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Theorem mplmonmul 21437
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 2736 . . 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 21436 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)
12 mplmonmul.x . . . 4 (𝜑𝑌𝐷)
131, 2, 6, 7, 5, 8, 9, 12mplmon 21436 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) ∈ 𝐵)
141, 2, 3, 4, 5, 11, 13mplmul 21415 . 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 2740 . . . . 5 (𝑦 = 𝑘 → (𝑦 = (𝑋f + 𝑌) ↔ 𝑘 = (𝑋f + 𝑌)))
1615ifbid 4509 . . . 4 (𝑦 = 𝑘 → if(𝑦 = (𝑋f + 𝑌), 1 , 0 ) = if(𝑘 = (𝑋f + 𝑌), 1 , 0 ))
1716cbvmptv 5218 . . 3 (𝑦𝐷 ↦ if(𝑦 = (𝑋f + 𝑌), 1 , 0 )) = (𝑘𝐷 ↦ if(𝑘 = (𝑋f + 𝑌), 1 , 0 ))
18 simpr 485 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑋 ∈ {𝑥𝐷𝑥r𝑘})
1918snssd 4769 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → {𝑋} ⊆ {𝑥𝐷𝑥r𝑘})
2019resmptd 5994 . . . . . . . 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 7373 . . . . . . 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 724 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑅 ∈ Ring)
23 ringmnd 19974 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
2422, 23syl 17 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑅 ∈ Mnd)
2510ad2antrr 724 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑋𝐷)
26 iftrue 4492 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1 , 0 ) = 1 )
27 eqid 2736 . . . . . . . . . . . . 13 (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))
287fvexi 6856 . . . . . . . . . . . . 13 1 ∈ V
2926, 27, 28fvmpt 6948 . . . . . . . . . . . 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 4037 . . . . . . . . . . . . 13 {𝑥𝐷𝑥r𝑘} ⊆ 𝐷
32 simplr 767 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑘𝐷)
33 eqid 2736 . . . . . . . . . . . . . . 15 {𝑥𝐷𝑥r𝑘} = {𝑥𝐷𝑥r𝑘}
345, 33psrbagconcl 21336 . . . . . . . . . . . . . 14 ((𝑘𝐷𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑋) ∈ {𝑥𝐷𝑥r𝑘})
3532, 18, 34syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑋) ∈ {𝑥𝐷𝑥r𝑘})
3631, 35sselid 3942 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑋) ∈ 𝐷)
37 eqeq1 2740 . . . . . . . . . . . . . 14 (𝑦 = (𝑘f𝑋) → (𝑦 = 𝑌 ↔ (𝑘f𝑋) = 𝑌))
3837ifbid 4509 . . . . . . . . . . . . 13 (𝑦 = (𝑘f𝑋) → if(𝑦 = 𝑌, 1 , 0 ) = if((𝑘f𝑋) = 𝑌, 1 , 0 ))
39 eqid 2736 . . . . . . . . . . . . 13 (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))
406fvexi 6856 . . . . . . . . . . . . . 14 0 ∈ V
4128, 40ifex 4536 . . . . . . . . . . . . 13 if((𝑘f𝑋) = 𝑌, 1 , 0 ) ∈ V
4238, 39, 41fvmpt 6948 . . . . . . . . . . . 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 7375 . . . . . . . . . 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 2736 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘𝑅)
4645, 7ringidcl 19989 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
4745, 6ring0cl 19990 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
4846, 47ifcld 4532 . . . . . . . . . . . 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 19992 . . . . . . . . . . 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 584 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ( 1 (.r𝑅)if((𝑘f𝑋) = 𝑌, 1 , 0 )) = if((𝑘f𝑋) = 𝑌, 1 , 0 ))
525psrbagf 21320 . . . . . . . . . . . . . . . . . 18 (𝑘𝐷𝑘:𝐼⟶ℕ0)
5332, 52syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑘:𝐼⟶ℕ0)
5453ffvelcdmda 7035 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
5510adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐷) → 𝑋𝐷)
565psrbagf 21320 . . . . . . . . . . . . . . . . . . 19 (𝑋𝐷𝑋:𝐼⟶ℕ0)
5755, 56syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑋:𝐼⟶ℕ0)
5857ffvelcdmda 7035 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
5958adantlr 713 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
605psrbagf 21320 . . . . . . . . . . . . . . . . . . . 20 (𝑌𝐷𝑌:𝐼⟶ℕ0)
6112, 60syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑌:𝐼⟶ℕ0)
6261adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑌:𝐼⟶ℕ0)
6362ffvelcdmda 7035 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
6463adantlr 713 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
65 nn0cn 12423 . . . . . . . . . . . . . . . . 17 ((𝑘𝑧) ∈ ℕ0 → (𝑘𝑧) ∈ ℂ)
66 nn0cn 12423 . . . . . . . . . . . . . . . . 17 ((𝑋𝑧) ∈ ℕ0 → (𝑋𝑧) ∈ ℂ)
67 nn0cn 12423 . . . . . . . . . . . . . . . . 17 ((𝑌𝑧) ∈ ℕ0 → (𝑌𝑧) ∈ ℂ)
68 subadd 11404 . . . . . . . . . . . . . . . . 17 (((𝑘𝑧) ∈ ℂ ∧ (𝑋𝑧) ∈ ℂ ∧ (𝑌𝑧) ∈ ℂ) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
6965, 66, 67, 68syl3an 1160 . . . . . . . . . . . . . . . 16 (((𝑘𝑧) ∈ ℕ0 ∧ (𝑋𝑧) ∈ ℕ0 ∧ (𝑌𝑧) ∈ ℕ0) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
7054, 59, 64, 69syl3anc 1371 . . . . . . . . . . . . . . 15 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) ∧ 𝑧𝐼) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
71 eqcom 2743 . . . . . . . . . . . . . . 15 (((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧) ↔ (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧)))
7270, 71bitrdi 286 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) ∧ 𝑧𝐼) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
7372ralbidva 3172 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
74 mpteqb 6967 . . . . . . . . . . . . . 14 (∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) ∈ V → ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ ∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧)))
75 ovexd 7392 . . . . . . . . . . . . . 14 (𝑧𝐼 → ((𝑘𝑧) − (𝑋𝑧)) ∈ V)
7674, 75mprg 3070 . . . . . . . . . . . . 13 ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ ∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧))
77 mpteqb 6967 . . . . . . . . . . . . . 14 (∀𝑧𝐼 (𝑘𝑧) ∈ V → ((𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
78 fvexd 6857 . . . . . . . . . . . . . 14 (𝑧𝐼 → (𝑘𝑧) ∈ V)
7977, 78mprg 3070 . . . . . . . . . . . . 13 ((𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧)))
8073, 76, 793bitr4g 313 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ (𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧)))))
818ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝐼𝑊)
8253feqmptd 6910 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑘 = (𝑧𝐼 ↦ (𝑘𝑧)))
8357feqmptd 6910 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐷) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
8483adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
8581, 54, 59, 82, 84offval2 7637 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑋) = (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))))
8662feqmptd 6910 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐷) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
8786adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
8885, 87eqeq12d 2752 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑘f𝑋) = 𝑌 ↔ (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧))))
898adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐷) → 𝐼𝑊)
9089, 58, 63, 83, 86offval2 7637 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐷) → (𝑋f + 𝑌) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))))
9190adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑋f + 𝑌) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))))
9282, 91eqeq12d 2752 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘 = (𝑋f + 𝑌) ↔ (𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧)))))
9380, 88, 923bitr4d 310 . . . . . . . . . . 11 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑘f𝑋) = 𝑌𝑘 = (𝑋f + 𝑌)))
9493ifbid 4509 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → if((𝑘f𝑋) = 𝑌, 1 , 0 ) = if(𝑘 = (𝑋f + 𝑌), 1 , 0 ))
9544, 51, 943eqtrd 2780 . . . . . . . . 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 2839 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → if(𝑘 = (𝑋f + 𝑌), 1 , 0 ) ∈ (Base‘𝑅))
9795, 96eqeltrd 2838 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋))) ∈ (Base‘𝑅))
98 fveq2 6842 . . . . . . . . . 10 (𝑗 = 𝑋 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋))
99 oveq2 7365 . . . . . . . . . . 11 (𝑗 = 𝑋 → (𝑘f𝑗) = (𝑘f𝑋))
10099fveq2d 6846 . . . . . . . . . 10 (𝑗 = 𝑋 → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)) = ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑋)))
10198, 100oveq12d 7375 . . . . . . . . 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 19731 . . . . . . . 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 1371 . . . . . . 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 2780 . . . . . 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 18539 . . . . . . 7 (𝑅 Σg ∅) = 0
106 disjsn 4672 . . . . . . . . 9 (({𝑥𝐷𝑥r𝑘} ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ {𝑥𝐷𝑥r𝑘})
1079ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑅 ∈ Ring)
1081, 45, 2, 5, 11mplelf 21404 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
109108ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
110 simpr 485 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑗 ∈ {𝑥𝐷𝑥r𝑘})
11131, 110sselid 3942 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑗𝐷)
112109, 111ffvelcdmd 7036 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) ∈ (Base‘𝑅))
1131, 45, 2, 5, 13mplelf 21404 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
114113ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
115 simplr 767 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → 𝑘𝐷)
1165, 33psrbagconcl 21336 . . . . . . . . . . . . . . . 16 ((𝑘𝐷𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑗) ∈ {𝑥𝐷𝑥r𝑘})
117115, 110, 116syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑗) ∈ {𝑥𝐷𝑥r𝑘})
11831, 117sselid 3942 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (𝑘f𝑗) ∈ 𝐷)
119114, 118ffvelcdmd 7036 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)) ∈ (Base‘𝑅))
12045, 3ringcl 19981 . . . . . . . . . . . . 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 1371 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) ∈ (Base‘𝑅))
122121fmpttd 7063 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))):{𝑥𝐷𝑥r𝑘}⟶(Base‘𝑅))
123 ffn 6668 . . . . . . . . . . 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 6621 . . . . . . . . . . 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 477 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ ({𝑥𝐷𝑥r𝑘} ∩ {𝑋}) = ∅) → ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋}) = ∅)
127106, 126sylan2br 595 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ↾ {𝑋}) = ∅)
128127oveq2d 7373 . . . . . . 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 5108 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥r ≤ (𝑋f + 𝑌) ↔ 𝑋r ≤ (𝑋f + 𝑌)))
13058nn0red 12474 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℝ)
131 nn0addge1 12459 . . . . . . . . . . . . . 14 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℕ0) → (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
132130, 63, 131syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
133132ralrimiva 3143 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
134 ovexd 7392 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → ((𝑋𝑧) + (𝑌𝑧)) ∈ V)
13589, 58, 134, 83, 90ofrfval2 7638 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → (𝑋r ≤ (𝑋f + 𝑌) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧))))
136133, 135mpbird 256 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 𝑋r ≤ (𝑋f + 𝑌))
137129, 55, 136elrabd 3647 . . . . . . . . . 10 ((𝜑𝑘𝐷) → 𝑋 ∈ {𝑥𝐷𝑥r ≤ (𝑋f + 𝑌)})
138 breq2 5109 . . . . . . . . . . . 12 (𝑘 = (𝑋f + 𝑌) → (𝑥r𝑘𝑥r ≤ (𝑋f + 𝑌)))
139138rabbidv 3415 . . . . . . . . . . 11 (𝑘 = (𝑋f + 𝑌) → {𝑥𝐷𝑥r𝑘} = {𝑥𝐷𝑥r ≤ (𝑋f + 𝑌)})
140139eleq2d 2823 . . . . . . . . . 10 (𝑘 = (𝑋f + 𝑌) → (𝑋 ∈ {𝑥𝐷𝑥r𝑘} ↔ 𝑋 ∈ {𝑥𝐷𝑥r ≤ (𝑋f + 𝑌)}))
141137, 140syl5ibrcom 246 . . . . . . . . 9 ((𝜑𝑘𝐷) → (𝑘 = (𝑋f + 𝑌) → 𝑋 ∈ {𝑥𝐷𝑥r𝑘}))
142141con3dimp 409 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → ¬ 𝑘 = (𝑋f + 𝑌))
143142iffalsed 4497 . . . . . . 7 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥r𝑘}) → if(𝑘 = (𝑋f + 𝑌), 1 , 0 ) = 0 )
144105, 128, 1433eqtr4a 2802 . . . . . 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 811 . . . . 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 481 . . . . . . 7 ((𝜑𝑘𝐷) → 𝑅 ∈ Ring)
147 ringcmn 20003 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
148146, 147syl 17 . . . . . 6 ((𝜑𝑘𝐷) → 𝑅 ∈ CMnd)
1495psrbaglefi 21334 . . . . . . 7 (𝑘𝐷 → {𝑥𝐷𝑥r𝑘} ∈ Fin)
150149adantl 482 . . . . . 6 ((𝜑𝑘𝐷) → {𝑥𝐷𝑥r𝑘} ∈ Fin)
151 ssdif 4099 . . . . . . . . . . . 12 ({𝑥𝐷𝑥r𝑘} ⊆ 𝐷 → ({𝑥𝐷𝑥r𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋}))
15231, 151ax-mp 5 . . . . . . . . . . 11 ({𝑥𝐷𝑥r𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋})
153152sseli 3940 . . . . . . . . . 10 (𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋}) → 𝑗 ∈ (𝐷 ∖ {𝑋}))
154108adantr 481 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
155 eldifsni 4750 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦𝑋)
156155adantl 482 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦𝑋)
157156neneqd 2948 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋)
158157iffalsed 4497 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 )
159 ovex 7390 . . . . . . . . . . . . . 14 (ℕ0m 𝐼) ∈ V
1605, 159rabex2 5291 . . . . . . . . . . . . 13 𝐷 ∈ V
161160a1i 11 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → 𝐷 ∈ V)
162158, 161suppss2 8131 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋})
16340a1i 11 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 0 ∈ V)
164154, 162, 161, 163suppssr 8127 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ (𝐷 ∖ {𝑋})) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
165153, 164sylan2 593 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋})) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
166165oveq1d 7372 . . . . . . . 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 4086 . . . . . . . . 9 (𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋}) → 𝑗 ∈ {𝑥𝐷𝑥r𝑘})
16845, 3, 6ringlz 20011 . . . . . . . . . 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 584 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥r𝑘}) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) = 0 )
170167, 169sylan2 593 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋})) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) = 0 )
171166, 170eqtrd 2776 . . . . . . 7 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥r𝑘} ∖ {𝑋})) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))) = 0 )
172160rabex 5289 . . . . . . . 8 {𝑥𝐷𝑥r𝑘} ∈ V
173172a1i 11 . . . . . . 7 ((𝜑𝑘𝐷) → {𝑥𝐷𝑥r𝑘} ∈ V)
174171, 173suppss2 8131 . . . . . 6 ((𝜑𝑘𝐷) → ((𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) supp 0 ) ⊆ {𝑋})
175160mptrabex 7175 . . . . . . . . 9 (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗)))) ∈ V
176 funmpt 6539 . . . . . . . . 9 Fun (𝑗 ∈ {𝑥𝐷𝑥r𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘f𝑗))))
177175, 176, 403pm3.2i 1339 . . . . . . . 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 8988 . . . . . . . 8 {𝑋} ∈ Fin
180179a1i 11 . . . . . . 7 ((𝜑𝑘𝐷) → {𝑋} ∈ Fin)
181 suppssfifsupp 9320 . . . . . . 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 835 . . . . . 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 19690 . . . . 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 2778 . . . 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 5205 . . 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 2788 . 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 2779 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 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  {crab 3407  Vcvv 3445  cdif 3907  cin 3909  wss 3910  c0 4282  ifcif 4486  {csn 4586   class class class wbr 5105  cmpt 5188  ccnv 5632  cres 5635  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  r cofr 7616   supp csupp 8092  m cmap 8765  Fincfn 8883   finSupp cfsupp 9305  cc 11049  cr 11050   + caddc 11054  cle 11190  cmin 11385  cn 12153  0cn0 12413  Basecbs 17083  .rcmulr 17134  0gc0g 17321   Σg cgsu 17322  Mndcmnd 18556  CMndccmn 19562  1rcur 19913  Ringcrg 19964   mPoly cmpl 21308
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-tset 17152  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-psr 21311  df-mpl 21313
This theorem is referenced by:  mplcoe3  21439  mplcoe5  21441  mplmon2mul  21477
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