Theorem dprdfcntz 20091

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 2763	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	1, 2, 3, 4, 5	dprdff 20088	. . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺))
7	6	ffnd 6706	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐼)
8	6	ffvelcdmda 7079	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (Base‘𝐺))
9		simpr 489	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧)
10	9	fveq2d 6885	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → (𝐹‘𝑦) = (𝐹‘𝑧))
11	9	equcomd 2049	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → 𝑧 = 𝑦)
12	11	fveq2d 6885	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → (𝐹‘𝑧) = (𝐹‘𝑦))
13	10, 12	oveq12d 7428	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
14	2	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → 𝐺dom DProd 𝑆)
15	3	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → dom 𝑆 = 𝐼)
16		simpllr 787	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → 𝑦 ∈ 𝐼)
17		simplr 780	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → 𝑧 ∈ 𝐼)
18		simpr 489	. . . . . . . . . . 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 20084	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑧)))
21	1, 2, 3, 4	dprdfcl 20089	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
22	21	ad2antrr 738	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
23	20, 22	sseldd 3938	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝐹‘𝑦) ∈ (𝑍‘(𝑆‘𝑧)))
24	1, 2, 3, 4	dprdfcl 20089	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) ∈ (𝑆‘𝑧))
25	24	ad4ant13 763	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝐹‘𝑧) ∈ (𝑆‘𝑧))
26		eqid 2763	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
27	26, 19	cntzi 19403	. . . . . . . . 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 595	. . . . . . . 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 3045	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
30	29	ralrimiva 3157	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ∀𝑧 ∈ 𝐼 ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
31	7	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝐹 Fn 𝐼)
32		oveq2 7418	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝐹‘𝑦)(+g‘𝐺)𝑥) = ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)))
33		oveq1 7417	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥(+g‘𝐺)(𝐹‘𝑦)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
34	32, 33	eqeq12d 2779	. . . . . . . 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 7083	. . . . . . 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 18	. . . . . 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 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ∀𝑥 ∈ ran 𝐹((𝐹‘𝑦)(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)(𝐹‘𝑦)))
38	6	frnd 6714	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
39	38	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ran 𝐹 ⊆ (Base‘𝐺))
40	5, 26, 19	elcntz 19396	. . . . . 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 18	. . . . 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 725	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (𝑍‘ran 𝐹))
43	42	ralrimiva 3157	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐼 (𝐹‘𝑦) ∈ (𝑍‘ran 𝐹))
44		ffnfv 7114	. . 3 ⊢ (𝐹:𝐼⟶(𝑍‘ran 𝐹) ↔ (𝐹 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝐹‘𝑦) ∈ (𝑍‘ran 𝐹)))
45	7, 43, 44	sylanbrc 594	. 2 ⊢ (𝜑 → 𝐹:𝐼⟶(𝑍‘ran 𝐹))
46	45	frnd 6714	1 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  {crab 3416   ⊆ wss 3905   class class class wbr 5109  dom cdm 5661  ran crn 5662   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Xcixp 8891   finSupp cfsupp 9317  Basecbs 17273  +_gcplusg 17314  Cntzccntz 19389   DProd cdprd 20069

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-ixp 8892  df-subg 19193  df-cntz 19391  df-dprd 20071

This theorem is used by:  dprdssv  20092  dprdfinv  20095  dprdfadd  20096  dprdfeq0  20098  dprdlub  20102  dmdprdsplitlem  20113  dpjidcl  20134