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Theorem dprd2dlem2 20262

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 7424	. . 3 ⊢ ((1^st ‘𝑋)𝑆(2^nd ‘𝑋)) = (𝑆‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
2		dprd2d.1	. . . . . . . 8 ⊢ (𝜑 → Rel 𝐴)
3		1st2nd 8051	. . . . . . . 8 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
4	2, 3	sylan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
5		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
6	4, 5	eqeltrrd 2861	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴)
7		df-br 5103	. . . . . 6 ⊢ ((1^st ‘𝑋)𝐴(2^nd ‘𝑋) ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴)
8	6, 7	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1^st ‘𝑋)𝐴(2^nd ‘𝑋))
9	2	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → Rel 𝐴)
10		elrelimasn 6083	. . . . . 6 ⊢ (Rel 𝐴 → ((2^nd ‘𝑋) ∈ (𝐴 “ {(1^st ‘𝑋)}) ↔ (1^st ‘𝑋)𝐴(2^nd ‘𝑋)))
11	9, 10	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((2^nd ‘𝑋) ∈ (𝐴 “ {(1^st ‘𝑋)}) ↔ (1^st ‘𝑋)𝐴(2^nd ‘𝑋)))
12	8, 11	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (2^nd ‘𝑋) ∈ (𝐴 “ {(1^st ‘𝑋)}))
13		oveq2 7429	. . . . 5 ⊢ (𝑗 = (2^nd ‘𝑋) → ((1^st ‘𝑋)𝑆𝑗) = ((1^st ‘𝑋)𝑆(2^nd ‘𝑋)))
14		eqid 2760	. . . . 5 ⊢ (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))
15		ovex 7454	. . . . 5 ⊢ ((1^st ‘𝑋)𝑆𝑗) ∈ V
16	13, 14, 15	fvmpt3i 7000	. . . 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 18	. . 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 6890	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) = (𝑆‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
19	1, 17, 18	3eqtr4a 2821	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))‘(2^nd ‘𝑋)) = (𝑆‘𝑋))
20		sneq 4590	. . . . . . 7 ⊢ (𝑖 = (1^st ‘𝑋) → {𝑖} = {(1^st ‘𝑋)})
21	20	imaeq2d 6051	. . . . . 6 ⊢ (𝑖 = (1^st ‘𝑋) → (𝐴 “ {𝑖}) = (𝐴 “ {(1^st ‘𝑋)}))
22		oveq1 7428	. . . . . 6 ⊢ (𝑖 = (1^st ‘𝑋) → (𝑖𝑆𝑗) = ((1^st ‘𝑋)𝑆𝑗))
23	21, 22	mpteq12dv 5191	. . . . 5 ⊢ (𝑖 = (1^st ‘𝑋) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)))
24	23	breq2d 5114	. . . 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 3154	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
27	26	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
28		dprd2d.3	. . . . . 6 ⊢ (𝜑 → dom 𝐴 ⊆ 𝐼)
29	28	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → dom 𝐴 ⊆ 𝐼)
30		1stdm 8052	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝐴) → (1^st ‘𝑋) ∈ dom 𝐴)
31	2, 30	sylan 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1^st ‘𝑋) ∈ dom 𝐴)
32	29, 31	sseldd 3909	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1^st ‘𝑋) ∈ 𝐼)
33	24, 27, 32	rspcdva 3577	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)))
34	15, 14	dmmpti 6684	. . . 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 20247	. 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 3943	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ∀wral 3076 ⊆ wss 3895 {csn 4584 ⟨cop 4834 class class class wbr 5102 ↦ cmpt 5185 dom cdm 5650 “ cima 5653 Rel wrel 5655 ⟶wf 6535 ‘cfv 6539 (class class class)co 7421 1^st c1st 8000 2^nd c2nd 8001 mrClscmrc 17759 SubGrpcsubg 19336 DProd cdprd 20215

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8181 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fsupp 9358 df-oi 9508 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-nn 12340 df-2 12409 df-n0 12611 df-z 12698 df-uz 12970 df-fz 13644 df-fzo 13793 df-seq 14149 df-hash 14479 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-0g 17618 df-gsum 17619 df-mre 17762 df-mrc 17763 df-acs 17765 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-grp 19153 df-mulg 19284 df-subg 19339 df-cntz 19537 df-cmn 20002 df-dprd 20217

This theorem is used by: dprd2dlem1 20263 dprd2da 20264

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