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Theorem dprdssv 18843

Description: The internal direct product of a family of subgroups is a subset of the base. (Contributed by Mario Carneiro, 25-Apr-2016.)

**Hypothesis**
Ref	Expression
dprdssv.b	⊢ 𝐵 = (Base‘𝐺)

**Assertion**
Ref	Expression
dprdssv	⊢ (𝐺 DProd 𝑆) ⊆ 𝐵

Proof of Theorem dprdssv Dummy variables 𝑥 𝑓 ℎ 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2793	. . . 4 ⊢ dom 𝑆 = dom 𝑆
2		eqid 2793	. . . . 5 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
3		eqid 2793	. . . . 5 ⊢ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)} = {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}
4	2, 3	eldprd 18831	. . . 4 ⊢ (dom 𝑆 = dom 𝑆 → (𝑥 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}𝑥 = (𝐺 Σ_g 𝑓))))
5	1, 4	ax-mp 5	. . 3 ⊢ (𝑥 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}𝑥 = (𝐺 Σ_g 𝑓)))
6		dprdssv.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
7		eqid 2793	. . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
8		dprdgrp 18832	. . . . . . . . 9 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
9		grpmnd 17856	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
10	8, 9	syl 17	. . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Mnd)
11	10	adantr 481	. . . . . . 7 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝐺 ∈ Mnd)
12		reldmdprd 18824	. . . . . . . . . 10 ⊢ Rel dom DProd
13	12	brrelex2i 5487	. . . . . . . . 9 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V)
14	13	dmexd 7462	. . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V)
15	14	adantr 481	. . . . . . 7 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → dom 𝑆 ∈ V)
16		simpl 483	. . . . . . . 8 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝐺dom DProd 𝑆)
17		eqidd 2794	. . . . . . . 8 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → dom 𝑆 = dom 𝑆)
18		simpr 485	. . . . . . . 8 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)})
19	3, 16, 17, 18, 6	dprdff 18839	. . . . . . 7 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑓:dom 𝑆⟶𝐵)
20	3, 16, 17, 18, 7	dprdfcntz 18842	. . . . . . 7 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → ran 𝑓 ⊆ ((Cntz‘𝐺)‘ran 𝑓))
21	3, 16, 17, 18	dprdffsupp 18841	. . . . . . 7 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑓 finSupp (0_g‘𝐺))
22	6, 2, 7, 11, 15, 19, 20, 21	gsumzcl 18740	. . . . . 6 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → (𝐺 Σ_g 𝑓) ∈ 𝐵)
23		eleq1 2868	. . . . . 6 ⊢ (𝑥 = (𝐺 Σ_g 𝑓) → (𝑥 ∈ 𝐵 ↔ (𝐺 Σ_g 𝑓) ∈ 𝐵))
24	22, 23	syl5ibrcom 248	. . . . 5 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → (𝑥 = (𝐺 Σ_g 𝑓) → 𝑥 ∈ 𝐵))
25	24	rexlimdva 3244	. . . 4 ⊢ (𝐺dom DProd 𝑆 → (∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}𝑥 = (𝐺 Σ_g 𝑓) → 𝑥 ∈ 𝐵))
26	25	imp 407	. . 3 ⊢ ((𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}𝑥 = (𝐺 Σ_g 𝑓)) → 𝑥 ∈ 𝐵)
27	5, 26	sylbi 218	. 2 ⊢ (𝑥 ∈ (𝐺 DProd 𝑆) → 𝑥 ∈ 𝐵)
28	27	ssriv 3888	1 ⊢ (𝐺 DProd 𝑆) ⊆ 𝐵

Colors of variables: wff setvar class

Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ∃wrex 3104 {crab 3107 Vcvv 3432 ⊆ wss 3854 class class class wbr 4956 dom cdm 5435 ‘cfv 6217 (class class class)co 7007 Xcixp 8300 finSupp cfsupp 8669 Basecbs 16300 0_gc0g 16530 Σ_g cgsu 16531 Mndcmnd 17721 Grpcgrp 17849 Cntzccntz 18174 DProd cdprd 18820

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-supp 7673 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-ixp 8301 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-fsupp 8670 df-oi 8810 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-n0 11735 df-z 11819 df-uz 12083 df-fz 12732 df-fzo 12873 df-seq 13208 df-hash 13529 df-0g 16532 df-gsum 16533 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-grp 17852 df-subg 18018 df-cntz 18176 df-dprd 18822

This theorem is referenced by: dprdfsub 18848 dprdf11 18850 dprdsubg 18851 dprdspan 18854 dprdcntz2 18865 dprd2da 18869 dmdprdsplit2lem 18872 ablfac1c 18898 ablfac1eulem 18899 ablfac1eu 18900

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