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

Theorem mplmonmul 19861
 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 2778 . . 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 19860 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)
12 mplmonmul.x . . . 4 (𝜑𝑌𝐷)
131, 2, 6, 7, 5, 8, 9, 12mplmon 19860 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) ∈ 𝐵)
141, 2, 3, 4, 5, 11, 13mplmul 19840 . 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 2782 . . . . 5 (𝑦 = 𝑘 → (𝑦 = (𝑋𝑓 + 𝑌) ↔ 𝑘 = (𝑋𝑓 + 𝑌)))
1615ifbid 4329 . . . 4 (𝑦 = 𝑘 → if(𝑦 = (𝑋𝑓 + 𝑌), 1 , 0 ) = if(𝑘 = (𝑋𝑓 + 𝑌), 1 , 0 ))
1716cbvmptv 4985 . . 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 4571 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → {𝑋} ⊆ {𝑥𝐷𝑥𝑟𝑘})
2019resmptd 5702 . . . . . . . 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 6938 . . . . . . 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 716 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑅 ∈ Ring)
23 ringmnd 18943 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
2422, 23syl 17 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑅 ∈ Mnd)
2510ad2antrr 716 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑋𝐷)
26 iftrue 4313 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1 , 0 ) = 1 )
27 eqid 2778 . . . . . . . . . . . . 13 (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))
287fvexi 6460 . . . . . . . . . . . . 13 1 ∈ V
2926, 27, 28fvmpt 6542 . . . . . . . . . . . 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 3908 . . . . . . . . . . . . 13 {𝑥𝐷𝑥𝑟𝑘} ⊆ 𝐷
328ad2antrr 716 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝐼𝑊)
33 simplr 759 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑘𝐷)
34 eqid 2778 . . . . . . . . . . . . . . 15 {𝑥𝐷𝑥𝑟𝑘} = {𝑥𝐷𝑥𝑟𝑘}
355, 34psrbagconcl 19770 . . . . . . . . . . . . . 14 ((𝐼𝑊𝑘𝐷𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑋) ∈ {𝑥𝐷𝑥𝑟𝑘})
3632, 33, 18, 35syl3anc 1439 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑋) ∈ {𝑥𝐷𝑥𝑟𝑘})
3731, 36sseldi 3819 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑋) ∈ 𝐷)
38 eqeq1 2782 . . . . . . . . . . . . . 14 (𝑦 = (𝑘𝑓𝑋) → (𝑦 = 𝑌 ↔ (𝑘𝑓𝑋) = 𝑌))
3938ifbid 4329 . . . . . . . . . . . . 13 (𝑦 = (𝑘𝑓𝑋) → if(𝑦 = 𝑌, 1 , 0 ) = if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ))
40 eqid 2778 . . . . . . . . . . . . 13 (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))
416fvexi 6460 . . . . . . . . . . . . . 14 0 ∈ V
4228, 41ifex 4355 . . . . . . . . . . . . 13 if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ) ∈ V
4339, 40, 42fvmpt 6542 . . . . . . . . . . . 12 ((𝑘𝑓𝑋) ∈ 𝐷 → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑋)) = if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ))
4437, 43syl 17 . . . . . . . . . . 11 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑋)) = if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ))
4530, 44oveq12d 6940 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑋))) = ( 1 (.r𝑅)if((𝑘𝑓𝑋) = 𝑌, 1 , 0 )))
46 eqid 2778 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘𝑅)
4746, 7ringidcl 18955 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
4846, 6ring0cl 18956 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
4947, 48ifcld 4352 . . . . . . . . . . . 12 (𝑅 ∈ Ring → if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅))
5022, 49syl 17 . . . . . . . . . . 11 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅))
5146, 3, 7ringlidm 18958 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅)) → ( 1 (.r𝑅)if((𝑘𝑓𝑋) = 𝑌, 1 , 0 )) = if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ))
5222, 50, 51syl2anc 579 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ( 1 (.r𝑅)if((𝑘𝑓𝑋) = 𝑌, 1 , 0 )) = if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ))
535psrbagf 19762 . . . . . . . . . . . . . . . . . 18 ((𝐼𝑊𝑘𝐷) → 𝑘:𝐼⟶ℕ0)
5432, 33, 53syl2anc 579 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑘:𝐼⟶ℕ0)
5554ffvelrnda 6623 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
568adantr 474 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐷) → 𝐼𝑊)
5710adantr 474 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐷) → 𝑋𝐷)
585psrbagf 19762 . . . . . . . . . . . . . . . . . . 19 ((𝐼𝑊𝑋𝐷) → 𝑋:𝐼⟶ℕ0)
5956, 57, 58syl2anc 579 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑋:𝐼⟶ℕ0)
6059ffvelrnda 6623 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
6160adantlr 705 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
625psrbagf 19762 . . . . . . . . . . . . . . . . . . . 20 ((𝐼𝑊𝑌𝐷) → 𝑌:𝐼⟶ℕ0)
638, 12, 62syl2anc 579 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑌:𝐼⟶ℕ0)
6463adantr 474 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑌:𝐼⟶ℕ0)
6564ffvelrnda 6623 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
6665adantlr 705 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
67 nn0cn 11653 . . . . . . . . . . . . . . . . 17 ((𝑘𝑧) ∈ ℕ0 → (𝑘𝑧) ∈ ℂ)
68 nn0cn 11653 . . . . . . . . . . . . . . . . 17 ((𝑋𝑧) ∈ ℕ0 → (𝑋𝑧) ∈ ℂ)
69 nn0cn 11653 . . . . . . . . . . . . . . . . 17 ((𝑌𝑧) ∈ ℕ0 → (𝑌𝑧) ∈ ℂ)
70 subadd 10625 . . . . . . . . . . . . . . . . 17 (((𝑘𝑧) ∈ ℂ ∧ (𝑋𝑧) ∈ ℂ ∧ (𝑌𝑧) ∈ ℂ) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
7167, 68, 69, 70syl3an 1160 . . . . . . . . . . . . . . . 16 (((𝑘𝑧) ∈ ℕ0 ∧ (𝑋𝑧) ∈ ℕ0 ∧ (𝑌𝑧) ∈ ℕ0) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
7255, 61, 66, 71syl3anc 1439 . . . . . . . . . . . . . . 15 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) ∧ 𝑧𝐼) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧)))
73 eqcom 2785 . . . . . . . . . . . . . . 15 (((𝑋𝑧) + (𝑌𝑧)) = (𝑘𝑧) ↔ (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧)))
7472, 73syl6bb 279 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) ∧ 𝑧𝐼) → (((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
7574ralbidva 3167 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
76 mpteqb 6560 . . . . . . . . . . . . . 14 (∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) ∈ V → ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ ∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧)))
77 ovexd 6956 . . . . . . . . . . . . . 14 (𝑧𝐼 → ((𝑘𝑧) − (𝑋𝑧)) ∈ V)
7876, 77mprg 3108 . . . . . . . . . . . . 13 ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ ∀𝑧𝐼 ((𝑘𝑧) − (𝑋𝑧)) = (𝑌𝑧))
79 mpteqb 6560 . . . . . . . . . . . . . 14 (∀𝑧𝐼 (𝑘𝑧) ∈ V → ((𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧))))
80 fvex 6459 . . . . . . . . . . . . . . 15 (𝑘𝑧) ∈ V
8180a1i 11 . . . . . . . . . . . . . 14 (𝑧𝐼 → (𝑘𝑧) ∈ V)
8279, 81mprg 3108 . . . . . . . . . . . . 13 ((𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))) ↔ ∀𝑧𝐼 (𝑘𝑧) = ((𝑋𝑧) + (𝑌𝑧)))
8375, 78, 823bitr4g 306 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧)) ↔ (𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧)))))
8454feqmptd 6509 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑘 = (𝑧𝐼 ↦ (𝑘𝑧)))
8559feqmptd 6509 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐷) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
8685adantr 474 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
8732, 55, 61, 84, 86offval2 7191 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑋) = (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))))
8864feqmptd 6509 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐷) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
8988adantr 474 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
9087, 89eqeq12d 2793 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑘𝑓𝑋) = 𝑌 ↔ (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑋𝑧))) = (𝑧𝐼 ↦ (𝑌𝑧))))
9156, 60, 65, 85, 88offval2 7191 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐷) → (𝑋𝑓 + 𝑌) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))))
9291adantr 474 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑋𝑓 + 𝑌) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧))))
9384, 92eqeq12d 2793 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘 = (𝑋𝑓 + 𝑌) ↔ (𝑧𝐼 ↦ (𝑘𝑧)) = (𝑧𝐼 ↦ ((𝑋𝑧) + (𝑌𝑧)))))
9483, 90, 933bitr4d 303 . . . . . . . . . . 11 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑘𝑓𝑋) = 𝑌𝑘 = (𝑋𝑓 + 𝑌)))
9594ifbid 4329 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → if((𝑘𝑓𝑋) = 𝑌, 1 , 0 ) = if(𝑘 = (𝑋𝑓 + 𝑌), 1 , 0 ))
9645, 52, 953eqtrd 2818 . . . . . . . . 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 ))
9795, 50eqeltrrd 2860 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → if(𝑘 = (𝑋𝑓 + 𝑌), 1 , 0 ) ∈ (Base‘𝑅))
9896, 97eqeltrd 2859 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑋))) ∈ (Base‘𝑅))
99 fveq2 6446 . . . . . . . . . 10 (𝑗 = 𝑋 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋))
100 oveq2 6930 . . . . . . . . . . 11 (𝑗 = 𝑋 → (𝑘𝑓𝑗) = (𝑘𝑓𝑋))
101100fveq2d 6450 . . . . . . . . . 10 (𝑗 = 𝑋 → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)) = ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑋)))
10299, 101oveq12d 6940 . . . . . . . . 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))))
10346, 102gsumsn 18740 . . . . . . . 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))))
10424, 25, 98, 103syl3anc 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))))
10521, 104, 963eqtrd 2818 . . . . . 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 ))
1066gsum0 17664 . . . . . . 7 (𝑅 Σg ∅) = 0
107 disjsn 4478 . . . . . . . . 9 (({𝑥𝐷𝑥𝑟𝑘} ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘})
1089ad2antrr 716 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑅 ∈ Ring)
1091, 46, 2, 5, 11mplelf 19830 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
110109ad2antrr 716 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
111 simpr 479 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘})
11231, 111sseldi 3819 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑗𝐷)
113110, 112ffvelrnd 6624 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) ∈ (Base‘𝑅))
1141, 46, 2, 5, 13mplelf 19830 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
115114ad2antrr 716 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
1168ad2antrr 716 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝐼𝑊)
117 simplr 759 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → 𝑘𝐷)
1185, 34psrbagconcl 19770 . . . . . . . . . . . . . . . 16 ((𝐼𝑊𝑘𝐷𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑗) ∈ {𝑥𝐷𝑥𝑟𝑘})
119116, 117, 111, 118syl3anc 1439 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑗) ∈ {𝑥𝐷𝑥𝑟𝑘})
12031, 119sseldi 3819 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (𝑘𝑓𝑗) ∈ 𝐷)
121115, 120ffvelrnd 6624 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)) ∈ (Base‘𝑅))
12246, 3ringcl 18948 . . . . . . . . . . . . 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))
123108, 113, 121, 122syl3anc 1439 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))) ∈ (Base‘𝑅))
124123fmpttd 6649 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))):{𝑥𝐷𝑥𝑟𝑘}⟶(Base‘𝑅))
125 ffn 6291 . . . . . . . . . . 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})
126 fnresdisj 6247 . . . . . . . . . . 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}) = ∅))
127124, 125, 1263syl 18 . . . . . . . . . 10 ((𝜑𝑘𝐷) → (({𝑥𝐷𝑥𝑟𝑘} ∩ {𝑋}) = ∅ ↔ ((𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) ↾ {𝑋}) = ∅))
128127biimpa 470 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ ({𝑥𝐷𝑥𝑟𝑘} ∩ {𝑋}) = ∅) → ((𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) ↾ {𝑋}) = ∅)
129107, 128sylan2br 588 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ((𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) ↾ {𝑋}) = ∅)
130129oveq2d 6938 . . . . . . 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 ∅))
13160nn0red 11703 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℝ)
132 nn0addge1 11690 . . . . . . . . . . . . . 14 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℕ0) → (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
133131, 65, 132syl2anc 579 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
134133ralrimiva 3148 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧)))
135 ovexd 6956 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑧𝐼) → ((𝑋𝑧) + (𝑌𝑧)) ∈ V)
13656, 60, 135, 85, 91ofrfval2 7192 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → (𝑋𝑟 ≤ (𝑋𝑓 + 𝑌) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝑋𝑧) + (𝑌𝑧))))
137134, 136mpbird 249 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 𝑋𝑟 ≤ (𝑋𝑓 + 𝑌))
138 breq1 4889 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥𝑟 ≤ (𝑋𝑓 + 𝑌) ↔ 𝑋𝑟 ≤ (𝑋𝑓 + 𝑌)))
139138elrab 3572 . . . . . . . . . . 11 (𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝑋𝑓 + 𝑌)} ↔ (𝑋𝐷𝑋𝑟 ≤ (𝑋𝑓 + 𝑌)))
14057, 137, 139sylanbrc 578 . . . . . . . . . 10 ((𝜑𝑘𝐷) → 𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝑋𝑓 + 𝑌)})
141 breq2 4890 . . . . . . . . . . . 12 (𝑘 = (𝑋𝑓 + 𝑌) → (𝑥𝑟𝑘𝑥𝑟 ≤ (𝑋𝑓 + 𝑌)))
142141rabbidv 3386 . . . . . . . . . . 11 (𝑘 = (𝑋𝑓 + 𝑌) → {𝑥𝐷𝑥𝑟𝑘} = {𝑥𝐷𝑥𝑟 ≤ (𝑋𝑓 + 𝑌)})
143142eleq2d 2845 . . . . . . . . . 10 (𝑘 = (𝑋𝑓 + 𝑌) → (𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘} ↔ 𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝑋𝑓 + 𝑌)}))
144140, 143syl5ibrcom 239 . . . . . . . . 9 ((𝜑𝑘𝐷) → (𝑘 = (𝑋𝑓 + 𝑌) → 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}))
145144con3dimp 399 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ¬ 𝑘 = (𝑋𝑓 + 𝑌))
146145iffalsed 4318 . . . . . . 7 (((𝜑𝑘𝐷) ∧ ¬ 𝑋 ∈ {𝑥𝐷𝑥𝑟𝑘}) → if(𝑘 = (𝑋𝑓 + 𝑌), 1 , 0 ) = 0 )
147106, 130, 1463eqtr4a 2840 . . . . . 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 ))
148105, 147pm2.61dan 803 . . . . 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 ))
1499adantr 474 . . . . . . 7 ((𝜑𝑘𝐷) → 𝑅 ∈ Ring)
150 ringcmn 18968 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
151149, 150syl 17 . . . . . 6 ((𝜑𝑘𝐷) → 𝑅 ∈ CMnd)
1525psrbaglefi 19769 . . . . . . 7 ((𝐼𝑊𝑘𝐷) → {𝑥𝐷𝑥𝑟𝑘} ∈ Fin)
1538, 152sylan 575 . . . . . 6 ((𝜑𝑘𝐷) → {𝑥𝐷𝑥𝑟𝑘} ∈ Fin)
154 ssdif 3968 . . . . . . . . . . . 12 ({𝑥𝐷𝑥𝑟𝑘} ⊆ 𝐷 → ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋}))
15531, 154ax-mp 5 . . . . . . . . . . 11 ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋})
156155sseli 3817 . . . . . . . . . 10 (𝑗 ∈ ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋}) → 𝑗 ∈ (𝐷 ∖ {𝑋}))
157109adantr 474 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
158 eldifsni 4553 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦𝑋)
159158adantl 475 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦𝑋)
160159neneqd 2974 . . . . . . . . . . . . 13 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋)
161160iffalsed 4318 . . . . . . . . . . . 12 (((𝜑𝑘𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 )
162 ovex 6954 . . . . . . . . . . . . . 14 (ℕ0𝑚 𝐼) ∈ V
1635, 162rabex2 5051 . . . . . . . . . . . . 13 𝐷 ∈ V
164163a1i 11 . . . . . . . . . . . 12 ((𝜑𝑘𝐷) → 𝐷 ∈ V)
165161, 164suppss2 7611 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋})
16641a1i 11 . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 0 ∈ V)
167157, 165, 164, 166suppssr 7608 . . . . . . . . . 10 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ (𝐷 ∖ {𝑋})) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
168156, 167sylan2 586 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋})) → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
169168oveq1d 6937 . . . . . . . 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))))
170 eldifi 3955 . . . . . . . . 9 (𝑗 ∈ ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋}) → 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘})
17146, 3, 6ringlz 18974 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ ((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)) ∈ (Base‘𝑅)) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))) = 0 )
172108, 121, 171syl2anc 579 . . . . . . . . 9 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘}) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))) = 0 )
173170, 172sylan2 586 . . . . . . . 8 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋})) → ( 0 (.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))) = 0 )
174169, 173eqtrd 2814 . . . . . . 7 (((𝜑𝑘𝐷) ∧ 𝑗 ∈ ({𝑥𝐷𝑥𝑟𝑘} ∖ {𝑋})) → (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))) = 0 )
175163rabex 5049 . . . . . . . 8 {𝑥𝐷𝑥𝑟𝑘} ∈ V
176175a1i 11 . . . . . . 7 ((𝜑𝑘𝐷) → {𝑥𝐷𝑥𝑟𝑘} ∈ V)
177174, 176suppss2 7611 . . . . . 6 ((𝜑𝑘𝐷) → ((𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) supp 0 ) ⊆ {𝑋})
178163mptrabex 6760 . . . . . . . . 9 (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) ∈ V
179 funmpt 6173 . . . . . . . . 9 Fun (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗))))
180178, 179, 413pm3.2i 1395 . . . . . . . 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)
181180a1i 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))
182 snfi 8326 . . . . . . . 8 {𝑋} ∈ Fin
183182a1i 11 . . . . . . 7 ((𝜑𝑘𝐷) → {𝑋} ∈ Fin)
184 suppssfifsupp 8578 . . . . . . 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 )
185181, 183, 177, 184syl12anc 827 . . . . . 6 ((𝜑𝑘𝐷) → (𝑗 ∈ {𝑥𝐷𝑥𝑟𝑘} ↦ (((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r𝑅)((𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘𝑓𝑗)))) finSupp 0 )
18646, 6, 151, 153, 124, 177, 185gsumres 18700 . . . . 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))))))
187148, 186eqtr3d 2816 . . . 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))))))
188187mpteq2dva 4979 . . 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)))))))
18917, 188syl5eq 2826 . 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)))))))
19014, 189eqtr4d 2817 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 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090  {crab 3094  Vcvv 3398   ∖ cdif 3789   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141  ifcif 4307  {csn 4398   class class class wbr 4886   ↦ cmpt 4965  ◡ccnv 5354   ↾ cres 5357   “ cima 5358  Fun wfun 6129   Fn wfn 6130  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922   ∘𝑓 cof 7172   ∘𝑟 cofr 7173   supp csupp 7576   ↑𝑚 cmap 8140  Fincfn 8241   finSupp cfsupp 8563  ℂcc 10270  ℝcr 10271   + caddc 10275   ≤ cle 10412   − cmin 10606  ℕcn 11374  ℕ0cn0 11642  Basecbs 16255  .rcmulr 16339  0gc0g 16486   Σg cgsu 16487  Mndcmnd 17680  CMndccmn 18579  1rcur 18888  Ringcrg 18934   mPoly cmpl 19750 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-ofr 7175  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-ixp 8195  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fsupp 8564  df-oi 8704  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-seq 13120  df-hash 13436  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-mulr 16352  df-sca 16354  df-vsca 16355  df-tset 16357  df-0g 16488  df-gsum 16489  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-grp 17812  df-minusg 17813  df-mulg 17928  df-cntz 18133  df-cmn 18581  df-abl 18582  df-mgp 18877  df-ur 18889  df-ring 18936  df-psr 19753  df-mpl 19755 This theorem is referenced by:  mplcoe3  19863  mplcoe5  19865  mplmon2mul  19897
 Copyright terms: Public domain W3C validator