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Theorem dprdfcntz 20035
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 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
dprdff.1 (𝜑𝐺dom DProd 𝑆)
dprdff.2 (𝜑 → dom 𝑆 = 𝐼)
dprdff.3 (𝜑𝐹𝑊)
dprdfcntz.z 𝑍 = (Cntz‘𝐺)
Assertion
Ref Expression
dprdfcntz (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
Distinct variable groups:   ,𝐹   ,𝑖,𝐼   0 ,   𝑆,,𝑖
Allowed substitution hints:   𝜑(,𝑖)   𝐹(𝑖)   𝐺(,𝑖)   𝑊(,𝑖)   0 (𝑖)   𝑍(,𝑖)

Proof of Theorem dprdfcntz
Dummy variables 𝑦 𝑧 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dprdff.w . . . . 5 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
2 dprdff.1 . . . . 5 (𝜑𝐺dom DProd 𝑆)
3 dprdff.2 . . . . 5 (𝜑 → dom 𝑆 = 𝐼)
4 dprdff.3 . . . . 5 (𝜑𝐹𝑊)
5 eqid 2737 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
61, 2, 3, 4, 5dprdff 20032 . . . 4 (𝜑𝐹:𝐼⟶(Base‘𝐺))
76ffnd 6737 . . 3 (𝜑𝐹 Fn 𝐼)
86ffvelcdmda 7104 . . . . 5 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ (Base‘𝐺))
9 simpr 484 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧)
109fveq2d 6910 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → (𝐹𝑦) = (𝐹𝑧))
119equcomd 2018 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → 𝑧 = 𝑦)
1211fveq2d 6910 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → (𝐹𝑧) = (𝐹𝑦))
1310, 12oveq12d 7449 . . . . . . . 8 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
142ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝐺dom DProd 𝑆)
153ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → dom 𝑆 = 𝐼)
16 simpllr 776 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝑦𝐼)
17 simplr 769 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝑧𝐼)
18 simpr 484 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝑦𝑧)
19 dprdfcntz.z . . . . . . . . . . 11 𝑍 = (Cntz‘𝐺)
2014, 15, 16, 17, 18, 19dprdcntz 20028 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝑆𝑦) ⊆ (𝑍‘(𝑆𝑧)))
211, 2, 3, 4dprdfcl 20033 . . . . . . . . . . 11 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ (𝑆𝑦))
2221ad2antrr 726 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝐹𝑦) ∈ (𝑆𝑦))
2320, 22sseldd 3984 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝐹𝑦) ∈ (𝑍‘(𝑆𝑧)))
241, 2, 3, 4dprdfcl 20033 . . . . . . . . . 10 ((𝜑𝑧𝐼) → (𝐹𝑧) ∈ (𝑆𝑧))
2524ad4ant13 751 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝐹𝑧) ∈ (𝑆𝑧))
26 eqid 2737 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
2726, 19cntzi 19347 . . . . . . . . 9 (((𝐹𝑦) ∈ (𝑍‘(𝑆𝑧)) ∧ (𝐹𝑧) ∈ (𝑆𝑧)) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
2823, 25, 27syl2anc 584 . . . . . . . 8 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
2913, 28pm2.61dane 3029 . . . . . . 7 (((𝜑𝑦𝐼) ∧ 𝑧𝐼) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
3029ralrimiva 3146 . . . . . 6 ((𝜑𝑦𝐼) → ∀𝑧𝐼 ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
317adantr 480 . . . . . . 7 ((𝜑𝑦𝐼) → 𝐹 Fn 𝐼)
32 oveq2 7439 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → ((𝐹𝑦)(+g𝐺)𝑥) = ((𝐹𝑦)(+g𝐺)(𝐹𝑧)))
33 oveq1 7438 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → (𝑥(+g𝐺)(𝐹𝑦)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
3432, 33eqeq12d 2753 . . . . . . . 8 (𝑥 = (𝐹𝑧) → (((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)) ↔ ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦))))
3534ralrn 7108 . . . . . . 7 (𝐹 Fn 𝐼 → (∀𝑥 ∈ ran 𝐹((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)) ↔ ∀𝑧𝐼 ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦))))
3631, 35syl 17 . . . . . 6 ((𝜑𝑦𝐼) → (∀𝑥 ∈ ran 𝐹((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)) ↔ ∀𝑧𝐼 ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦))))
3730, 36mpbird 257 . . . . 5 ((𝜑𝑦𝐼) → ∀𝑥 ∈ ran 𝐹((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)))
386frnd 6744 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
3938adantr 480 . . . . . 6 ((𝜑𝑦𝐼) → ran 𝐹 ⊆ (Base‘𝐺))
405, 26, 19elcntz 19340 . . . . . 6 (ran 𝐹 ⊆ (Base‘𝐺) → ((𝐹𝑦) ∈ (𝑍‘ran 𝐹) ↔ ((𝐹𝑦) ∈ (Base‘𝐺) ∧ ∀𝑥 ∈ ran 𝐹((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)))))
4139, 40syl 17 . . . . 5 ((𝜑𝑦𝐼) → ((𝐹𝑦) ∈ (𝑍‘ran 𝐹) ↔ ((𝐹𝑦) ∈ (Base‘𝐺) ∧ ∀𝑥 ∈ ran 𝐹((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)))))
428, 37, 41mpbir2and 713 . . . 4 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ (𝑍‘ran 𝐹))
4342ralrimiva 3146 . . 3 (𝜑 → ∀𝑦𝐼 (𝐹𝑦) ∈ (𝑍‘ran 𝐹))
44 ffnfv 7139 . . 3 (𝐹:𝐼⟶(𝑍‘ran 𝐹) ↔ (𝐹 Fn 𝐼 ∧ ∀𝑦𝐼 (𝐹𝑦) ∈ (𝑍‘ran 𝐹)))
457, 43, 44sylanbrc 583 . 2 (𝜑𝐹:𝐼⟶(𝑍‘ran 𝐹))
4645frnd 6744 1 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  {crab 3436  wss 3951   class class class wbr 5143  dom cdm 5685  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Xcixp 8937   finSupp cfsupp 9401  Basecbs 17247  +gcplusg 17297  Cntzccntz 19333   DProd cdprd 20013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-ixp 8938  df-subg 19141  df-cntz 19335  df-dprd 20015
This theorem is referenced by:  dprdssv  20036  dprdfinv  20039  dprdfadd  20040  dprdfeq0  20042  dprdlub  20046  dmdprdsplitlem  20057  dpjidcl  20078
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