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Theorem dprdfcntz 19794
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 2736 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
61, 2, 3, 4, 5dprdff 19791 . . . 4 (𝜑𝐹:𝐼⟶(Base‘𝐺))
76ffnd 6669 . . 3 (𝜑𝐹 Fn 𝐼)
86ffvelcdmda 7035 . . . . 5 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ (Base‘𝐺))
9 simpr 485 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧)
109fveq2d 6846 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → (𝐹𝑦) = (𝐹𝑧))
119equcomd 2022 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → 𝑧 = 𝑦)
1211fveq2d 6846 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → (𝐹𝑧) = (𝐹𝑦))
1310, 12oveq12d 7375 . . . . . . . 8 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
142ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝐺dom DProd 𝑆)
153ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → dom 𝑆 = 𝐼)
16 simpllr 774 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝑦𝐼)
17 simplr 767 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝑧𝐼)
18 simpr 485 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝑦𝑧)
19 dprdfcntz.z . . . . . . . . . . 11 𝑍 = (Cntz‘𝐺)
2014, 15, 16, 17, 18, 19dprdcntz 19787 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝑆𝑦) ⊆ (𝑍‘(𝑆𝑧)))
211, 2, 3, 4dprdfcl 19792 . . . . . . . . . . 11 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ (𝑆𝑦))
2221ad2antrr 724 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝐹𝑦) ∈ (𝑆𝑦))
2320, 22sseldd 3945 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝐹𝑦) ∈ (𝑍‘(𝑆𝑧)))
241, 2, 3, 4dprdfcl 19792 . . . . . . . . . 10 ((𝜑𝑧𝐼) → (𝐹𝑧) ∈ (𝑆𝑧))
2524ad4ant13 749 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝐹𝑧) ∈ (𝑆𝑧))
26 eqid 2736 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
2726, 19cntzi 19109 . . . . . . . . 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 3032 . . . . . . 7 (((𝜑𝑦𝐼) ∧ 𝑧𝐼) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
3029ralrimiva 3143 . . . . . 6 ((𝜑𝑦𝐼) → ∀𝑧𝐼 ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
317adantr 481 . . . . . . 7 ((𝜑𝑦𝐼) → 𝐹 Fn 𝐼)
32 oveq2 7365 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → ((𝐹𝑦)(+g𝐺)𝑥) = ((𝐹𝑦)(+g𝐺)(𝐹𝑧)))
33 oveq1 7364 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → (𝑥(+g𝐺)(𝐹𝑦)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
3432, 33eqeq12d 2752 . . . . . . . 8 (𝑥 = (𝐹𝑧) → (((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)) ↔ ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦))))
3534ralrn 7038 . . . . . . 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 256 . . . . 5 ((𝜑𝑦𝐼) → ∀𝑥 ∈ ran 𝐹((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)))
386frnd 6676 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
3938adantr 481 . . . . . 6 ((𝜑𝑦𝐼) → ran 𝐹 ⊆ (Base‘𝐺))
405, 26, 19elcntz 19102 . . . . . 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 711 . . . 4 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ (𝑍‘ran 𝐹))
4342ralrimiva 3143 . . 3 (𝜑 → ∀𝑦𝐼 (𝐹𝑦) ∈ (𝑍‘ran 𝐹))
44 ffnfv 7066 . . 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 6676 1 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  {crab 3407  wss 3910   class class class wbr 5105  dom cdm 5633  ran crn 5634   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  Xcixp 8835   finSupp cfsupp 9305  Basecbs 17083  +gcplusg 17133  Cntzccntz 19095   DProd cdprd 19772
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-ixp 8836  df-subg 18925  df-cntz 19097  df-dprd 19774
This theorem is referenced by:  dprdssv  19795  dprdfinv  19798  dprdfadd  19799  dprdfeq0  19801  dprdlub  19805  dmdprdsplitlem  19816  dpjidcl  19837
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