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Theorem dmdprdsplitlem 19907
Description: Lemma for dmdprdsplit 19917. (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 19877 . . . . . . 7 (πœ‘ β†’ 𝑆:𝐼⟢(SubGrpβ€˜πΊ))
5 dmdprdsplitlem.3 . . . . . . 7 (πœ‘ β†’ 𝐴 βŠ† 𝐼)
64, 5fssresd 6759 . . . . . 6 (πœ‘ β†’ (𝑆 β†Ύ 𝐴):𝐴⟢(SubGrpβ€˜πΊ))
7 fdm 6727 . . . . . 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 19874 . . . . . 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 19882 . . . . . . . . 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 6961 . . . . . . . 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 6041 . . . . . . . . 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 4535 . . . . . . . . . 10 (𝑛 ∈ 𝐴 β†’ if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ) = (π‘“β€˜π‘›))
2726mpteq2ia 5252 . . . . . . . . 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 7425 . . . . . 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 19875 . . . . . . . 8 (𝐺dom DProd 𝑆 β†’ 𝐺 ∈ Grp)
34 grpmnd 18826 . . . . . . . 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 19871 . . . . . . . 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 19883 . . . . . . . . . . 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 6911 . . . . . . . . . . . 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 7087 . . . . . . . . . . . 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 19014 . . . . . . . . . . . 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 4564 . . . . . . . . 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 7226 . . . . . . . . . . 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 6587 . . . . . . . . . . 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 19884 . . . . . . . . . 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 4128 . . . . . . . . . . . . . . . 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 3960 . . . . . . . . . . . . . 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 4006 . . . . . . . . . . . . . . . 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 5325 . . . . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . 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 8181 . . . . . . . . . . . . . . 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 4560 . . . . . . . . . . . 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 4569 . . . . . . . . . . . 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 8185 . . . . . . . . . 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 9379 . . . . . . . . . 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 19881 . . . . . . . 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 19882 . . . . . . 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 19885 . . . . . . 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 4128 . . . . . . . . . 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 4540 . . . . . . . 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 8185 . . . . . . 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 19777 . . . . . 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 19893 . . . . 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 6894 . . 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 4127 . . . . 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 6892 . . . . . 6 (𝑛 = 𝑋 β†’ (π‘“β€˜π‘›) = (π‘“β€˜π‘‹))
9492, 93ifbieq1d 4553 . . . . 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 6905 . . . . . 6 (π‘“β€˜π‘›) ∈ V
9796, 65ifex 4579 . . . . 5 if(𝑛 ∈ 𝐴, (π‘“β€˜π‘›), 0 ) ∈ V
9894, 95, 97fvmpt3i 7004 . . . 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 4128 . . . . 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 4540 . . 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 3162 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 3071  {crab 3433  Vcvv 3475   βˆ– cdif 3946   βŠ† wss 3949  ifcif 4529   class class class wbr 5149   ↦ cmpt 5232  dom cdm 5677   β†Ύ cres 5679  Fun wfun 6538  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   supp csupp 8146  Xcixp 8891   finSupp cfsupp 9361  Basecbs 17144  0gc0g 17385   Ξ£g cgsu 17386  Mndcmnd 18625  Grpcgrp 18819  SubGrpcsubg 19000  Cntzccntz 19179   DProd cdprd 19863
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9362  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-seq 13967  df-hash 14291  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-0g 17387  df-gsum 17388  df-mre 17530  df-mrc 17531  df-acs 17533  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671  df-submnd 18672  df-grp 18822  df-minusg 18823  df-sbg 18824  df-mulg 18951  df-subg 19003  df-ghm 19090  df-gim 19133  df-cntz 19181  df-oppg 19210  df-cmn 19650  df-dprd 19865
This theorem is referenced by:  dprddisj2  19909
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