Theorem dprdfcntz 20075

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 2765	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	1, 2, 3, 4, 5	dprdff 20072	. . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺))
7	6	ffnd 6696	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐼)
8	6	ffvelcdmda 7069	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (Base‘𝐺))
9		simpr 489	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧)
10	9	fveq2d 6875	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → (𝐹‘𝑦) = (𝐹‘𝑧))
11	9	equcomd 2042	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → 𝑧 = 𝑦)
12	11	fveq2d 6875	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → (𝐹‘𝑧) = (𝐹‘𝑦))
13	10, 12	oveq12d 7418	. . . . . . . 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 20068	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑧)))
21	1, 2, 3, 4	dprdfcl 20073	. . . . . . . . . . 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 3940	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝐹‘𝑦) ∈ (𝑍‘(𝑆‘𝑧)))
24	1, 2, 3, 4	dprdfcl 20073	. . . . . . . . . 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 2765	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
27	26, 19	cntzi 19387	. . . . . . . . 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 3047	. . . . . . 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 7408	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝐹‘𝑦)(+g‘𝐺)𝑥) = ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)))
33		oveq1 7407	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥(+g‘𝐺)(𝐹‘𝑦)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
34	32, 33	eqeq12d 2781	. . . . . . . 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 7073	. . . . . . 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 6704	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
39	38	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ran 𝐹 ⊆ (Base‘𝐺))
40	5, 26, 19	elcntz 19380	. . . . . 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 7104	. . 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 6704	1 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  {crab 3417   ⊆ wss 3907   class class class wbr 5104  dom cdm 5651  ran crn 5652   Fn wfn 6520  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  Xcixp 8883   finSupp cfsupp 9309  Basecbs 17257  +_gcplusg 17298  Cntzccntz 19373   DProd cdprd 20053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-ixp 8884  df-subg 19177  df-cntz 19375  df-dprd 20055

This theorem is referenced by:  dprdssv  20076  dprdfinv  20079  dprdfadd  20080  dprdfeq0  20082  dprdlub  20086  dmdprdsplitlem  20097  dpjidcl  20118