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Theorem dprdfcntz 19885
Description: A function on the elements of an internal direct product has pairwise commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
Hypotheses
Ref Expression
dprdff.w π‘Š = {β„Ž ∈ X𝑖 ∈ 𝐼 (π‘†β€˜π‘–) ∣ β„Ž finSupp 0 }
dprdff.1 (πœ‘ β†’ 𝐺dom DProd 𝑆)
dprdff.2 (πœ‘ β†’ dom 𝑆 = 𝐼)
dprdff.3 (πœ‘ β†’ 𝐹 ∈ π‘Š)
dprdfcntz.z 𝑍 = (Cntzβ€˜πΊ)
Assertion
Ref Expression
dprdfcntz (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
Distinct variable groups:   β„Ž,𝐹   β„Ž,𝑖,𝐼   0 ,β„Ž   𝑆,β„Ž,𝑖
Allowed substitution hints:   πœ‘(β„Ž,𝑖)   𝐹(𝑖)   𝐺(β„Ž,𝑖)   π‘Š(β„Ž,𝑖)   0 (𝑖)   𝑍(β„Ž,𝑖)

Proof of Theorem dprdfcntz
Dummy variables 𝑦 𝑧 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dprdff.w . . . . 5 π‘Š = {β„Ž ∈ X𝑖 ∈ 𝐼 (π‘†β€˜π‘–) ∣ β„Ž finSupp 0 }
2 dprdff.1 . . . . 5 (πœ‘ β†’ 𝐺dom DProd 𝑆)
3 dprdff.2 . . . . 5 (πœ‘ β†’ dom 𝑆 = 𝐼)
4 dprdff.3 . . . . 5 (πœ‘ β†’ 𝐹 ∈ π‘Š)
5 eqid 2733 . . . . 5 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
61, 2, 3, 4, 5dprdff 19882 . . . 4 (πœ‘ β†’ 𝐹:𝐼⟢(Baseβ€˜πΊ))
76ffnd 6719 . . 3 (πœ‘ β†’ 𝐹 Fn 𝐼)
86ffvelcdmda 7087 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (πΉβ€˜π‘¦) ∈ (Baseβ€˜πΊ))
9 simpr 486 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) β†’ 𝑦 = 𝑧)
109fveq2d 6896 . . . . . . . . 9 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘§))
119equcomd 2023 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) β†’ 𝑧 = 𝑦)
1211fveq2d 6896 . . . . . . . . 9 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘¦))
1310, 12oveq12d 7427 . . . . . . . 8 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) β†’ ((πΉβ€˜π‘¦)(+gβ€˜πΊ)(πΉβ€˜π‘§)) = ((πΉβ€˜π‘§)(+gβ€˜πΊ)(πΉβ€˜π‘¦)))
142ad3antrrr 729 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ 𝐺dom DProd 𝑆)
153ad3antrrr 729 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ dom 𝑆 = 𝐼)
16 simpllr 775 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ 𝑦 ∈ 𝐼)
17 simplr 768 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ 𝑧 ∈ 𝐼)
18 simpr 486 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ 𝑦 β‰  𝑧)
19 dprdfcntz.z . . . . . . . . . . 11 𝑍 = (Cntzβ€˜πΊ)
2014, 15, 16, 17, 18, 19dprdcntz 19878 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ (π‘†β€˜π‘¦) βŠ† (π‘β€˜(π‘†β€˜π‘§)))
211, 2, 3, 4dprdfcl 19883 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (πΉβ€˜π‘¦) ∈ (π‘†β€˜π‘¦))
2221ad2antrr 725 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ (πΉβ€˜π‘¦) ∈ (π‘†β€˜π‘¦))
2320, 22sseldd 3984 . . . . . . . . 9 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ (πΉβ€˜π‘¦) ∈ (π‘β€˜(π‘†β€˜π‘§)))
241, 2, 3, 4dprdfcl 19883 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ 𝐼) β†’ (πΉβ€˜π‘§) ∈ (π‘†β€˜π‘§))
2524ad4ant13 750 . . . . . . . . 9 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ (πΉβ€˜π‘§) ∈ (π‘†β€˜π‘§))
26 eqid 2733 . . . . . . . . . 10 (+gβ€˜πΊ) = (+gβ€˜πΊ)
2726, 19cntzi 19193 . . . . . . . . 9 (((πΉβ€˜π‘¦) ∈ (π‘β€˜(π‘†β€˜π‘§)) ∧ (πΉβ€˜π‘§) ∈ (π‘†β€˜π‘§)) β†’ ((πΉβ€˜π‘¦)(+gβ€˜πΊ)(πΉβ€˜π‘§)) = ((πΉβ€˜π‘§)(+gβ€˜πΊ)(πΉβ€˜π‘¦)))
2823, 25, 27syl2anc 585 . . . . . . . 8 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ ((πΉβ€˜π‘¦)(+gβ€˜πΊ)(πΉβ€˜π‘§)) = ((πΉβ€˜π‘§)(+gβ€˜πΊ)(πΉβ€˜π‘¦)))
2913, 28pm2.61dane 3030 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) β†’ ((πΉβ€˜π‘¦)(+gβ€˜πΊ)(πΉβ€˜π‘§)) = ((πΉβ€˜π‘§)(+gβ€˜πΊ)(πΉβ€˜π‘¦)))
3029ralrimiva 3147 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ βˆ€π‘§ ∈ 𝐼 ((πΉβ€˜π‘¦)(+gβ€˜πΊ)(πΉβ€˜π‘§)) = ((πΉβ€˜π‘§)(+gβ€˜πΊ)(πΉβ€˜π‘¦)))
317adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ 𝐹 Fn 𝐼)
32 oveq2 7417 . . . . . . . . 9 (π‘₯ = (πΉβ€˜π‘§) β†’ ((πΉβ€˜π‘¦)(+gβ€˜πΊ)π‘₯) = ((πΉβ€˜π‘¦)(+gβ€˜πΊ)(πΉβ€˜π‘§)))
33 oveq1 7416 . . . . . . . . 9 (π‘₯ = (πΉβ€˜π‘§) β†’ (π‘₯(+gβ€˜πΊ)(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘§)(+gβ€˜πΊ)(πΉβ€˜π‘¦)))
3432, 33eqeq12d 2749 . . . . . . . 8 (π‘₯ = (πΉβ€˜π‘§) β†’ (((πΉβ€˜π‘¦)(+gβ€˜πΊ)π‘₯) = (π‘₯(+gβ€˜πΊ)(πΉβ€˜π‘¦)) ↔ ((πΉβ€˜π‘¦)(+gβ€˜πΊ)(πΉβ€˜π‘§)) = ((πΉβ€˜π‘§)(+gβ€˜πΊ)(πΉβ€˜π‘¦))))
3534ralrn 7090 . . . . . . 7 (𝐹 Fn 𝐼 β†’ (βˆ€π‘₯ ∈ ran 𝐹((πΉβ€˜π‘¦)(+gβ€˜πΊ)π‘₯) = (π‘₯(+gβ€˜πΊ)(πΉβ€˜π‘¦)) ↔ βˆ€π‘§ ∈ 𝐼 ((πΉβ€˜π‘¦)(+gβ€˜πΊ)(πΉβ€˜π‘§)) = ((πΉβ€˜π‘§)(+gβ€˜πΊ)(πΉβ€˜π‘¦))))
3631, 35syl 17 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (βˆ€π‘₯ ∈ ran 𝐹((πΉβ€˜π‘¦)(+gβ€˜πΊ)π‘₯) = (π‘₯(+gβ€˜πΊ)(πΉβ€˜π‘¦)) ↔ βˆ€π‘§ ∈ 𝐼 ((πΉβ€˜π‘¦)(+gβ€˜πΊ)(πΉβ€˜π‘§)) = ((πΉβ€˜π‘§)(+gβ€˜πΊ)(πΉβ€˜π‘¦))))
3730, 36mpbird 257 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ βˆ€π‘₯ ∈ ran 𝐹((πΉβ€˜π‘¦)(+gβ€˜πΊ)π‘₯) = (π‘₯(+gβ€˜πΊ)(πΉβ€˜π‘¦)))
386frnd 6726 . . . . . . 7 (πœ‘ β†’ ran 𝐹 βŠ† (Baseβ€˜πΊ))
3938adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ ran 𝐹 βŠ† (Baseβ€˜πΊ))
405, 26, 19elcntz 19186 . . . . . 6 (ran 𝐹 βŠ† (Baseβ€˜πΊ) β†’ ((πΉβ€˜π‘¦) ∈ (π‘β€˜ran 𝐹) ↔ ((πΉβ€˜π‘¦) ∈ (Baseβ€˜πΊ) ∧ βˆ€π‘₯ ∈ ran 𝐹((πΉβ€˜π‘¦)(+gβ€˜πΊ)π‘₯) = (π‘₯(+gβ€˜πΊ)(πΉβ€˜π‘¦)))))
4139, 40syl 17 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ ((πΉβ€˜π‘¦) ∈ (π‘β€˜ran 𝐹) ↔ ((πΉβ€˜π‘¦) ∈ (Baseβ€˜πΊ) ∧ βˆ€π‘₯ ∈ ran 𝐹((πΉβ€˜π‘¦)(+gβ€˜πΊ)π‘₯) = (π‘₯(+gβ€˜πΊ)(πΉβ€˜π‘¦)))))
428, 37, 41mpbir2and 712 . . . 4 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (πΉβ€˜π‘¦) ∈ (π‘β€˜ran 𝐹))
4342ralrimiva 3147 . . 3 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐼 (πΉβ€˜π‘¦) ∈ (π‘β€˜ran 𝐹))
44 ffnfv 7118 . . 3 (𝐹:𝐼⟢(π‘β€˜ran 𝐹) ↔ (𝐹 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (πΉβ€˜π‘¦) ∈ (π‘β€˜ran 𝐹)))
457, 43, 44sylanbrc 584 . 2 (πœ‘ β†’ 𝐹:𝐼⟢(π‘β€˜ran 𝐹))
4645frnd 6726 1 (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  {crab 3433   βŠ† wss 3949   class class class wbr 5149  dom cdm 5677  ran crn 5678   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Xcixp 8891   finSupp cfsupp 9361  Basecbs 17144  +gcplusg 17197  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
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-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-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  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-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-ixp 8892  df-subg 19003  df-cntz 19181  df-dprd 19865
This theorem is referenced by:  dprdssv  19886  dprdfinv  19889  dprdfadd  19890  dprdfeq0  19892  dprdlub  19896  dmdprdsplitlem  19907  dpjidcl  19928
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