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Theorem dprdfcntz 19073
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 2826 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
61, 2, 3, 4, 5dprdff 19070 . . . 4 (𝜑𝐹:𝐼⟶(Base‘𝐺))
76ffnd 6514 . . 3 (𝜑𝐹 Fn 𝐼)
86ffvelrnda 6849 . . . . 5 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ (Base‘𝐺))
9 simpr 485 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧)
109fveq2d 6673 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → (𝐹𝑦) = (𝐹𝑧))
119equcomd 2019 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → 𝑧 = 𝑦)
1211fveq2d 6673 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → (𝐹𝑧) = (𝐹𝑦))
1310, 12oveq12d 7168 . . . . . . . 8 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
142ad3antrrr 726 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝐺dom DProd 𝑆)
153ad3antrrr 726 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → dom 𝑆 = 𝐼)
16 simpllr 772 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝑦𝐼)
17 simplr 765 . . . . . . . . . . 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 19066 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝑆𝑦) ⊆ (𝑍‘(𝑆𝑧)))
211, 2, 3, 4dprdfcl 19071 . . . . . . . . . . 11 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ (𝑆𝑦))
2221ad2antrr 722 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝐹𝑦) ∈ (𝑆𝑦))
2320, 22sseldd 3972 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝐹𝑦) ∈ (𝑍‘(𝑆𝑧)))
241, 2, 3, 4dprdfcl 19071 . . . . . . . . . 10 ((𝜑𝑧𝐼) → (𝐹𝑧) ∈ (𝑆𝑧))
2524ad4ant13 747 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝐹𝑧) ∈ (𝑆𝑧))
26 eqid 2826 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
2726, 19cntzi 18404 . . . . . . . . 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 3109 . . . . . . 7 (((𝜑𝑦𝐼) ∧ 𝑧𝐼) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
3029ralrimiva 3187 . . . . . 6 ((𝜑𝑦𝐼) → ∀𝑧𝐼 ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
317adantr 481 . . . . . . 7 ((𝜑𝑦𝐼) → 𝐹 Fn 𝐼)
32 oveq2 7158 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → ((𝐹𝑦)(+g𝐺)𝑥) = ((𝐹𝑦)(+g𝐺)(𝐹𝑧)))
33 oveq1 7157 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → (𝑥(+g𝐺)(𝐹𝑦)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
3432, 33eqeq12d 2842 . . . . . . . 8 (𝑥 = (𝐹𝑧) → (((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)) ↔ ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦))))
3534ralrn 6852 . . . . . . 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 258 . . . . 5 ((𝜑𝑦𝐼) → ∀𝑥 ∈ ran 𝐹((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)))
386frnd 6520 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
3938adantr 481 . . . . . 6 ((𝜑𝑦𝐼) → ran 𝐹 ⊆ (Base‘𝐺))
405, 26, 19elcntz 18397 . . . . . 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 709 . . . 4 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ (𝑍‘ran 𝐹))
4342ralrimiva 3187 . . 3 (𝜑 → ∀𝑦𝐼 (𝐹𝑦) ∈ (𝑍‘ran 𝐹))
44 ffnfv 6880 . . 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 6520 1 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wne 3021  wral 3143  {crab 3147  wss 3940   class class class wbr 5063  dom cdm 5554  ran crn 5555   Fn wfn 6349  wf 6350  cfv 6354  (class class class)co 7150  Xcixp 8455   finSupp cfsupp 8827  Basecbs 16478  +gcplusg 16560  Cntzccntz 18390   DProd cdprd 19051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-ixp 8456  df-subg 18221  df-cntz 18392  df-dprd 19053
This theorem is referenced by:  dprdssv  19074  dprdfinv  19077  dprdfadd  19078  dprdfeq0  19080  dprdlub  19084  dmdprdsplitlem  19095  dpjidcl  19116
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