 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 18615 . . . . . . 7 (𝑅 Σg ∅) = 0
106 disjsn 4715 . . . . . . . . 9 (({𝑥𝐷𝑥r𝑘} ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ {𝑥𝐷𝑥r𝑘})
1079ad2antrr 723 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑅 ∈ Ring)
1081, 45, 2, 5, 11mplelf 21868 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
109108ad2antrr 723 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
110 simpr 484 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑗 ∈ {𝑥𝐷𝑥r𝑘})
11131, 110sselid 3980 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑗𝐷)
112109, 111ffvelcdmd 7087 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) ∈ (Base‘𝑅))
1131, 45, 2, 5, 13mplelf 21868 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
114113ad2antrr 723 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
115 simplr 766 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑘𝐷)
1165, 33psrbagconcl 21797 . . . . . . . . . . . . . . . 16 ((𝑘𝐷𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑗) ∈ {𝑥𝐷𝑥r𝑘})
117115, 110, 116syl2anc 583 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑗) ∈ {𝑥𝐷𝑥r𝑘})
11831, 117sselid 3980 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑗) ∈ 𝐷)
119114, 118ffvelcdmd 7087 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)) ∈ (Base‘𝑅))
12045, 3ringcl 20151 . . . . . . . . . . . . 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))
121107, 112, 119, 120syl3anc 1370 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) ∈ (Base‘𝑅))
122121fmpttd 7116 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))):{𝑥𝐷𝑥r𝑘}⟶(Base‘𝑅))
123 ffn 6717 . . . . . . . . . . 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})
124 fnresdisj 6670 . . . . . . . . . . 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}) = ∅))
125122, 123, 1243syl 18 . . . . . . . . . 10 ((𝜑𝑘𝐷) → (({𝑥𝐷𝑥r𝑘} ∩ {𝑋}) = ∅ ↔ ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋}) = ∅))
126125biimpa 476 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ ({𝑥𝐷𝑥r𝑘} ∩ {𝑋}) = ∅) → ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋}) = ∅)
127106, 126sylan2br 594 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋}) = ∅)
128127oveq2d 7428 . . . . . . 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 ∅))
129 breq1 5151 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥r ≤ (𝑋f + 𝑌) ↔ 𝑋r ≤ (𝑋f + 𝑌)))
13058nn0red 12540 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℝ)
131 nn0addge1 12525 . . . . . . . . . . . . . 14 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℕ0) → (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
132130, 63, 131syl2anc 583 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
133132ralrimiva 3145 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
134 ovexd 7447 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → ((𝑋𝑧) + (𝑌𝑧)) ∈ V)
13589, 58, 134, 83, 90ofrfval2 7695 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → (𝑋r ≤ (𝑋f + 𝑌) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧))))
136133, 135mpbird 257 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 𝑋r ≤ (𝑋f + 𝑌))
137129, 55, 136elrabd 3685 . . . . . . . . . 10 ((𝜑𝑘𝐷) → 𝑋 ∈ {𝑥𝐷𝑥r ≤ (𝑋f + 𝑌)})
138 breq2 5152 . . . . . . . . . . . 12 (𝑘 = (𝑋f + 𝑌) → (𝑥r𝑘𝑥r ≤ (𝑋f + 𝑌)))
139138rabbidv 3439 . . . . . . . . . . 11 (𝑘 = (𝑋f + 𝑌) → {𝑥𝐷𝑥r𝑘} = {𝑥𝐷𝑥r ≤ (𝑋f + 𝑌)})
140139eleq2d 2818 . . . . . . . . . 10 (𝑘 = (𝑋f + 𝑌) → (𝑋 ∈ {𝑥𝐷𝑥r𝑘} ↔ 𝑋 ∈ {𝑥𝐷𝑥r ≤ (𝑋f + 𝑌)}))
141137, 140syl5ibrcom 246 . . . . . . . . 9 ((𝜑𝑘𝐷) → (𝑘 = (𝑋f + 𝑌) → 𝑋 ∈ {𝑥𝐷𝑥r𝑘}))
142141con3dimp 408 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ¬ 𝑘 = (𝑋f + 𝑌))
143142iffalsed 4539 . . . . . . 7 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → if(𝑘 = (𝑋f + 𝑌), 1 , 0 ) = 0 )
144105, 128, 1433eqtr4a 2797 . . . . . 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 ))
145104, 144pm2.61dan 810 . . . . 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 ))
1469adantr 480 . . . . . . 7 ((𝜑𝑘𝐷) → 𝑅 ∈ Ring)
147 ringcmn 20177 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
148146, 147syl 17 . . . . . 6 ((𝜑𝑘𝐷) → 𝑅 ∈ CMnd)
1495psrbaglefi 21795 . . . . . . 7 (𝑘𝐷 → {𝑥𝐷𝑥r𝑘} ∈ Fin)
150149adantl 481 . . . . . 6 ((𝜑𝑘𝐷) → {𝑥𝐷𝑥r𝑘} ∈ Fin)
151 ssdif 4139 . . . . . . . . . . . 12 ({𝑥𝐷𝑥r𝑘} ⊆ 𝐷 → ({𝑥𝐷𝑥r𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋}))
15231, 151ax-mp 5 . . . . . . . . . . 11 ({𝑥𝐷𝑥r𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋})
153152sseli 3978 . . . . . . . . . 10 (𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋}) → 𝑗 ∈ (𝐷 ∖ {𝑋}))
154108adantr 480 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
155 eldifsni 4793 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦𝑋)
156155adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦𝑋)
157156neneqd 2944 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋)
158157iffalsed 4539 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 )
159 ovex 7445 . . . . . . . . . . . . . 14 (ℕ0m 𝐼) ∈ V
1605, 159rabex2 5334 . . . . . . . . . . . . 13 𝐷 ∈ V
161160a1i 11 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → 𝐷 ∈ V)
162158, 161suppss2 8191 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋})
16340a1i 11 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 0 ∈ V)
164154, 162, 161, 163suppssr 8186 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ (𝐷 ∖ {𝑋})) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
165153, 164sylan2 592 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋})) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
166165oveq1d 7427 . . . . . . . 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))))
167 eldifi 4126 . . . . . . . . 9 (𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋}) → 𝑗 ∈ {𝑥𝐷𝑥r𝑘})
16845, 3, 6ringlz 20188 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)) ∈ (Base‘𝑅)) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) = 0 )
169107, 119, 168syl2anc 583 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) = 0 )
170167, 169sylan2 592 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋})) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) = 0 )
171166, 170eqtrd 2771 . . . . . . 7 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋})) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) = 0 )
172160rabex 5332 . . . . . . . 8 {𝑥𝐷𝑥r𝑘} ∈ V
173172a1i 11 . . . . . . 7 ((𝜑𝑘𝐷) → {𝑥𝐷𝑥r𝑘} ∈ V)
174171, 173suppss2 8191 . . . . . 6 ((𝜑𝑘𝐷) → ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) supp 0 ) ⊆ {𝑋})
175160mptrabex 7229 . . . . . . . . 9 (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ∈ V
176 funmpt 6586 . . . . . . . . 9 Fun (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))))
177175, 176, 403pm3.2i 1338 . . . . . . . 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)
178177a1i 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))
179 snfi 9050 . . . . . . . 8 {𝑋} ∈ Fin
180179a1i 11 . . . . . . 7 ((𝜑𝑘𝐷) → {𝑋} ∈ Fin)
181 suppssfifsupp 9384 . . . . . . 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 )
182178, 180, 174, 181syl12anc 834 . . . . . 6 ((𝜑𝑘𝐷) → (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) finSupp 0 )
18345, 6, 148, 150, 122, 174, 182gsumres 19829 . . . . 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))))))
184145, 183eqtr3d 2773 . . . 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))))))
185184mpteq2dva 5248 . . 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)))))))
18617, 185eqtrid 2783 . 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)))))))
18714, 186eqtr4d 2774 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 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wne 2939  wral 3060  {crab 3431  Vcvv 3473  cdif 3945  cin 3947  wss 3948  c0 4322  ifcif 4528  {csn 4628   class class class wbr 5148  cmpt 5231  ccnv 5675  cres 5678  cima 5679  Fun wfun 6537   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7412  f cof 7672  r cofr 7673   supp csupp 8151  m cmap 8826  Fincfn 8945   finSupp cfsupp 9367  cc 11114  cr 11115   + caddc 11119  cle 11256  cmin 11451  cn 12219  0cn0 12479  Basecbs 17151  .rcmulr 17205  0gc0g 17392   Σg cgsu 17393  Mndcmnd 18665  CMndccmn 19696  1rcur 20082  Ringcrg 20134   mPoly cmpl 21769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  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-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-ofr 7675  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8152  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-map 8828  df-pm 8829  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9368  df-oi 9511  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-uz 12830  df-fz 13492  df-fzo 13635  df-seq 13974  df-hash 14298  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-sca 17220  df-vsca 17221  df-tset 17223  df-0g 17394  df-gsum 17395  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-grp 18864  df-minusg 18865  df-mulg 18994  df-cntz 19229  df-cmn 19698  df-abl 19699  df-mgp 20036  df-rng 20054  df-ur 20083  df-ring 20136  df-psr 21772  df-mpl 21774
This theorem is referenced by:  mplcoe3  21904  mplcoe5  21906  mplmon2mul  21941
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