Theorem dprdfcntz 19896

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 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
dprdff.1	⊢ (𝜑 → 𝐺dom DProd 𝑆)
dprdff.2	⊢ (𝜑 → dom 𝑆 = 𝐼)
dprdff.3	⊢ (𝜑 → 𝐹 ∈ 𝑊)
dprdfcntz.z	⊢ 𝑍 = (Cntz‘𝐺)

Assertion

Ref	Expression
dprdfcntz	⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))

Distinct variable groups:   ℎ,𝐹   ℎ,𝑖,𝐼   0 ,ℎ   𝑆,ℎ,𝑖

Allowed substitution hints:   𝜑(ℎ,𝑖)   𝐹(𝑖)   𝐺(ℎ,𝑖)   𝑊(ℎ,𝑖)   0 (𝑖)   𝑍(ℎ,𝑖)

Proof of Theorem dprdfcntz

Dummy variables 𝑦 𝑧 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dprdff.w	. . . . 5 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
2		dprdff.1	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
3		dprdff.2	. . . . 5 ⊢ (𝜑 → dom 𝑆 = 𝐼)
4		dprdff.3	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
5		eqid 2729	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	1, 2, 3, 4, 5	dprdff 19893	. . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺))
7	6	ffnd 6653	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐼)
8	6	ffvelcdmda 7018	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (Base‘𝐺))
9		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧)
10	9	fveq2d 6826	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → (𝐹‘𝑦) = (𝐹‘𝑧))
11	9	equcomd 2019	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → 𝑧 = 𝑦)
12	11	fveq2d 6826	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → (𝐹‘𝑧) = (𝐹‘𝑦))
13	10, 12	oveq12d 7367	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
14	2	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → 𝐺dom DProd 𝑆)
15	3	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → dom 𝑆 = 𝐼)
16		simpllr 775	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → 𝑦 ∈ 𝐼)
17		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → 𝑧 ∈ 𝐼)
18		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → 𝑦 ≠ 𝑧)
19		dprdfcntz.z	. . . . . . . . . . 11 ⊢ 𝑍 = (Cntz‘𝐺)
20	14, 15, 16, 17, 18, 19	dprdcntz 19889	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑧)))
21	1, 2, 3, 4	dprdfcl 19894	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
22	21	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
23	20, 22	sseldd 3936	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝐹‘𝑦) ∈ (𝑍‘(𝑆‘𝑧)))
24	1, 2, 3, 4	dprdfcl 19894	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) ∈ (𝑆‘𝑧))
25	24	ad4ant13 751	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝐹‘𝑧) ∈ (𝑆‘𝑧))
26		eqid 2729	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
27	26, 19	cntzi 19208	. . . . . . . . 9 ⊢ (((𝐹‘𝑦) ∈ (𝑍‘(𝑆‘𝑧)) ∧ (𝐹‘𝑧) ∈ (𝑆‘𝑧)) → ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
28	23, 25, 27	syl2anc 584	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
29	13, 28	pm2.61dane 3012	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
30	29	ralrimiva 3121	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ∀𝑧 ∈ 𝐼 ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
31	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝐹 Fn 𝐼)
32		oveq2 7357	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝐹‘𝑦)(+g‘𝐺)𝑥) = ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)))
33		oveq1 7356	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥(+g‘𝐺)(𝐹‘𝑦)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
34	32, 33	eqeq12d 2745	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑧) → (((𝐹‘𝑦)(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)(𝐹‘𝑦)) ↔ ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦))))
35	34	ralrn 7022	. . . . . . 7 ⊢ (𝐹 Fn 𝐼 → (∀𝑥 ∈ ran 𝐹((𝐹‘𝑦)(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)(𝐹‘𝑦)) ↔ ∀𝑧 ∈ 𝐼 ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦))))
36	31, 35	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (∀𝑥 ∈ ran 𝐹((𝐹‘𝑦)(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)(𝐹‘𝑦)) ↔ ∀𝑧 ∈ 𝐼 ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦))))
37	30, 36	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ∀𝑥 ∈ ran 𝐹((𝐹‘𝑦)(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)(𝐹‘𝑦)))
38	6	frnd 6660	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
39	38	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ran 𝐹 ⊆ (Base‘𝐺))
40	5, 26, 19	elcntz 19201	. . . . . 6 ⊢ (ran 𝐹 ⊆ (Base‘𝐺) → ((𝐹‘𝑦) ∈ (𝑍‘ran 𝐹) ↔ ((𝐹‘𝑦) ∈ (Base‘𝐺) ∧ ∀𝑥 ∈ ran 𝐹((𝐹‘𝑦)(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)(𝐹‘𝑦)))))
41	39, 40	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((𝐹‘𝑦) ∈ (𝑍‘ran 𝐹) ↔ ((𝐹‘𝑦) ∈ (Base‘𝐺) ∧ ∀𝑥 ∈ ran 𝐹((𝐹‘𝑦)(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)(𝐹‘𝑦)))))
42	8, 37, 41	mpbir2and 713	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (𝑍‘ran 𝐹))
43	42	ralrimiva 3121	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐼 (𝐹‘𝑦) ∈ (𝑍‘ran 𝐹))
44		ffnfv 7053	. . 3 ⊢ (𝐹:𝐼⟶(𝑍‘ran 𝐹) ↔ (𝐹 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝐹‘𝑦) ∈ (𝑍‘ran 𝐹)))
45	7, 43, 44	sylanbrc 583	. 2 ⊢ (𝜑 → 𝐹:𝐼⟶(𝑍‘ran 𝐹))
46	45	frnd 6660	1 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3394   ⊆ wss 3903   class class class wbr 5092  dom cdm 5619  ran crn 5620   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  Xcixp 8824   finSupp cfsupp 9251  Basecbs 17120  +_gcplusg 17161  Cntzccntz 19194   DProd cdprd 19874

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-ixp 8825  df-subg 19002  df-cntz 19196  df-dprd 19876

This theorem is referenced by:  dprdssv  19897  dprdfinv  19900  dprdfadd  19901  dprdfeq0  19903  dprdlub  19907  dmdprdsplitlem  19918  dpjidcl  19939