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Theorem dmdprdsplitlem 19824
Description: Lemma for dmdprdsplit 19834. (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 19794 . . . . . . 7 (πœ‘ β†’ 𝑆:𝐼⟢(SubGrpβ€˜πΊ))
5 dmdprdsplitlem.3 . . . . . . 7 (πœ‘ β†’ 𝐴 βŠ† 𝐼)
64, 5fssresd 6713 . . . . . 6 (πœ‘ β†’ (𝑆 β†Ύ 𝐴):𝐴⟢(SubGrpβ€˜πΊ))
7 fdm 6681 . . . . . 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 2733 . . . . . . 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 19791 . . . . . 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 497 . . 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 482 . 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 772 . . . . . 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 496 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺dom DProd (𝑆 β†Ύ 𝐴))
1716ad2antrr 725 . . . . . . . . . 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 725 . . . . . . . . . 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 770 . . . . . . . . . 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 2733 . . . . . . . . . 10 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
229, 17, 19, 20, 21dprdff 19799 . . . . . . . . 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 6914 . . . . . . . 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 725 . . . . . . . . . 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 5998 . . . . . . . . 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 4496 . . . . . . . . . 10 (𝑛 ∈ 𝐴 β†’ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ) = (π‘“β€˜π‘›))
2726mpteq2ia 5212 . . . . . . . . 9 (𝑛 ∈ 𝐴 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) = (𝑛 ∈ 𝐴 ↦ (π‘“β€˜π‘›))
2825, 27eqtrdi 2789 . . . . . . . 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 2776 . . . . . . 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 7377 . . . . . 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 2733 . . . . . . 7 (Cntzβ€˜πΊ) = (Cntzβ€˜πΊ)
322ad2antrr 725 . . . . . . . 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 19792 . . . . . . . 8 (𝐺dom DProd 𝑆 β†’ 𝐺 ∈ Grp)
34 grpmnd 18763 . . . . . . . 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 19788 . . . . . . . 8 (πœ‘ β†’ 𝐼 ∈ V)
3736ad2antrr 725 . . . . . . 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 725 . . . . . . . 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 482 . . . . . . . . . . . 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 482 . . . . . . . . . . . 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 776 . . . . . . . . . . . 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 19800 . . . . . . . . . . 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 6865 . . . . . . . . . . . 12 (𝑛 ∈ 𝐴 β†’ ((𝑆 β†Ύ 𝐴)β€˜π‘›) = (π‘†β€˜π‘›))
4544adantl 483 . . . . . . . . . . 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 2836 . . . . . . . . . 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 725 . . . . . . . . . . . . 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 7039 . . . . . . . . . . . 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 18944 . . . . . . . . . . . 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 482 . . . . . . . . . 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 4525 . . . . . . . . 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 7178 . . . . . . . . . . 11 (πœ‘ β†’ (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) ∈ V)
5453ad2antrr 725 . . . . . . . . . 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 6543 . . . . . . . . . . 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 19801 . . . . . . . . . 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 486 . . . . . . . . . . . . . . 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 4091 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (𝐼 βˆ– (𝑓 supp 0 )) β†’ Β¬ 𝑛 ∈ (𝑓 supp 0 ))
6059ad2antlr 726 . . . . . . . . . . . . . . 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 3925 . . . . . . . . . . . . . 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 3971 . . . . . . . . . . . . . . . 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 5285 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝐴 ∈ V)
6463ad2antrr 725 . . . . . . . . . . . . . . . 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 6860 . . . . . . . . . . . . . . . . 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 8131 . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . 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 592 . . . . . . . . . . . . 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 4521 . . . . . . . . . . . 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 4530 . . . . . . . . . . . 12 if(𝑛 ∈ 𝐴, 0 , 0 ) = 0
7270, 71eqtrdi 2789 . . . . . . . . . . 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 8135 . . . . . . . . . 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 9329 . . . . . . . . . 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 838 . . . . . . . . 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 19798 . . . . . . . 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 19799 . . . . . . 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 19802 . . . . . . 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 4091 . . . . . . . . . 10 (𝑛 ∈ (𝐼 βˆ– 𝐴) β†’ Β¬ 𝑛 ∈ 𝐴)
8079adantl 483 . . . . . . . . 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 4501 . . . . . . . 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 8135 . . . . . . 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 19694 . . . . . 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 2777 . . . . 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 725 . . . . . 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 19810 . . . . 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 6848 . . 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 4090 . . . . 5 (𝑋 ∈ (𝐼 βˆ– 𝐴) β†’ 𝑋 ∈ 𝐼)
9190ad2antlr 726 . . . 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 2822 . . . . . 6 (𝑛 = 𝑋 β†’ (𝑛 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
93 fveq2 6846 . . . . . 6 (𝑛 = 𝑋 β†’ (π‘“β€˜π‘›) = (π‘“β€˜π‘‹))
9492, 93ifbieq1d 4514 . . . . 5 (𝑛 = 𝑋 β†’ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ) = if(𝑋 ∈ 𝐴, (π‘“β€˜π‘‹), 0 ))
95 eqid 2733 . . . . 5 (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 )) = (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ))
96 fvex 6859 . . . . . 6 (π‘“β€˜π‘›) ∈ V
9796, 65ifex 4540 . . . . 5 if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ) ∈ V
9894, 95, 97fvmpt3i 6957 . . . 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 4091 . . . . 5 (𝑋 ∈ (𝐼 βˆ– 𝐴) β†’ Β¬ 𝑋 ∈ 𝐴)
101100ad2antlr 726 . . . 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 4501 . . 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 2777 . 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 3155 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 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3070  {crab 3406  Vcvv 3447   βˆ– cdif 3911   βŠ† wss 3914  ifcif 4490   class class class wbr 5109   ↦ cmpt 5192  dom cdm 5637   β†Ύ cres 5639  Fun wfun 6494  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   supp csupp 8096  Xcixp 8841   finSupp cfsupp 9311  Basecbs 17091  0gc0g 17329   Ξ£g cgsu 17330  Mndcmnd 18564  Grpcgrp 18756  SubGrpcsubg 18930  Cntzccntz 19103   DProd cdprd 19780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7621  df-om 7807  df-1st 7925  df-2nd 7926  df-supp 8097  df-tpos 8161  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-ixp 8842  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-fsupp 9312  df-oi 9454  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-fzo 13577  df-seq 13916  df-hash 14240  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-0g 17331  df-gsum 17332  df-mre 17474  df-mrc 17475  df-acs 17477  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-mhm 18609  df-submnd 18610  df-grp 18759  df-minusg 18760  df-sbg 18761  df-mulg 18881  df-subg 18933  df-ghm 19014  df-gim 19057  df-cntz 19105  df-oppg 19132  df-cmn 19572  df-dprd 19782
This theorem is referenced by:  dprddisj2  19826
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