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Theorem dprd2dlem2 20008

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 7402	. . 3 ⊢ ((1^st ‘𝑋)𝑆(2^nd ‘𝑋)) = (𝑆‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
2		dprd2d.1	. . . . . . . 8 ⊢ (𝜑 → Rel 𝐴)
3		1st2nd 8032	. . . . . . . 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 2834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴)
7		df-br 5117	. . . . . 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 6070	. . . . . 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 7407	. . . . 5 ⊢ (𝑗 = (2^nd ‘𝑋) → ((1^st ‘𝑋)𝑆𝑗) = ((1^st ‘𝑋)𝑆(2^nd ‘𝑋)))
14		eqid 2734	. . . . 5 ⊢ (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))
15		ovex 7432	. . . . 5 ⊢ ((1^st ‘𝑋)𝑆𝑗) ∈ V
16	13, 14, 15	fvmpt3i 6987	. . . 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 6876	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) = (𝑆‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
19	1, 17, 18	3eqtr4a 2795	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))‘(2^nd ‘𝑋)) = (𝑆‘𝑋))
20		sneq 4609	. . . . . . 7 ⊢ (𝑖 = (1^st ‘𝑋) → {𝑖} = {(1^st ‘𝑋)})
21	20	imaeq2d 6044	. . . . . 6 ⊢ (𝑖 = (1^st ‘𝑋) → (𝐴 “ {𝑖}) = (𝐴 “ {(1^st ‘𝑋)}))
22		oveq1 7406	. . . . . 6 ⊢ (𝑖 = (1^st ‘𝑋) → (𝑖𝑆𝑗) = ((1^st ‘𝑋)𝑆𝑗))
23	21, 22	mpteq12dv 5204	. . . . 5 ⊢ (𝑖 = (1^st ‘𝑋) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)))
24	23	breq2d 5128	. . . 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 3130	. . . . 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 8033	. . . . . 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 3957	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1^st ‘𝑋) ∈ 𝐼)
33	24, 27, 32	rspcdva 3600	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)))
34	15, 14	dmmpti 6678	. . . 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 19993	. 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 3992	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 1539 ∈ wcel 2107 ∀wral 3050 ⊆ wss 3924 {csn 4599 ⟨cop 4605 class class class wbr 5116 ↦ cmpt 5198 dom cdm 5651 “ cima 5654 Rel wrel 5656 ⟶wf 6523 ‘cfv 6527 (class class class)co 7399 1^st c1st 7980 2^nd c2nd 7981 mrClscmrc 17580 SubGrpcsubg 19088 DProd cdprd 19961

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5246 ax-sep 5263 ax-nul 5273 ax-pow 5332 ax-pr 5399 ax-un 7723 ax-cnex 11177 ax-resscn 11178 ax-1cn 11179 ax-icn 11180 ax-addcl 11181 ax-addrcl 11182 ax-mulcl 11183 ax-mulrcl 11184 ax-mulcom 11185 ax-addass 11186 ax-mulass 11187 ax-distr 11188 ax-i2m1 11189 ax-1ne0 11190 ax-1rid 11191 ax-rnegex 11192 ax-rrecex 11193 ax-cnre 11194 ax-pre-lttri 11195 ax-pre-lttrn 11196 ax-pre-ltadd 11197 ax-pre-mulgt0 11198

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3357 df-reu 3358 df-rab 3414 df-v 3459 df-sbc 3764 df-csb 3873 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-pss 3944 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-op 4606 df-uni 4881 df-int 4920 df-iun 4966 df-iin 4967 df-br 5117 df-opab 5179 df-mpt 5199 df-tr 5227 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 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-pred 6287 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6480 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-isom 6536 df-riota 7356 df-ov 7402 df-oprab 7403 df-mpo 7404 df-om 7856 df-1st 7982 df-2nd 7983 df-supp 8154 df-frecs 8274 df-wrecs 8305 df-recs 8379 df-rdg 8418 df-1o 8474 df-2o 8475 df-er 8713 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9368 df-oi 9516 df-card 9945 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11460 df-neg 11461 df-nn 12233 df-2 12295 df-n0 12494 df-z 12581 df-uz 12845 df-fz 13514 df-fzo 13661 df-seq 14009 df-hash 14337 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17214 df-ress 17237 df-plusg 17269 df-0g 17440 df-gsum 17441 df-mre 17583 df-mrc 17584 df-acs 17586 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-grp 18904 df-mulg 19036 df-subg 19091 df-cntz 19285 df-cmn 19748 df-dprd 19963

This theorem is referenced by: dprd2dlem1 20009 dprd2da 20010

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