Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dprdcntz Structured version   Visualization version   GIF version

Theorem dprdcntz 18401
 Description: The function 𝑆 is a family having pairwise commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdcntz.1 (𝜑𝐺dom DProd 𝑆)
dprdcntz.2 (𝜑 → dom 𝑆 = 𝐼)
dprdcntz.3 (𝜑𝑋𝐼)
dprdcntz.4 (𝜑𝑌𝐼)
dprdcntz.5 (𝜑𝑋𝑌)
dprdcntz.z 𝑍 = (Cntz‘𝐺)
Assertion
Ref Expression
dprdcntz (𝜑 → (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑌)))

Proof of Theorem dprdcntz
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dprdcntz.4 . . 3 (𝜑𝑌𝐼)
2 dprdcntz.5 . . . 4 (𝜑𝑋𝑌)
32necomd 2848 . . 3 (𝜑𝑌𝑋)
4 eldifsn 4315 . . 3 (𝑌 ∈ (𝐼 ∖ {𝑋}) ↔ (𝑌𝐼𝑌𝑋))
51, 3, 4sylanbrc 698 . 2 (𝜑𝑌 ∈ (𝐼 ∖ {𝑋}))
6 dprdcntz.3 . . 3 (𝜑𝑋𝐼)
7 dprdcntz.1 . . . . . 6 (𝜑𝐺dom DProd 𝑆)
8 dprdcntz.2 . . . . . . . 8 (𝜑 → dom 𝑆 = 𝐼)
97, 8dprddomcld 18394 . . . . . . 7 (𝜑𝐼 ∈ V)
10 dprdcntz.z . . . . . . . 8 𝑍 = (Cntz‘𝐺)
11 eqid 2621 . . . . . . . 8 (0g𝐺) = (0g𝐺)
12 eqid 2621 . . . . . . . 8 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
1310, 11, 12dmdprd 18391 . . . . . . 7 ((𝐼 ∈ V ∧ dom 𝑆 = 𝐼) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)}))))
149, 8, 13syl2anc 693 . . . . . 6 (𝜑 → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)}))))
157, 14mpbid 222 . . . . 5 (𝜑 → (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)})))
1615simp3d 1074 . . . 4 (𝜑 → ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)}))
17 simpl 473 . . . . 5 ((∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)}) → ∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)))
1817ralimi 2951 . . . 4 (∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)}) → ∀𝑥𝐼𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)))
1916, 18syl 17 . . 3 (𝜑 → ∀𝑥𝐼𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)))
20 sneq 4185 . . . . . 6 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2120difeq2d 3726 . . . . 5 (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
22 fveq2 6189 . . . . . 6 (𝑥 = 𝑋 → (𝑆𝑥) = (𝑆𝑋))
2322sseq1d 3630 . . . . 5 (𝑥 = 𝑋 → ((𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ↔ (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑦))))
2421, 23raleqbidv 3150 . . . 4 (𝑥 = 𝑋 → (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ↔ ∀𝑦 ∈ (𝐼 ∖ {𝑋})(𝑆𝑋) ⊆ (𝑍‘(𝑆𝑦))))
2524rspcv 3303 . . 3 (𝑋𝐼 → (∀𝑥𝐼𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) → ∀𝑦 ∈ (𝐼 ∖ {𝑋})(𝑆𝑋) ⊆ (𝑍‘(𝑆𝑦))))
266, 19, 25sylc 65 . 2 (𝜑 → ∀𝑦 ∈ (𝐼 ∖ {𝑋})(𝑆𝑋) ⊆ (𝑍‘(𝑆𝑦)))
27 fveq2 6189 . . . . 5 (𝑦 = 𝑌 → (𝑆𝑦) = (𝑆𝑌))
2827fveq2d 6193 . . . 4 (𝑦 = 𝑌 → (𝑍‘(𝑆𝑦)) = (𝑍‘(𝑆𝑌)))
2928sseq2d 3631 . . 3 (𝑦 = 𝑌 → ((𝑆𝑋) ⊆ (𝑍‘(𝑆𝑦)) ↔ (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑌))))
3029rspcv 3303 . 2 (𝑌 ∈ (𝐼 ∖ {𝑋}) → (∀𝑦 ∈ (𝐼 ∖ {𝑋})(𝑆𝑋) ⊆ (𝑍‘(𝑆𝑦)) → (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑌))))
315, 26, 30sylc 65 1 (𝜑 → (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1482   ∈ wcel 1989   ≠ wne 2793  ∀wral 2911  Vcvv 3198   ∖ cdif 3569   ∩ cin 3571   ⊆ wss 3572  {csn 4175  ∪ cuni 4434   class class class wbr 4651  dom cdm 5112   “ cima 5115  ⟶wf 5882  ‘cfv 5886  0gc0g 16094  mrClscmrc 16237  Grpcgrp 17416  SubGrpcsubg 17582  Cntzccntz 17742   DProd cdprd 18386 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-ixp 7906  df-dprd 18388 This theorem is referenced by:  dprdfcntz  18408  dprdfadd  18413  dprdres  18421  dprdss  18422  dprdf1o  18425  dprdcntz2  18431  dprd2da  18435  dmdprdsplit2lem  18438
 Copyright terms: Public domain W3C validator