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Theorem dprd2dlem2 19643

Description: The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)

**Hypotheses**
Ref	Expression
dprd2d.1	⊢ (𝜑 → Rel 𝐴)
dprd2d.2	⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
dprd2d.3	⊢ (𝜑 → dom 𝐴 ⊆ 𝐼)
dprd2d.4	⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
dprd2d.5	⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
dprd2d.k	⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))

**Assertion**
Ref	Expression
dprd2dlem2	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))))

Distinct variable groups: 𝑖,𝑗,𝐴 𝑖,𝐺,𝑗 𝑖,𝐼 𝑖,𝐾 𝜑,𝑖,𝑗 𝑆,𝑖,𝑗 𝑖,𝑋,𝑗 Allowed substitution hints: 𝐼(𝑗) 𝐾(𝑗)

**Proof of Theorem dprd2dlem2**
Step	Hyp	Ref	Expression
1		df-ov 7278	. . 3 ⊢ ((1^st ‘𝑋)𝑆(2^nd ‘𝑋)) = (𝑆‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
2		dprd2d.1	. . . . . . . 8 ⊢ (𝜑 → Rel 𝐴)
3		1st2nd 7880	. . . . . . . 8 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
4	2, 3	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
5		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
6	4, 5	eqeltrrd 2840	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴)
7		df-br 5075	. . . . . 6 ⊢ ((1^st ‘𝑋)𝐴(2^nd ‘𝑋) ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴)
8	6, 7	sylibr 233	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1^st ‘𝑋)𝐴(2^nd ‘𝑋))
9	2	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → Rel 𝐴)
10		elrelimasn 5993	. . . . . 6 ⊢ (Rel 𝐴 → ((2^nd ‘𝑋) ∈ (𝐴 “ {(1^st ‘𝑋)}) ↔ (1^st ‘𝑋)𝐴(2^nd ‘𝑋)))
11	9, 10	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((2^nd ‘𝑋) ∈ (𝐴 “ {(1^st ‘𝑋)}) ↔ (1^st ‘𝑋)𝐴(2^nd ‘𝑋)))
12	8, 11	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (2^nd ‘𝑋) ∈ (𝐴 “ {(1^st ‘𝑋)}))
13		oveq2 7283	. . . . 5 ⊢ (𝑗 = (2^nd ‘𝑋) → ((1^st ‘𝑋)𝑆𝑗) = ((1^st ‘𝑋)𝑆(2^nd ‘𝑋)))
14		eqid 2738	. . . . 5 ⊢ (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))
15		ovex 7308	. . . . 5 ⊢ ((1^st ‘𝑋)𝑆𝑗) ∈ V
16	13, 14, 15	fvmpt3i 6880	. . . 4 ⊢ ((2^nd ‘𝑋) ∈ (𝐴 “ {(1^st ‘𝑋)}) → ((𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))‘(2^nd ‘𝑋)) = ((1^st ‘𝑋)𝑆(2^nd ‘𝑋)))
17	12, 16	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))‘(2^nd ‘𝑋)) = ((1^st ‘𝑋)𝑆(2^nd ‘𝑋)))
18	4	fveq2d 6778	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) = (𝑆‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
19	1, 17, 18	3eqtr4a 2804	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))‘(2^nd ‘𝑋)) = (𝑆‘𝑋))
20		sneq 4571	. . . . . . 7 ⊢ (𝑖 = (1^st ‘𝑋) → {𝑖} = {(1^st ‘𝑋)})
21	20	imaeq2d 5969	. . . . . 6 ⊢ (𝑖 = (1^st ‘𝑋) → (𝐴 “ {𝑖}) = (𝐴 “ {(1^st ‘𝑋)}))
22		oveq1 7282	. . . . . 6 ⊢ (𝑖 = (1^st ‘𝑋) → (𝑖𝑆𝑗) = ((1^st ‘𝑋)𝑆𝑗))
23	21, 22	mpteq12dv 5165	. . . . 5 ⊢ (𝑖 = (1^st ‘𝑋) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)))
24	23	breq2d 5086	. . . 4 ⊢ (𝑖 = (1^st ‘𝑋) → (𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ↔ 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))))
25		dprd2d.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
26	25	ralrimiva 3103	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
27	26	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
28		dprd2d.3	. . . . . 6 ⊢ (𝜑 → dom 𝐴 ⊆ 𝐼)
29	28	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → dom 𝐴 ⊆ 𝐼)
30		1stdm 7881	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝐴) → (1^st ‘𝑋) ∈ dom 𝐴)
31	2, 30	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1^st ‘𝑋) ∈ dom 𝐴)
32	29, 31	sseldd 3922	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1^st ‘𝑋) ∈ 𝐼)
33	24, 27, 32	rspcdva 3562	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)))
34	15, 14	dmmpti 6577	. . . 4 ⊢ dom (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)) = (𝐴 “ {(1^st ‘𝑋)})
35	34	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → dom (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)) = (𝐴 “ {(1^st ‘𝑋)}))
36	33, 35, 12	dprdub 19628	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))‘(2^nd ‘𝑋)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))))
37	19, 36	eqsstrrd 3960	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ⊆ wss 3887 {csn 4561 ⟨cop 4567 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 “ cima 5592 Rel wrel 5594 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 mrClscmrc 17292 SubGrpcsubg 18749 DProd cdprd 19596

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-0g 17152 df-gsum 17153 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-mulg 18701 df-subg 18752 df-cntz 18923 df-cmn 19388 df-dprd 19598

This theorem is referenced by: dprd2dlem1 19644 dprd2da 19645

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