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Theorem dprd2dlem2 20028

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 7413	. . 3 ⊢ ((1^st ‘𝑋)𝑆(2^nd ‘𝑋)) = (𝑆‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
2		dprd2d.1	. . . . . . . 8 ⊢ (𝜑 → Rel 𝐴)
3		1st2nd 8043	. . . . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
6	4, 5	eqeltrrd 2836	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴)
7		df-br 5125	. . . . . 6 ⊢ ((1^st ‘𝑋)𝐴(2^nd ‘𝑋) ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴)
8	6, 7	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1^st ‘𝑋)𝐴(2^nd ‘𝑋))
9	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → Rel 𝐴)
10		elrelimasn 6078	. . . . . 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 257	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (2^nd ‘𝑋) ∈ (𝐴 “ {(1^st ‘𝑋)}))
13		oveq2 7418	. . . . 5 ⊢ (𝑗 = (2^nd ‘𝑋) → ((1^st ‘𝑋)𝑆𝑗) = ((1^st ‘𝑋)𝑆(2^nd ‘𝑋)))
14		eqid 2736	. . . . 5 ⊢ (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))
15		ovex 7443	. . . . 5 ⊢ ((1^st ‘𝑋)𝑆𝑗) ∈ V
16	13, 14, 15	fvmpt3i 6996	. . . 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 6885	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) = (𝑆‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
19	1, 17, 18	3eqtr4a 2797	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))‘(2^nd ‘𝑋)) = (𝑆‘𝑋))
20		sneq 4616	. . . . . . 7 ⊢ (𝑖 = (1^st ‘𝑋) → {𝑖} = {(1^st ‘𝑋)})
21	20	imaeq2d 6052	. . . . . 6 ⊢ (𝑖 = (1^st ‘𝑋) → (𝐴 “ {𝑖}) = (𝐴 “ {(1^st ‘𝑋)}))
22		oveq1 7417	. . . . . 6 ⊢ (𝑖 = (1^st ‘𝑋) → (𝑖𝑆𝑗) = ((1^st ‘𝑋)𝑆𝑗))
23	21, 22	mpteq12dv 5212	. . . . 5 ⊢ (𝑖 = (1^st ‘𝑋) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)))
24	23	breq2d 5136	. . . 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 3133	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
27	26	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
28		dprd2d.3	. . . . . 6 ⊢ (𝜑 → dom 𝐴 ⊆ 𝐼)
29	28	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → dom 𝐴 ⊆ 𝐼)
30		1stdm 8044	. . . . . 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 3964	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1^st ‘𝑋) ∈ 𝐼)
33	24, 27, 32	rspcdva 3607	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)))
34	15, 14	dmmpti 6687	. . . 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 20013	. 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 3999	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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3052 ⊆ wss 3931 {csn 4606 ⟨cop 4612 class class class wbr 5124 ↦ cmpt 5206 dom cdm 5659 “ cima 5662 Rel wrel 5664 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 1^st c1st 7991 2^nd c2nd 7992 mrClscmrc 17600 SubGrpcsubg 19108 DProd cdprd 19981

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-n0 12507 df-z 12594 df-uz 12858 df-fz 13530 df-fzo 13677 df-seq 14025 df-hash 14354 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-0g 17460 df-gsum 17461 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-grp 18924 df-mulg 19056 df-subg 19111 df-cntz 19305 df-cmn 19768 df-dprd 19983

This theorem is referenced by: dprd2dlem1 20029 dprd2da 20030

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