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Theorem dprdlub 20015

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 2734	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
4		eqid 2734	. . . 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 19992	. . 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 584	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)} ↦ (𝐺 Σ_g 𝑓)))
7		eqid 2734	. . . . 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 19994	. . . . . 6 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
10		grpmnd 18928	. . . . . 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 19990	. . . . . 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 19136	. . . . . 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 2734	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
21	4, 8, 18, 19, 20	dprdff 20001	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑓:𝐼⟶(Base‘𝐺))
22	21	ffnd 6717	. . . . . 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 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑆‘𝑘) ⊆ 𝑇)
25	4, 8, 18, 19	dprdfcl 20002	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ (𝑆‘𝑘))
26	24, 25	sseldd 3964	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ 𝑇)
27	26	ralrimiva 3133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → ∀𝑘 ∈ 𝐼 (𝑓‘𝑘) ∈ 𝑇)
28		ffnfv 7119	. . . . . 6 ⊢ (𝑓:𝐼⟶𝑇 ↔ (𝑓 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑓‘𝑘) ∈ 𝑇))
29	22, 27, 28	sylanbrc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑓:𝐼⟶𝑇)
30	4, 8, 18, 19, 7	dprdfcntz 20004	. . . . 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 20003	. . . . 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 19905	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → (𝐺 Σ_g 𝑓) ∈ 𝑇)
33	32	fmpttd 7115	. . 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 6724	. 2 ⊢ (𝜑 → ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)} ↦ (𝐺 Σ_g 𝑓)) ⊆ 𝑇)
35	6, 34	eqsstrd 3998	1 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3050 {crab 3419 Vcvv 3463 ⊆ wss 3931 class class class wbr 5123 ↦ cmpt 5205 dom cdm 5665 ran crn 5666 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7413 Xcixp 8919 finSupp cfsupp 9383 Basecbs 17230 0_gc0g 17456 Σ_g cgsu 17457 Mndcmnd 18717 SubMndcsubmnd 18765 Grpcgrp 18921 SubGrpcsubg 19108 Cntzccntz 19303 DProd cdprd 19982

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 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214

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 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-int 4927 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-se 5618 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-1st 7996 df-2nd 7997 df-supp 8168 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8727 df-ixp 8920 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-fsupp 9384 df-oi 9532 df-card 9961 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-nn 12249 df-2 12311 df-n0 12510 df-z 12597 df-uz 12861 df-fz 13530 df-fzo 13677 df-seq 14025 df-hash 14353 df-sets 17184 df-slot 17202 df-ndx 17214 df-base 17231 df-ress 17254 df-plusg 17287 df-0g 17458 df-gsum 17459 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-grp 18924 df-minusg 18925 df-subg 19111 df-cntz 19305 df-dprd 19984

This theorem is referenced by: dprdspan 20016 dprdz 20019 dprdcntz2 20027 dprd2dlem1 20030 dprdsplit 20037 ablfac1eu 20062

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