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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
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