Theorem dprdfcntz 19954

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 2730	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	1, 2, 3, 4, 5	dprdff 19951	. . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺))
7	6	ffnd 6692	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐼)
8	6	ffvelcdmda 7059	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (Base‘𝐺))
9		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧)
10	9	fveq2d 6865	. . . . . . . . 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 6865	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 = 𝑧) → (𝐹‘𝑧) = (𝐹‘𝑦))
13	10, 12	oveq12d 7408	. . . . . . . 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 19947	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑧)))
21	1, 2, 3, 4	dprdfcl 19952	. . . . . . . . . . 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 3950	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) ∧ 𝑦 ≠ 𝑧) → (𝐹‘𝑦) ∈ (𝑍‘(𝑆‘𝑧)))
24	1, 2, 3, 4	dprdfcl 19952	. . . . . . . . . 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 2730	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
27	26, 19	cntzi 19268	. . . . . . . . 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 3013	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
30	29	ralrimiva 3126	. . . . . 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 7398	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝐹‘𝑦)(+g‘𝐺)𝑥) = ((𝐹‘𝑦)(+g‘𝐺)(𝐹‘𝑧)))
33		oveq1 7397	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥(+g‘𝐺)(𝐹‘𝑦)) = ((𝐹‘𝑧)(+g‘𝐺)(𝐹‘𝑦)))
34	32, 33	eqeq12d 2746	. . . . . . . 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 7063	. . . . . . 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 6699	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
39	38	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ran 𝐹 ⊆ (Base‘𝐺))
40	5, 26, 19	elcntz 19261	. . . . . 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 3126	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐼 (𝐹‘𝑦) ∈ (𝑍‘ran 𝐹))
44		ffnfv 7094	. . 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 6699	1 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  {crab 3408   ⊆ wss 3917   class class class wbr 5110  dom cdm 5641  ran crn 5642   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Xcixp 8873   finSupp cfsupp 9319  Basecbs 17186  +_gcplusg 17227  Cntzccntz 19254   DProd cdprd 19932

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-ixp 8874  df-subg 19062  df-cntz 19256  df-dprd 19934

This theorem is referenced by:  dprdssv  19955  dprdfinv  19958  dprdfadd  19959  dprdfeq0  19961  dprdlub  19965  dmdprdsplitlem  19976  dpjidcl  19997