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Theorem dprdfcntz 18237
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 2610 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
61, 2, 3, 4, 5dprdff 18234 . . . 4 (𝜑𝐹:𝐼⟶(Base‘𝐺))
7 ffn 5958 . . . 4 (𝐹:𝐼⟶(Base‘𝐺) → 𝐹 Fn 𝐼)
86, 7syl 17 . . 3 (𝜑𝐹 Fn 𝐼)
96ffvelrnda 6267 . . . . 5 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ (Base‘𝐺))
10 simpr 476 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧)
1110fveq2d 6107 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → (𝐹𝑦) = (𝐹𝑧))
1210eqcomd 2616 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → 𝑧 = 𝑦)
1312fveq2d 6107 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → (𝐹𝑧) = (𝐹𝑦))
1411, 13oveq12d 6567 . . . . . . . 8 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦 = 𝑧) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
152ad3antrrr 762 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝐺dom DProd 𝑆)
163ad3antrrr 762 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → dom 𝑆 = 𝐼)
17 simpllr 795 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝑦𝐼)
18 simplr 788 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝑧𝐼)
19 simpr 476 . . . . . . . . . . 11 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → 𝑦𝑧)
20 dprdfcntz.z . . . . . . . . . . 11 𝑍 = (Cntz‘𝐺)
2115, 16, 17, 18, 19, 20dprdcntz 18230 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝑆𝑦) ⊆ (𝑍‘(𝑆𝑧)))
221, 2, 3, 4dprdfcl 18235 . . . . . . . . . . 11 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ (𝑆𝑦))
2322ad2antrr 758 . . . . . . . . . 10 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝐹𝑦) ∈ (𝑆𝑦))
2421, 23sseldd 3569 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝐹𝑦) ∈ (𝑍‘(𝑆𝑧)))
251, 2, 3, 4dprdfcl 18235 . . . . . . . . . . 11 ((𝜑𝑧𝐼) → (𝐹𝑧) ∈ (𝑆𝑧))
2625adantlr 747 . . . . . . . . . 10 (((𝜑𝑦𝐼) ∧ 𝑧𝐼) → (𝐹𝑧) ∈ (𝑆𝑧))
2726adantr 480 . . . . . . . . 9 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → (𝐹𝑧) ∈ (𝑆𝑧))
28 eqid 2610 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
2928, 20cntzi 17585 . . . . . . . . 9 (((𝐹𝑦) ∈ (𝑍‘(𝑆𝑧)) ∧ (𝐹𝑧) ∈ (𝑆𝑧)) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
3024, 27, 29syl2anc 691 . . . . . . . 8 ((((𝜑𝑦𝐼) ∧ 𝑧𝐼) ∧ 𝑦𝑧) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
3114, 30pm2.61dane 2869 . . . . . . 7 (((𝜑𝑦𝐼) ∧ 𝑧𝐼) → ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
3231ralrimiva 2949 . . . . . 6 ((𝜑𝑦𝐼) → ∀𝑧𝐼 ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
338adantr 480 . . . . . . 7 ((𝜑𝑦𝐼) → 𝐹 Fn 𝐼)
34 oveq2 6557 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → ((𝐹𝑦)(+g𝐺)𝑥) = ((𝐹𝑦)(+g𝐺)(𝐹𝑧)))
35 oveq1 6556 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → (𝑥(+g𝐺)(𝐹𝑦)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦)))
3634, 35eqeq12d 2625 . . . . . . . 8 (𝑥 = (𝐹𝑧) → (((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)) ↔ ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦))))
3736ralrn 6270 . . . . . . 7 (𝐹 Fn 𝐼 → (∀𝑥 ∈ ran 𝐹((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)) ↔ ∀𝑧𝐼 ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦))))
3833, 37syl 17 . . . . . 6 ((𝜑𝑦𝐼) → (∀𝑥 ∈ ran 𝐹((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)) ↔ ∀𝑧𝐼 ((𝐹𝑦)(+g𝐺)(𝐹𝑧)) = ((𝐹𝑧)(+g𝐺)(𝐹𝑦))))
3932, 38mpbird 246 . . . . 5 ((𝜑𝑦𝐼) → ∀𝑥 ∈ ran 𝐹((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)))
40 frn 5966 . . . . . . . 8 (𝐹:𝐼⟶(Base‘𝐺) → ran 𝐹 ⊆ (Base‘𝐺))
416, 40syl 17 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
4241adantr 480 . . . . . 6 ((𝜑𝑦𝐼) → ran 𝐹 ⊆ (Base‘𝐺))
435, 28, 20elcntz 17578 . . . . . 6 (ran 𝐹 ⊆ (Base‘𝐺) → ((𝐹𝑦) ∈ (𝑍‘ran 𝐹) ↔ ((𝐹𝑦) ∈ (Base‘𝐺) ∧ ∀𝑥 ∈ ran 𝐹((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)))))
4442, 43syl 17 . . . . 5 ((𝜑𝑦𝐼) → ((𝐹𝑦) ∈ (𝑍‘ran 𝐹) ↔ ((𝐹𝑦) ∈ (Base‘𝐺) ∧ ∀𝑥 ∈ ran 𝐹((𝐹𝑦)(+g𝐺)𝑥) = (𝑥(+g𝐺)(𝐹𝑦)))))
459, 39, 44mpbir2and 959 . . . 4 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ (𝑍‘ran 𝐹))
4645ralrimiva 2949 . . 3 (𝜑 → ∀𝑦𝐼 (𝐹𝑦) ∈ (𝑍‘ran 𝐹))
47 ffnfv 6295 . . 3 (𝐹:𝐼⟶(𝑍‘ran 𝐹) ↔ (𝐹 Fn 𝐼 ∧ ∀𝑦𝐼 (𝐹𝑦) ∈ (𝑍‘ran 𝐹)))
488, 46, 47sylanbrc 695 . 2 (𝜑𝐹:𝐼⟶(𝑍‘ran 𝐹))
49 frn 5966 . 2 (𝐹:𝐼⟶(𝑍‘ran 𝐹) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
5048, 49syl 17 1 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wne 2780  wral 2896  {crab 2900  wss 3540   class class class wbr 4583  dom cdm 5038  ran crn 5039   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  Xcixp 7794   finSupp cfsupp 8158  Basecbs 15695  +gcplusg 15768  Cntzccntz 17571   DProd cdprd 18215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-ixp 7795  df-subg 17414  df-cntz 17573  df-dprd 18217
This theorem is referenced by:  dprdssv  18238  dprdfinv  18241  dprdfadd  18242  dprdfeq0  18244  dprdlub  18248  dmdprdsplitlem  18259  dpjidcl  18280
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