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Theorem dprdfcntz 19138
 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 2798 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
61, 2, 3, 4, 5dprdff 19135 . . . 4 (𝜑𝐹:𝐼⟶(Base‘𝐺))
76ffnd 6491 . . 3 (𝜑𝐹 Fn 𝐼)
86ffvelrnda 6833 . . . . 5 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ (Base‘𝐺))
9 simpr 488 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧)
109fveq2d 6654 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → (𝐹𝑦) = (𝐹𝑧))
119equcomd 2026 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → 𝑧 = 𝑦)
1211fveq2d 6654 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → (𝐹𝑧) = (𝐹𝑦))
1310, 12oveq12d 7158 . . . . . . . 8 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
142ad3antrrr 729 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝐺dom DProd 𝑆)
153ad3antrrr 729 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → dom 𝑆 = 𝐼)
16 simpllr 775 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝑦𝐼)
17 simplr 768 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝑧𝐼)
18 simpr 488 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝑦𝑧)
19 dprdfcntz.z . . . . . . . . . . 11 𝑍 = (Cntz‘𝐺)
2014, 15, 16, 17, 18, 19dprdcntz 19131 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝑆𝑦) ⊆ (𝑍‘(𝑆𝑧)))
211, 2, 3, 4dprdfcl 19136 . . . . . . . . . . 11 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ (𝑆𝑦))
2221ad2antrr 725 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝐹𝑦) ∈ (𝑆𝑦))
2320, 22sseldd 3916 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝐹𝑦) ∈ (𝑍‘(𝑆𝑧)))
241, 2, 3, 4dprdfcl 19136 . . . . . . . . . 10 ((𝜑𝑧𝐼) → (𝐹𝑧) ∈ (𝑆𝑧))
2524ad4ant13 750 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝐹𝑧) ∈ (𝑆𝑧))
26 eqid 2798 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
2726, 19cntzi 18459 . . . . . . . . 9 (((𝐹𝑦) ∈ (𝑍‘(𝑆𝑧)) ∧ (𝐹𝑧) ∈ (𝑆𝑧)) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
2823, 25, 27syl2anc 587 . . . . . . . 8 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
2913, 28pm2.61dane 3074 . . . . . . 7 (((𝜑𝑦𝐼) ∧ 𝑧𝐼) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
3029ralrimiva 3149 . . . . . 6 ((𝜑𝑦𝐼) → ∀𝑧𝐼 ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
317adantr 484 . . . . . . 7 ((𝜑𝑦𝐼) → 𝐹 Fn 𝐼)
32 oveq2 7148 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → ((𝐹𝑦)(+g𝐺)𝑥) = ((𝐹𝑦)(+g𝐺)(𝐹𝑧)))
33 oveq1 7147 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → (𝑥(+g𝐺)(𝐹𝑦)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
3432, 33eqeq12d 2814 . . . . . . . 8 (𝑥 = (𝐹𝑧) → (((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)) ↔ ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦))))
3534ralrn 6836 . . . . . . 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 260 . . . . 5 ((𝜑𝑦𝐼) → ∀𝑥 ∈ ran 𝐹((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)))
386frnd 6497 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
3938adantr 484 . . . . . 6 ((𝜑𝑦𝐼) → ran 𝐹 ⊆ (Base‘𝐺))
405, 26, 19elcntz 18452 . . . . . 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 712 . . . 4 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ (𝑍‘ran 𝐹))
4342ralrimiva 3149 . . 3 (𝜑 → ∀𝑦𝐼 (𝐹𝑦) ∈ (𝑍‘ran 𝐹))
44 ffnfv 6864 . . 3 (𝐹:𝐼⟶(𝑍‘ran 𝐹) ↔ (𝐹 Fn 𝐼 ∧ ∀𝑦𝐼 (𝐹𝑦) ∈ (𝑍‘ran 𝐹)))
457, 43, 44sylanbrc 586 . 2 (𝜑𝐹:𝐼⟶(𝑍‘ran 𝐹))
4645frnd 6497 1 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  {crab 3110   ⊆ wss 3881   class class class wbr 5031  dom cdm 5520  ran crn 5521   Fn wfn 6322  ⟶wf 6323  ‘cfv 6327  (class class class)co 7140  Xcixp 8451   finSupp cfsupp 8824  Basecbs 16482  +gcplusg 16564  Cntzccntz 18445   DProd cdprd 19116 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7678  df-2nd 7679  df-ixp 8452  df-subg 18276  df-cntz 18447  df-dprd 19118 This theorem is referenced by:  dprdssv  19139  dprdfinv  19142  dprdfadd  19143  dprdfeq0  19145  dprdlub  19149  dmdprdsplitlem  19160  dpjidcl  19181
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