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Theorem dprdlub 20059

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 2761	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
4		eqid 2761	. . . 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 20036	. . 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 593	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)} ↦ (𝐺 Σ_g 𝑓)))
7		eqid 2761	. . . . 5 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
8	1	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝐺dom DProd 𝑆)
9		dprdgrp 20038	. . . . . 6 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
10		grpmnd 18973	. . . . . 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 20034	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
13	12	adantr 484	. . . . 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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑇 ∈ (SubGrp‘𝐺))
16		subgsubm 19181	. . . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → dom 𝑆 = 𝐼)
19		simpr 488	. . . . . . . 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 2761	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
21	4, 8, 18, 19, 20	dprdff 20045	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑓:𝐼⟶(Base‘𝐺))
22	21	ffnd 6687	. . . . . 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 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑆‘𝑘) ⊆ 𝑇)
25	4, 8, 18, 19	dprdfcl 20046	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ (𝑆‘𝑘))
26	24, 25	sseldd 3935	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ 𝑇)
27	26	ralrimiva 3153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → ∀𝑘 ∈ 𝐼 (𝑓‘𝑘) ∈ 𝑇)
28		ffnfv 7095	. . . . . 6 ⊢ (𝑓:𝐼⟶𝑇 ↔ (𝑓 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑓‘𝑘) ∈ 𝑇))
29	22, 27, 28	sylanbrc 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → 𝑓:𝐼⟶𝑇)
30	4, 8, 18, 19, 7	dprdfcntz 20048	. . . . 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 20047	. . . . 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 19949	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)}) → (𝐺 Σ_g 𝑓) ∈ 𝑇)
33	32	fmpttd 7091	. . 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 6695	. 2 ⊢ (𝜑 → ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐺)} ↦ (𝐺 Σ_g 𝑓)) ⊆ 𝑇)
35	6, 34	eqsstrd 3968	1 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ∀wral 3075 {crab 3413 Vcvv 3453 ⊆ wss 3902 class class class wbr 5097 ↦ cmpt 5178 dom cdm 5643 ran crn 5644 Fn wfn 6511 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 Xcixp 8873 finSupp cfsupp 9301 Basecbs 17236 0_gc0g 17459 Σ_g cgsu 17460 Mndcmnd 18759 SubMndcsubmnd 18807 Grpcgrp 18966 SubGrpcsubg 19153 Cntzccntz 19346 DProd cdprd 20026

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-supp 8135 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-er 8672 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9302 df-oi 9452 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-2 12274 df-n0 12476 df-z 12563 df-uz 12834 df-fz 13507 df-fzo 13654 df-seq 14009 df-hash 14338 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-0g 17461 df-gsum 17462 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18969 df-minusg 18970 df-subg 19156 df-cntz 19348 df-dprd 20028

This theorem is referenced by: dprdspan 20060 dprdz 20063 dprdcntz2 20071 dprd2dlem1 20074 dprdsplit 20081 ablfac1eu 20106

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