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Theorem gicsubgen 19273
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 19264 . . 3 (𝑅𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2 n0 4349 . . 3 ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
31, 2bitri 274 . 2 (𝑅𝑔 𝑆 ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
4 fvexd 6916 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ∈ V)
5 fvexd 6916 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑆) ∈ V)
6 vex 3466 . . . . . 6 𝑎 ∈ V
76imaex 7927 . . . . 5 (𝑎𝑏) ∈ V
872a1i 12 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑏 ∈ (SubGrp‘𝑅) → (𝑎𝑏) ∈ V))
96cnvex 7938 . . . . . 6 𝑎 ∈ V
109imaex 7927 . . . . 5 (𝑎𝑐) ∈ V
11102a1i 12 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑐 ∈ (SubGrp‘𝑆) → (𝑎𝑐) ∈ V))
12 gimghm 19258 . . . . . . . . 9 (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎 ∈ (𝑅 GrpHom 𝑆))
13 ghmima 19231 . . . . . . . . 9 ((𝑎 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑎𝑏) ∈ (SubGrp‘𝑆))
1412, 13sylan 578 . . . . . . . 8 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑎𝑏) ∈ (SubGrp‘𝑆))
15 eqid 2726 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
16 eqid 2726 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
1715, 16gimf1o 19257 . . . . . . . . . . 11 (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
18 f1of1 6842 . . . . . . . . . . 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 19121 . . . . . . . . . 10 (𝑏 ∈ (SubGrp‘𝑅) → 𝑏 ⊆ (Base‘𝑅))
21 f1imacnv 6859 . . . . . . . . . 10 ((𝑎:(Base‘𝑅)–1-1→(Base‘𝑆) ∧ 𝑏 ⊆ (Base‘𝑅)) → (𝑎 “ (𝑎𝑏)) = 𝑏)
2219, 20, 21syl2an 594 . . . . . . . . 9 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑎 “ (𝑎𝑏)) = 𝑏)
2322eqcomd 2732 . . . . . . . 8 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → 𝑏 = (𝑎 “ (𝑎𝑏)))
2414, 23jca 510 . . . . . . 7 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → ((𝑎𝑏) ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎 “ (𝑎𝑏))))
25 eleq1 2814 . . . . . . . 8 (𝑐 = (𝑎𝑏) → (𝑐 ∈ (SubGrp‘𝑆) ↔ (𝑎𝑏) ∈ (SubGrp‘𝑆)))
26 imaeq2 6065 . . . . . . . . 9 (𝑐 = (𝑎𝑏) → (𝑎𝑐) = (𝑎 “ (𝑎𝑏)))
2726eqeq2d 2737 . . . . . . . 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 246 . . . . . 6 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑐 = (𝑎𝑏) → (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐))))
3029impr 453 . . . . 5 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏))) → (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐)))
31 ghmpreima 19232 . . . . . . . . 9 ((𝑎 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑎𝑐) ∈ (SubGrp‘𝑅))
3212, 31sylan 578 . . . . . . . 8 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑎𝑐) ∈ (SubGrp‘𝑅))
33 f1ofo 6850 . . . . . . . . . . 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 19121 . . . . . . . . . 10 (𝑐 ∈ (SubGrp‘𝑆) → 𝑐 ⊆ (Base‘𝑆))
36 foimacnv 6860 . . . . . . . . . 10 ((𝑎:(Base‘𝑅)–onto→(Base‘𝑆) ∧ 𝑐 ⊆ (Base‘𝑆)) → (𝑎 “ (𝑎𝑐)) = 𝑐)
3734, 35, 36syl2an 594 . . . . . . . . 9 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑎 “ (𝑎𝑐)) = 𝑐)
3837eqcomd 2732 . . . . . . . 8 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → 𝑐 = (𝑎 “ (𝑎𝑐)))
3932, 38jca 510 . . . . . . 7 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → ((𝑎𝑐) ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ (𝑎𝑐))))
40 eleq1 2814 . . . . . . . 8 (𝑏 = (𝑎𝑐) → (𝑏 ∈ (SubGrp‘𝑅) ↔ (𝑎𝑐) ∈ (SubGrp‘𝑅)))
41 imaeq2 6065 . . . . . . . . 9 (𝑏 = (𝑎𝑐) → (𝑎𝑏) = (𝑎 “ (𝑎𝑐)))
4241eqeq2d 2737 . . . . . . . 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 246 . . . . . 6 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑏 = (𝑎𝑐) → (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏))))
4544impr 453 . . . . 5 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐))) → (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏)))
4630, 45impbida 799 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → ((𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏)) ↔ (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐))))
474, 5, 8, 11, 46en2d 9019 . . 3 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
4847exlimiv 1926 . 2 (∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
493, 48sylbi 216 1 (𝑅𝑔 𝑆 → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wex 1774  wcel 2099  wne 2930  Vcvv 3462  wss 3947  c0 4325   class class class wbr 5153  ccnv 5681  cima 5685  1-1wf1 6551  ontowfo 6552  1-1-ontowf1o 6553  cfv 6554  (class class class)co 7424  cen 8971  Basecbs 17213  SubGrpcsubg 19114   GrpHom cghm 19206   GrpIso cgim 19251  𝑔 cgic 19252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-plusg 17279  df-0g 17456  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-grp 18931  df-minusg 18932  df-subg 19117  df-ghm 19207  df-gim 19253  df-gic 19254
This theorem is referenced by: (None)
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