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Theorem dprdlub 20070

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 2740	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
4		eqid 2740	. . . 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 20047	. . 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 583	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)} ↦ (𝐺 Σ_g 𝑓)))
7		eqid 2740	. . . . 5 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
8	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝐺dom DProd 𝑆)
9		dprdgrp 20049	. . . . . 6 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
10		grpmnd 18980	. . . . . 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 20045	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
13	12	adantr 480	. . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑇 ∈ (SubGrp‘𝐺))
16		subgsubm 19188	. . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → dom 𝑆 = 𝐼)
19		simpr 484	. . . . . . . 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 2740	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
21	4, 8, 18, 19, 20	dprdff 20056	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑓:𝐼⟶(Base‘𝐺))
22	21	ffnd 6748	. . . . . 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 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑆‘𝑘) ⊆ 𝑇)
25	4, 8, 18, 19	dprdfcl 20057	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ (𝑆‘𝑘))
26	24, 25	sseldd 4009	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ 𝑇)
27	26	ralrimiva 3152	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → ∀𝑘 ∈ 𝐼 (𝑓‘𝑘) ∈ 𝑇)
28		ffnfv 7153	. . . . . 6 ⊢ (𝑓:𝐼⟶𝑇 ↔ (𝑓 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑓‘𝑘) ∈ 𝑇))
29	22, 27, 28	sylanbrc 582	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑓:𝐼⟶𝑇)
30	4, 8, 18, 19, 7	dprdfcntz 20059	. . . . 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 20058	. . . . 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 19960	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → (𝐺 Σ_g 𝑓) ∈ 𝑇)
33	32	fmpttd 7149	. . 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 6755	. 2 ⊢ (𝜑 → ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)} ↦ (𝐺 Σ_g 𝑓)) ⊆ 𝑇)
35	6, 34	eqsstrd 4047	1 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 {crab 3443 Vcvv 3488 ⊆ wss 3976 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ran crn 5701 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 Xcixp 8955 finSupp cfsupp 9431 Basecbs 17258 0_gc0g 17499 Σ_g cgsu 17500 Mndcmnd 18772 SubMndcsubmnd 18817 Grpcgrp 18973 SubGrpcsubg 19160 Cntzccntz 19355 DProd cdprd 20037

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-0g 17501 df-gsum 17502 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-grp 18976 df-minusg 18977 df-subg 19163 df-cntz 19357 df-dprd 20039

This theorem is referenced by: dprdspan 20071 dprdz 20074 dprdcntz2 20082 dprd2dlem1 20085 dprdsplit 20092 ablfac1eu 20117

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