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Theorem dprdlub 19150

Description: The direct product is smaller than any subgroup which contains the factors. (Contributed by Mario Carneiro, 25-Apr-2016.)

**Hypotheses**
Ref	Expression
dprdlub.1	⊢ (𝜑 → 𝐺dom DProd 𝑆)
dprdlub.2	⊢ (𝜑 → dom 𝑆 = 𝐼)
dprdlub.3	⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
dprdlub.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑆‘𝑘) ⊆ 𝑇)

**Assertion**
Ref	Expression
dprdlub	⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝑇)

Distinct variable groups: 𝑘,𝐺 𝑘,𝐼 𝜑,𝑘 𝑆,𝑘 𝑇,𝑘

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

Step	Hyp	Ref	Expression
1		dprdlub.1	. . 3 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
2		dprdlub.2	. . 3 ⊢ (𝜑 → dom 𝑆 = 𝐼)
3		eqid 2823	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
4		eqid 2823	. . . 4 ⊢ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)} = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}
5	3, 4	dprdval 19127	. . 3 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)} ↦ (𝐺 Σ_g 𝑓)))
6	1, 2, 5	syl2anc 586	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)} ↦ (𝐺 Σ_g 𝑓)))
7		eqid 2823	. . . . 5 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
8	1	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝐺dom DProd 𝑆)
9		dprdgrp 19129	. . . . . 6 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
10		grpmnd 18112	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
11	8, 9, 10	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝐺 ∈ Mnd)
12	1, 2	dprddomcld 19125	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
13	12	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝐼 ∈ V)
14		dprdlub.3	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
15	14	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑇 ∈ (SubGrp‘𝐺))
16		subgsubm 18303	. . . . . 6 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ∈ (SubMnd‘𝐺))
17	15, 16	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑇 ∈ (SubMnd‘𝐺))
18	2	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → dom 𝑆 = 𝐼)
19		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)})
20		eqid 2823	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
21	4, 8, 18, 19, 20	dprdff 19136	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑓:𝐼⟶(Base‘𝐺))
22	21	ffnd 6517	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑓 Fn 𝐼)
23		dprdlub.4	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑆‘𝑘) ⊆ 𝑇)
24	23	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑆‘𝑘) ⊆ 𝑇)
25	4, 8, 18, 19	dprdfcl 19137	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ (𝑆‘𝑘))
26	24, 25	sseldd 3970	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ 𝑇)
27	26	ralrimiva 3184	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → ∀𝑘 ∈ 𝐼 (𝑓‘𝑘) ∈ 𝑇)
28		ffnfv 6884	. . . . . 6 ⊢ (𝑓:𝐼⟶𝑇 ↔ (𝑓 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑓‘𝑘) ∈ 𝑇))
29	22, 27, 28	sylanbrc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑓:𝐼⟶𝑇)
30	4, 8, 18, 19, 7	dprdfcntz 19139	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → ran 𝑓 ⊆ ((Cntz‘𝐺)‘ran 𝑓))
31	4, 8, 18, 19	dprdffsupp 19138	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑓 finSupp (0_g‘𝐺))
32	3, 7, 11, 13, 17, 29, 30, 31	gsumzsubmcl 19040	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → (𝐺 Σ_g 𝑓) ∈ 𝑇)
33	32	fmpttd 6881	. . 3 ⊢ (𝜑 → (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)} ↦ (𝐺 Σ_g 𝑓)):{ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}⟶𝑇)
34	33	frnd 6523	. 2 ⊢ (𝜑 → ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)} ↦ (𝐺 Σ_g 𝑓)) ⊆ 𝑇)
35	6, 34	eqsstrd 4007	1 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 Vcvv 3496 ⊆ wss 3938 class class class wbr 5068 ↦ cmpt 5148 dom cdm 5557 ran crn 5558 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Xcixp 8463 finSupp cfsupp 8835 Basecbs 16485 0_gc0g 16715 Σ_g cgsu 16716 Mndcmnd 17913 SubMndcsubmnd 17957 Grpcgrp 18105 SubGrpcsubg 18275 Cntzccntz 18447 DProd cdprd 19117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-gsum 16718 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-subg 18278 df-cntz 18449 df-dprd 19119

This theorem is referenced by: dprdspan 19151 dprdz 19154 dprdcntz2 19162 dprd2dlem1 19165 dprdsplit 19172 ablfac1eu 19197

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