Theorem dprdfcntz 19979

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 2728	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	1, 2, 3, 4, 5	dprdff 19976	. . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺))
7	6	ffnd 6728	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐼)
8	6	ffvelcdmda 7099	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (Base‘𝐺))
9		simpr 483	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧)
10	9	fveq2d 6906	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → (𝐹‘𝑦) = (𝐹‘𝑧))
11	9	equcomd 2014	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → 𝑧 = 𝑦)
12	11	fveq2d 6906	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → (𝐹‘𝑧) = (𝐹‘𝑦))
13	10, 12	oveq12d 7444	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
14	2	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → 𝐺dom DProd 𝑆)
15	3	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → dom 𝑆 = 𝐼)
16		simpllr 774	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → 𝑦 ∈ 𝐼)
17		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → 𝑧 ∈ 𝐼)
18		simpr 483	. . . . . . . . . . 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 19972	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑧)))
21	1, 2, 3, 4	dprdfcl 19977	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
22	21	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
23	20, 22	sseldd 3983	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝐹‘𝑦) ∈ (𝑍‘(𝑆‘𝑧)))
24	1, 2, 3, 4	dprdfcl 19977	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) ∈ (𝑆‘𝑧))
25	24	ad4ant13 749	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝐹‘𝑧) ∈ (𝑆‘𝑧))
26		eqid 2728	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
27	26, 19	cntzi 19287	. . . . . . . . 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 582	. . . . . . . 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 3026	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
30	29	ralrimiva 3143	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ∀𝑧 ∈ 𝐼 ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
31	7	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝐹 Fn 𝐼)
32		oveq2 7434	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝐹‘𝑦)(+g‘𝐺)𝑥) = ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)))
33		oveq1 7433	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥(+g‘𝐺)(𝐹‘𝑦)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
34	32, 33	eqeq12d 2744	. . . . . . . 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 7103	. . . . . . 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 256	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ∀𝑥 ∈ ran 𝐹((𝐹‘𝑦)(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)(𝐹‘𝑦)))
38	6	frnd 6735	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
39	38	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ran 𝐹 ⊆ (Base‘𝐺))
40	5, 26, 19	elcntz 19280	. . . . . 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 711	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (𝑍‘ran 𝐹))
43	42	ralrimiva 3143	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐼 (𝐹‘𝑦) ∈ (𝑍‘ran 𝐹))
44		ffnfv 7134	. . 3 ⊢ (𝐹:𝐼⟶(𝑍‘ran 𝐹) ↔ (𝐹 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝐹‘𝑦) ∈ (𝑍‘ran 𝐹)))
45	7, 43, 44	sylanbrc 581	. 2 ⊢ (𝜑 → 𝐹:𝐼⟶(𝑍‘ran 𝐹))
46	45	frnd 6735	1 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2937  ∀wral 3058  {crab 3430   ⊆ wss 3949   class class class wbr 5152  dom cdm 5682  ran crn 5683   Fn wfn 6548  ⟶wf 6549  ‘cfv 6553  (class class class)co 7426  Xcixp 8922   finSupp cfsupp 9393  Basecbs 17187  +_gcplusg 17240  Cntzccntz 19273   DProd cdprd 19957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-ixp 8923  df-subg 19085  df-cntz 19275  df-dprd 19959

This theorem is referenced by:  dprdssv  19980  dprdfinv  19983  dprdfadd  19984  dprdfeq0  19986  dprdlub  19990  dmdprdsplitlem  20001  dpjidcl  20022

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