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

Theorem dprdfcntz 19887
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 2732 . . . . 5 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
61, 2, 3, 4, 5dprdff 19884 . . . 4 (πœ‘ β†’ 𝐹:𝐼⟢(Baseβ€˜πΊ))
76ffnd 6718 . . 3 (πœ‘ β†’ 𝐹 Fn 𝐼)
86ffvelcdmda 7086 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (πΉβ€˜π‘¦) ∈ (Baseβ€˜πΊ))
9 simpr 485 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) β†’ 𝑦 = 𝑧)
109fveq2d 6895 . . . . . . . . 9 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘§))
119equcomd 2022 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) β†’ 𝑧 = 𝑦)
1211fveq2d 6895 . . . . . . . . 9 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘¦))
1310, 12oveq12d 7429 . . . . . . . 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 728 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ 𝐺dom DProd 𝑆)
153ad3antrrr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ dom 𝑆 = 𝐼)
16 simpllr 774 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ 𝑦 ∈ 𝐼)
17 simplr 767 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ 𝑧 ∈ 𝐼)
18 simpr 485 . . . . . . . . . . 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 19880 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ (π‘†β€˜π‘¦) βŠ† (π‘β€˜(π‘†β€˜π‘§)))
211, 2, 3, 4dprdfcl 19885 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (πΉβ€˜π‘¦) ∈ (π‘†β€˜π‘¦))
2221ad2antrr 724 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ (πΉβ€˜π‘¦) ∈ (π‘†β€˜π‘¦))
2320, 22sseldd 3983 . . . . . . . . 9 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ (πΉβ€˜π‘¦) ∈ (π‘β€˜(π‘†β€˜π‘§)))
241, 2, 3, 4dprdfcl 19885 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ 𝐼) β†’ (πΉβ€˜π‘§) ∈ (π‘†β€˜π‘§))
2524ad4ant13 749 . . . . . . . . 9 ((((πœ‘ ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 β‰  𝑧) β†’ (πΉβ€˜π‘§) ∈ (π‘†β€˜π‘§))
26 eqid 2732 . . . . . . . . . 10 (+gβ€˜πΊ) = (+gβ€˜πΊ)
2726, 19cntzi 19195 . . . . . . . . 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 584 . . . . . . . 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 3029 . . . . . . 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 3146 . . . . . 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 481 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ 𝐹 Fn 𝐼)
32 oveq2 7419 . . . . . . . . 9 (π‘₯ = (πΉβ€˜π‘§) β†’ ((πΉβ€˜π‘¦)(+gβ€˜πΊ)π‘₯) = ((πΉβ€˜π‘¦)(+gβ€˜πΊ)(πΉβ€˜π‘§)))
33 oveq1 7418 . . . . . . . . 9 (π‘₯ = (πΉβ€˜π‘§) β†’ (π‘₯(+gβ€˜πΊ)(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘§)(+gβ€˜πΊ)(πΉβ€˜π‘¦)))
3432, 33eqeq12d 2748 . . . . . . . 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 7089 . . . . . . 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 256 . . . . 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 6725 . . . . . . 7 (πœ‘ β†’ ran 𝐹 βŠ† (Baseβ€˜πΊ))
3938adantr 481 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ ran 𝐹 βŠ† (Baseβ€˜πΊ))
405, 26, 19elcntz 19188 . . . . . 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 711 . . . 4 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (πΉβ€˜π‘¦) ∈ (π‘β€˜ran 𝐹))
4342ralrimiva 3146 . . 3 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐼 (πΉβ€˜π‘¦) ∈ (π‘β€˜ran 𝐹))
44 ffnfv 7120 . . 3 (𝐹:𝐼⟢(π‘β€˜ran 𝐹) ↔ (𝐹 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (πΉβ€˜π‘¦) ∈ (π‘β€˜ran 𝐹)))
457, 43, 44sylanbrc 583 . 2 (πœ‘ β†’ 𝐹:𝐼⟢(π‘β€˜ran 𝐹))
4645frnd 6725 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  {crab 3432   βŠ† wss 3948   class class class wbr 5148  dom cdm 5676  ran crn 5677   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  Xcixp 8893   finSupp cfsupp 9363  Basecbs 17146  +gcplusg 17199  Cntzccntz 19181   DProd cdprd 19865
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 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-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-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  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-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-ixp 8894  df-subg 19005  df-cntz 19183  df-dprd 19867
This theorem is referenced by:  dprdssv  19888  dprdfinv  19891  dprdfadd  19892  dprdfeq0  19894  dprdlub  19898  dmdprdsplitlem  19909  dpjidcl  19930
  Copyright terms: Public domain W3C validator