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

Theorem dmdprdsplitlem 19906
Description: Lemma for dmdprdsplit 19916. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
Hypotheses
Ref Expression
dmdprdsplitlem.0 0 = (0gβ€˜πΊ)
dmdprdsplitlem.w π‘Š = {β„Ž ∈ X𝑖 ∈ 𝐼 (π‘†β€˜π‘–) ∣ β„Ž finSupp 0 }
dmdprdsplitlem.1 (πœ‘ β†’ 𝐺dom DProd 𝑆)
dmdprdsplitlem.2 (πœ‘ β†’ dom 𝑆 = 𝐼)
dmdprdsplitlem.3 (πœ‘ β†’ 𝐴 βŠ† 𝐼)
dmdprdsplitlem.4 (πœ‘ β†’ 𝐹 ∈ π‘Š)
dmdprdsplitlem.5 (πœ‘ β†’ (𝐺 Ξ£g 𝐹) ∈ (𝐺 DProd (𝑆 β†Ύ 𝐴)))
Assertion
Ref Expression
dmdprdsplitlem ((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) β†’ (πΉβ€˜π‘‹) = 0 )
Distinct variable groups:   0 ,β„Ž   β„Ž,𝑖,𝐴   β„Ž,𝐺,𝑖   β„Ž,𝐼,𝑖   β„Ž,𝐹   𝑆,β„Ž,𝑖
Allowed substitution hints:   πœ‘(β„Ž,𝑖)   𝐹(𝑖)   π‘Š(β„Ž,𝑖)   𝑋(β„Ž,𝑖)   0 (𝑖)

Proof of Theorem dmdprdsplitlem
Dummy variables 𝑓 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmdprdsplitlem.5 . . . . 5 (πœ‘ β†’ (𝐺 Ξ£g 𝐹) ∈ (𝐺 DProd (𝑆 β†Ύ 𝐴)))
2 dmdprdsplitlem.1 . . . . . . . 8 (πœ‘ β†’ 𝐺dom DProd 𝑆)
3 dmdprdsplitlem.2 . . . . . . . 8 (πœ‘ β†’ dom 𝑆 = 𝐼)
42, 3dprdf2 19876 . . . . . . 7 (πœ‘ β†’ 𝑆:𝐼⟢(SubGrpβ€˜πΊ))
5 dmdprdsplitlem.3 . . . . . . 7 (πœ‘ β†’ 𝐴 βŠ† 𝐼)
64, 5fssresd 6758 . . . . . 6 (πœ‘ β†’ (𝑆 β†Ύ 𝐴):𝐴⟢(SubGrpβ€˜πΊ))
7 fdm 6726 . . . . . 6 ((𝑆 β†Ύ 𝐴):𝐴⟢(SubGrpβ€˜πΊ) β†’ dom (𝑆 β†Ύ 𝐴) = 𝐴)
8 dmdprdsplitlem.0 . . . . . . 7 0 = (0gβ€˜πΊ)
9 eqid 2732 . . . . . . 7 {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } = {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 }
108, 9eldprd 19873 . . . . . 6 (dom (𝑆 β†Ύ 𝐴) = 𝐴 β†’ ((𝐺 Ξ£g 𝐹) ∈ (𝐺 DProd (𝑆 β†Ύ 𝐴)) ↔ (𝐺dom DProd (𝑆 β†Ύ 𝐴) ∧ βˆƒπ‘“ ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))))
116, 7, 103syl 18 . . . . 5 (πœ‘ β†’ ((𝐺 Ξ£g 𝐹) ∈ (𝐺 DProd (𝑆 β†Ύ 𝐴)) ↔ (𝐺dom DProd (𝑆 β†Ύ 𝐴) ∧ βˆƒπ‘“ ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))))
121, 11mpbid 231 . . . 4 (πœ‘ β†’ (𝐺dom DProd (𝑆 β†Ύ 𝐴) ∧ βˆƒπ‘“ ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓)))
1312simprd 496 . . 3 (πœ‘ β†’ βˆƒπ‘“ ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))
1413adantr 481 . 2 ((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) β†’ βˆƒπ‘“ ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))
15 simprr 771 . . . . . 6 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))
1612simpld 495 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺dom DProd (𝑆 β†Ύ 𝐴))
1716ad2antrr 724 . . . . . . . . . 10 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 𝐺dom DProd (𝑆 β†Ύ 𝐴))
186, 7syl 17 . . . . . . . . . . 11 (πœ‘ β†’ dom (𝑆 β†Ύ 𝐴) = 𝐴)
1918ad2antrr 724 . . . . . . . . . 10 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ dom (𝑆 β†Ύ 𝐴) = 𝐴)
20 simprl 769 . . . . . . . . . 10 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 })
21 eqid 2732 . . . . . . . . . 10 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
229, 17, 19, 20, 21dprdff 19881 . . . . . . . . 9 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 𝑓:𝐴⟢(Baseβ€˜πΊ))
2322feqmptd 6960 . . . . . . . 8 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 𝑓 = (𝑛 ∈ 𝐴 ↦ (π‘“β€˜π‘›)))
245ad2antrr 724 . . . . . . . . . 10 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 𝐴 βŠ† 𝐼)
2524resmptd 6040 . . . . . . . . 9 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) β†Ύ 𝐴) = (𝑛 ∈ 𝐴 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )))
26 iftrue 4534 . . . . . . . . . 10 (𝑛 ∈ 𝐴 β†’ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ) = (π‘“β€˜π‘›))
2726mpteq2ia 5251 . . . . . . . . 9 (𝑛 ∈ 𝐴 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) = (𝑛 ∈ 𝐴 ↦ (π‘“β€˜π‘›))
2825, 27eqtrdi 2788 . . . . . . . 8 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) β†Ύ 𝐴) = (𝑛 ∈ 𝐴 ↦ (π‘“β€˜π‘›)))
2923, 28eqtr4d 2775 . . . . . . 7 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 𝑓 = ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) β†Ύ 𝐴))
3029oveq2d 7424 . . . . . 6 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ (𝐺 Ξ£g 𝑓) = (𝐺 Ξ£g ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) β†Ύ 𝐴)))
31 eqid 2732 . . . . . . 7 (Cntzβ€˜πΊ) = (Cntzβ€˜πΊ)
322ad2antrr 724 . . . . . . . 8 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 𝐺dom DProd 𝑆)
33 dprdgrp 19874 . . . . . . . 8 (𝐺dom DProd 𝑆 β†’ 𝐺 ∈ Grp)
34 grpmnd 18825 . . . . . . . 8 (𝐺 ∈ Grp β†’ 𝐺 ∈ Mnd)
3532, 33, 343syl 18 . . . . . . 7 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 𝐺 ∈ Mnd)
362, 3dprddomcld 19870 . . . . . . . 8 (πœ‘ β†’ 𝐼 ∈ V)
3736ad2antrr 724 . . . . . . 7 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 𝐼 ∈ V)
38 dmdprdsplitlem.w . . . . . . . 8 π‘Š = {β„Ž ∈ X𝑖 ∈ 𝐼 (π‘†β€˜π‘–) ∣ β„Ž finSupp 0 }
393ad2antrr 724 . . . . . . . 8 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ dom 𝑆 = 𝐼)
4017adantr 481 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ 𝐼) β†’ 𝐺dom DProd (𝑆 β†Ύ 𝐴))
4119adantr 481 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ 𝐼) β†’ dom (𝑆 β†Ύ 𝐴) = 𝐴)
42 simplrl 775 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ 𝐼) β†’ 𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 })
439, 40, 41, 42dprdfcl 19882 . . . . . . . . . . 11 (((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 ∈ 𝐴) β†’ (π‘“β€˜π‘›) ∈ ((𝑆 β†Ύ 𝐴)β€˜π‘›))
44 fvres 6910 . . . . . . . . . . . 12 (𝑛 ∈ 𝐴 β†’ ((𝑆 β†Ύ 𝐴)β€˜π‘›) = (π‘†β€˜π‘›))
4544adantl 482 . . . . . . . . . . 11 (((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 ∈ 𝐴) β†’ ((𝑆 β†Ύ 𝐴)β€˜π‘›) = (π‘†β€˜π‘›))
4643, 45eleqtrd 2835 . . . . . . . . . 10 (((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 ∈ 𝐴) β†’ (π‘“β€˜π‘›) ∈ (π‘†β€˜π‘›))
474ad2antrr 724 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 𝑆:𝐼⟢(SubGrpβ€˜πΊ))
4847ffvelcdmda 7086 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ 𝐼) β†’ (π‘†β€˜π‘›) ∈ (SubGrpβ€˜πΊ))
498subg0cl 19013 . . . . . . . . . . . 12 ((π‘†β€˜π‘›) ∈ (SubGrpβ€˜πΊ) β†’ 0 ∈ (π‘†β€˜π‘›))
5048, 49syl 17 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ 𝐼) β†’ 0 ∈ (π‘†β€˜π‘›))
5150adantr 481 . . . . . . . . . 10 (((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ 𝐼) ∧ Β¬ 𝑛 ∈ 𝐴) β†’ 0 ∈ (π‘†β€˜π‘›))
5246, 51ifclda 4563 . . . . . . . . 9 ((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ 𝐼) β†’ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ) ∈ (π‘†β€˜π‘›))
5336mptexd 7225 . . . . . . . . . . 11 (πœ‘ β†’ (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) ∈ V)
5453ad2antrr 724 . . . . . . . . . 10 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) ∈ V)
55 funmpt 6586 . . . . . . . . . . 11 Fun (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ))
5655a1i 11 . . . . . . . . . 10 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ Fun (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )))
579, 17, 19, 20dprdffsupp 19883 . . . . . . . . . 10 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 𝑓 finSupp 0 )
58 simpr 485 . . . . . . . . . . . . . . 15 (((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ (𝐼 βˆ– (𝑓 supp 0 ))) ∧ 𝑛 ∈ 𝐴) β†’ 𝑛 ∈ 𝐴)
59 eldifn 4127 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (𝐼 βˆ– (𝑓 supp 0 )) β†’ Β¬ 𝑛 ∈ (𝑓 supp 0 ))
6059ad2antlr 725 . . . . . . . . . . . . . . 15 (((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ (𝐼 βˆ– (𝑓 supp 0 ))) ∧ 𝑛 ∈ 𝐴) β†’ Β¬ 𝑛 ∈ (𝑓 supp 0 ))
6158, 60eldifd 3959 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ (𝐼 βˆ– (𝑓 supp 0 ))) ∧ 𝑛 ∈ 𝐴) β†’ 𝑛 ∈ (𝐴 βˆ– (𝑓 supp 0 )))
62 ssidd 4005 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ (𝑓 supp 0 ) βŠ† (𝑓 supp 0 ))
6336, 5ssexd 5324 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝐴 ∈ V)
6463ad2antrr 724 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 𝐴 ∈ V)
658fvexi 6905 . . . . . . . . . . . . . . . . 17 0 ∈ V
6665a1i 11 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 0 ∈ V)
6722, 62, 64, 66suppssr 8180 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ (𝐴 βˆ– (𝑓 supp 0 ))) β†’ (π‘“β€˜π‘›) = 0 )
6867adantlr 713 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ (𝐼 βˆ– (𝑓 supp 0 ))) ∧ 𝑛 ∈ (𝐴 βˆ– (𝑓 supp 0 ))) β†’ (π‘“β€˜π‘›) = 0 )
6961, 68syldan 591 . . . . . . . . . . . . 13 (((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ (𝐼 βˆ– (𝑓 supp 0 ))) ∧ 𝑛 ∈ 𝐴) β†’ (π‘“β€˜π‘›) = 0 )
7069ifeq1da 4559 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ (𝐼 βˆ– (𝑓 supp 0 ))) β†’ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ) = if(𝑛 ∈ 𝐴, 0 , 0 ))
71 ifid 4568 . . . . . . . . . . . 12 if(𝑛 ∈ 𝐴, 0 , 0 ) = 0
7270, 71eqtrdi 2788 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ (𝐼 βˆ– (𝑓 supp 0 ))) β†’ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ) = 0 )
7372, 37suppss2 8184 . . . . . . . . . 10 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) supp 0 ) βŠ† (𝑓 supp 0 ))
74 fsuppsssupp 9378 . . . . . . . . . 10 ((((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) ∈ V ∧ Fun (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ))) ∧ (𝑓 finSupp 0 ∧ ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) supp 0 ) βŠ† (𝑓 supp 0 ))) β†’ (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) finSupp 0 )
7554, 56, 57, 73, 74syl22anc 837 . . . . . . . . 9 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) finSupp 0 )
7638, 32, 39, 52, 75dprdwd 19880 . . . . . . . 8 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) ∈ π‘Š)
7738, 32, 39, 76, 21dprdff 19881 . . . . . . 7 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )):𝐼⟢(Baseβ€˜πΊ))
7838, 32, 39, 76, 31dprdfcntz 19884 . . . . . . 7 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ ran (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) βŠ† ((Cntzβ€˜πΊ)β€˜ran (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ))))
79 eldifn 4127 . . . . . . . . . 10 (𝑛 ∈ (𝐼 βˆ– 𝐴) β†’ Β¬ 𝑛 ∈ 𝐴)
8079adantl 482 . . . . . . . . 9 ((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ (𝐼 βˆ– 𝐴)) β†’ Β¬ 𝑛 ∈ 𝐴)
8180iffalsed 4539 . . . . . . . 8 ((((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) ∧ 𝑛 ∈ (𝐼 βˆ– 𝐴)) β†’ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ) = 0 )
8281, 37suppss2 8184 . . . . . . 7 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) supp 0 ) βŠ† 𝐴)
8321, 8, 31, 35, 37, 77, 78, 82, 75gsumzres 19776 . . . . . 6 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ (𝐺 Ξ£g ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) β†Ύ 𝐴)) = (𝐺 Ξ£g (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ))))
8415, 30, 833eqtrd 2776 . . . . 5 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ))))
85 dmdprdsplitlem.4 . . . . . . 7 (πœ‘ β†’ 𝐹 ∈ π‘Š)
8685ad2antrr 724 . . . . . 6 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 𝐹 ∈ π‘Š)
878, 38, 32, 39, 86, 76dprdf11 19892 . . . . 5 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ ((𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ))) ↔ 𝐹 = (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ))))
8884, 87mpbid 231 . . . 4 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 𝐹 = (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )))
8988fveq1d 6893 . . 3 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ (πΉβ€˜π‘‹) = ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ))β€˜π‘‹))
90 eldifi 4126 . . . . 5 (𝑋 ∈ (𝐼 βˆ– 𝐴) β†’ 𝑋 ∈ 𝐼)
9190ad2antlr 725 . . . 4 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ 𝑋 ∈ 𝐼)
92 eleq1 2821 . . . . . 6 (𝑛 = 𝑋 β†’ (𝑛 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
93 fveq2 6891 . . . . . 6 (𝑛 = 𝑋 β†’ (π‘“β€˜π‘›) = (π‘“β€˜π‘‹))
9492, 93ifbieq1d 4552 . . . . 5 (𝑛 = 𝑋 β†’ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ) = if(𝑋 ∈ 𝐴, (π‘“β€˜π‘‹), 0 ))
95 eqid 2732 . . . . 5 (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) = (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ))
96 fvex 6904 . . . . . 6 (π‘“β€˜π‘›) ∈ V
9796, 65ifex 4578 . . . . 5 if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ) ∈ V
9894, 95, 97fvmpt3i 7003 . . . 4 (𝑋 ∈ 𝐼 β†’ ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ))β€˜π‘‹) = if(𝑋 ∈ 𝐴, (π‘“β€˜π‘‹), 0 ))
9991, 98syl 17 . . 3 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ))β€˜π‘‹) = if(𝑋 ∈ 𝐴, (π‘“β€˜π‘‹), 0 ))
100 eldifn 4127 . . . . 5 (𝑋 ∈ (𝐼 βˆ– 𝐴) β†’ Β¬ 𝑋 ∈ 𝐴)
101100ad2antlr 725 . . . 4 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ Β¬ 𝑋 ∈ 𝐴)
102101iffalsed 4539 . . 3 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ if(𝑋 ∈ 𝐴, (π‘“β€˜π‘‹), 0 ) = 0 )
10389, 99, 1023eqtrd 2776 . 2 (((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) ∧ (𝑓 ∈ {β„Ž ∈ X𝑖 ∈ 𝐴 ((𝑆 β†Ύ 𝐴)β€˜π‘–) ∣ β„Ž finSupp 0 } ∧ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g 𝑓))) β†’ (πΉβ€˜π‘‹) = 0 )
10414, 103rexlimddv 3161 1 ((πœ‘ ∧ 𝑋 ∈ (𝐼 βˆ– 𝐴)) β†’ (πΉβ€˜π‘‹) = 0 )
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βˆ– cdif 3945   βŠ† wss 3948  ifcif 4528   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676   β†Ύ cres 5678  Fun wfun 6537  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   supp csupp 8145  Xcixp 8890   finSupp cfsupp 9360  Basecbs 17143  0gc0g 17384   Ξ£g cgsu 17385  Mndcmnd 18624  Grpcgrp 18818  SubGrpcsubg 18999  Cntzccntz 19178   DProd cdprd 19862
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  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 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7669  df-om 7855  df-1st 7974  df-2nd 7975  df-supp 8146  df-tpos 8210  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-ixp 8891  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-fsupp 9361  df-oi 9504  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-fzo 13627  df-seq 13966  df-hash 14290  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-0g 17386  df-gsum 17387  df-mre 17529  df-mrc 17530  df-acs 17532  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-mhm 18670  df-submnd 18671  df-grp 18821  df-minusg 18822  df-sbg 18823  df-mulg 18950  df-subg 19002  df-ghm 19089  df-gim 19132  df-cntz 19180  df-oppg 19209  df-cmn 19649  df-dprd 19864
This theorem is referenced by:  dprddisj2  19908
  Copyright terms: Public domain W3C validator