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Theorem gicsubgen 18159
Description: A less trivial example of a group invariant: cardinality of the subgroup lattice. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Assertion
Ref Expression
gicsubgen (𝑅𝑔 𝑆 → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))

Proof of Theorem gicsubgen
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brgic 18150 . . 3 (𝑅𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2 n0 4230 . . 3 ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
31, 2bitri 276 . 2 (𝑅𝑔 𝑆 ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
4 fvexd 6553 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ∈ V)
5 fvexd 6553 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑆) ∈ V)
6 vex 3440 . . . . . 6 𝑎 ∈ V
76imaex 7477 . . . . 5 (𝑎𝑏) ∈ V
872a1i 12 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑏 ∈ (SubGrp‘𝑅) → (𝑎𝑏) ∈ V))
96cnvex 7486 . . . . . 6 𝑎 ∈ V
109imaex 7477 . . . . 5 (𝑎𝑐) ∈ V
11102a1i 12 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑐 ∈ (SubGrp‘𝑆) → (𝑎𝑐) ∈ V))
12 gimghm 18145 . . . . . . . . 9 (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎 ∈ (𝑅 GrpHom 𝑆))
13 ghmima 18120 . . . . . . . . 9 ((𝑎 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑎𝑏) ∈ (SubGrp‘𝑆))
1412, 13sylan 580 . . . . . . . 8 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑎𝑏) ∈ (SubGrp‘𝑆))
15 eqid 2795 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
16 eqid 2795 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
1715, 16gimf1o 18144 . . . . . . . . . . 11 (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
18 f1of1 6482 . . . . . . . . . . 11 (𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆) → 𝑎:(Base‘𝑅)–1-1→(Base‘𝑆))
1917, 18syl 17 . . . . . . . . . 10 (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎:(Base‘𝑅)–1-1→(Base‘𝑆))
2015subgss 18034 . . . . . . . . . 10 (𝑏 ∈ (SubGrp‘𝑅) → 𝑏 ⊆ (Base‘𝑅))
21 f1imacnv 6499 . . . . . . . . . 10 ((𝑎:(Base‘𝑅)–1-1→(Base‘𝑆) ∧ 𝑏 ⊆ (Base‘𝑅)) → (𝑎 “ (𝑎𝑏)) = 𝑏)
2219, 20, 21syl2an 595 . . . . . . . . 9 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑎 “ (𝑎𝑏)) = 𝑏)
2322eqcomd 2801 . . . . . . . 8 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → 𝑏 = (𝑎 “ (𝑎𝑏)))
2414, 23jca 512 . . . . . . 7 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → ((𝑎𝑏) ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎 “ (𝑎𝑏))))
25 eleq1 2870 . . . . . . . 8 (𝑐 = (𝑎𝑏) → (𝑐 ∈ (SubGrp‘𝑆) ↔ (𝑎𝑏) ∈ (SubGrp‘𝑆)))
26 imaeq2 5802 . . . . . . . . 9 (𝑐 = (𝑎𝑏) → (𝑎𝑐) = (𝑎 “ (𝑎𝑏)))
2726eqeq2d 2805 . . . . . . . 8 (𝑐 = (𝑎𝑏) → (𝑏 = (𝑎𝑐) ↔ 𝑏 = (𝑎 “ (𝑎𝑏))))
2825, 27anbi12d 630 . . . . . . 7 (𝑐 = (𝑎𝑏) → ((𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐)) ↔ ((𝑎𝑏) ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎 “ (𝑎𝑏)))))
2924, 28syl5ibrcom 248 . . . . . 6 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑐 = (𝑎𝑏) → (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐))))
3029impr 455 . . . . 5 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏))) → (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐)))
31 ghmpreima 18121 . . . . . . . . 9 ((𝑎 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑎𝑐) ∈ (SubGrp‘𝑅))
3212, 31sylan 580 . . . . . . . 8 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑎𝑐) ∈ (SubGrp‘𝑅))
33 f1ofo 6490 . . . . . . . . . . 11 (𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆) → 𝑎:(Base‘𝑅)–onto→(Base‘𝑆))
3417, 33syl 17 . . . . . . . . . 10 (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎:(Base‘𝑅)–onto→(Base‘𝑆))
3516subgss 18034 . . . . . . . . . 10 (𝑐 ∈ (SubGrp‘𝑆) → 𝑐 ⊆ (Base‘𝑆))
36 foimacnv 6500 . . . . . . . . . 10 ((𝑎:(Base‘𝑅)–onto→(Base‘𝑆) ∧ 𝑐 ⊆ (Base‘𝑆)) → (𝑎 “ (𝑎𝑐)) = 𝑐)
3734, 35, 36syl2an 595 . . . . . . . . 9 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑎 “ (𝑎𝑐)) = 𝑐)
3837eqcomd 2801 . . . . . . . 8 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → 𝑐 = (𝑎 “ (𝑎𝑐)))
3932, 38jca 512 . . . . . . 7 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → ((𝑎𝑐) ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ (𝑎𝑐))))
40 eleq1 2870 . . . . . . . 8 (𝑏 = (𝑎𝑐) → (𝑏 ∈ (SubGrp‘𝑅) ↔ (𝑎𝑐) ∈ (SubGrp‘𝑅)))
41 imaeq2 5802 . . . . . . . . 9 (𝑏 = (𝑎𝑐) → (𝑎𝑏) = (𝑎 “ (𝑎𝑐)))
4241eqeq2d 2805 . . . . . . . 8 (𝑏 = (𝑎𝑐) → (𝑐 = (𝑎𝑏) ↔ 𝑐 = (𝑎 “ (𝑎𝑐))))
4340, 42anbi12d 630 . . . . . . 7 (𝑏 = (𝑎𝑐) → ((𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏)) ↔ ((𝑎𝑐) ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ (𝑎𝑐)))))
4439, 43syl5ibrcom 248 . . . . . 6 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑏 = (𝑎𝑐) → (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏))))
4544impr 455 . . . . 5 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐))) → (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏)))
4630, 45impbida 797 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → ((𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏)) ↔ (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐))))
474, 5, 8, 11, 46en2d 8393 . . 3 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
4847exlimiv 1908 . 2 (∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
493, 48sylbi 218 1 (𝑅𝑔 𝑆 → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wex 1761  wcel 2081  wne 2984  Vcvv 3437  wss 3859  c0 4211   class class class wbr 4962  ccnv 5442  cima 5446  1-1wf1 6222  ontowfo 6223  1-1-ontowf1o 6224  cfv 6225  (class class class)co 7016  cen 8354  Basecbs 16312  SubGrpcsubg 18027   GrpHom cghm 18096   GrpIso cgim 18138  𝑔 cgic 18139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-0g 16544  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-grp 17864  df-minusg 17865  df-subg 18030  df-ghm 18097  df-gim 18140  df-gic 18141
This theorem is referenced by: (None)
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