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Theorem gicsubgen 19069
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 19060 . . 3 (𝑅𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2 n0 4307 . . 3 ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
31, 2bitri 275 . 2 (𝑅𝑔 𝑆 ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
4 fvexd 6858 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ∈ V)
5 fvexd 6858 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑆) ∈ V)
6 vex 3450 . . . . . 6 𝑎 ∈ V
76imaex 7854 . . . . 5 (𝑎𝑏) ∈ V
872a1i 12 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑏 ∈ (SubGrp‘𝑅) → (𝑎𝑏) ∈ V))
96cnvex 7863 . . . . . 6 𝑎 ∈ V
109imaex 7854 . . . . 5 (𝑎𝑐) ∈ V
11102a1i 12 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑐 ∈ (SubGrp‘𝑆) → (𝑎𝑐) ∈ V))
12 gimghm 19055 . . . . . . . . 9 (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎 ∈ (𝑅 GrpHom 𝑆))
13 ghmima 19030 . . . . . . . . 9 ((𝑎 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑎𝑏) ∈ (SubGrp‘𝑆))
1412, 13sylan 581 . . . . . . . 8 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑎𝑏) ∈ (SubGrp‘𝑆))
15 eqid 2737 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
16 eqid 2737 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
1715, 16gimf1o 19054 . . . . . . . . . . 11 (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
18 f1of1 6784 . . . . . . . . . . 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 18930 . . . . . . . . . 10 (𝑏 ∈ (SubGrp‘𝑅) → 𝑏 ⊆ (Base‘𝑅))
21 f1imacnv 6801 . . . . . . . . . 10 ((𝑎:(Base‘𝑅)–1-1→(Base‘𝑆) ∧ 𝑏 ⊆ (Base‘𝑅)) → (𝑎 “ (𝑎𝑏)) = 𝑏)
2219, 20, 21syl2an 597 . . . . . . . . 9 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑎 “ (𝑎𝑏)) = 𝑏)
2322eqcomd 2743 . . . . . . . 8 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → 𝑏 = (𝑎 “ (𝑎𝑏)))
2414, 23jca 513 . . . . . . 7 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → ((𝑎𝑏) ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎 “ (𝑎𝑏))))
25 eleq1 2826 . . . . . . . 8 (𝑐 = (𝑎𝑏) → (𝑐 ∈ (SubGrp‘𝑆) ↔ (𝑎𝑏) ∈ (SubGrp‘𝑆)))
26 imaeq2 6010 . . . . . . . . 9 (𝑐 = (𝑎𝑏) → (𝑎𝑐) = (𝑎 “ (𝑎𝑏)))
2726eqeq2d 2748 . . . . . . . 8 (𝑐 = (𝑎𝑏) → (𝑏 = (𝑎𝑐) ↔ 𝑏 = (𝑎 “ (𝑎𝑏))))
2825, 27anbi12d 632 . . . . . . 7 (𝑐 = (𝑎𝑏) → ((𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐)) ↔ ((𝑎𝑏) ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎 “ (𝑎𝑏)))))
2924, 28syl5ibrcom 247 . . . . . 6 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑐 = (𝑎𝑏) → (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐))))
3029impr 456 . . . . 5 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏))) → (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐)))
31 ghmpreima 19031 . . . . . . . . 9 ((𝑎 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑎𝑐) ∈ (SubGrp‘𝑅))
3212, 31sylan 581 . . . . . . . 8 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑎𝑐) ∈ (SubGrp‘𝑅))
33 f1ofo 6792 . . . . . . . . . . 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 18930 . . . . . . . . . 10 (𝑐 ∈ (SubGrp‘𝑆) → 𝑐 ⊆ (Base‘𝑆))
36 foimacnv 6802 . . . . . . . . . 10 ((𝑎:(Base‘𝑅)–onto→(Base‘𝑆) ∧ 𝑐 ⊆ (Base‘𝑆)) → (𝑎 “ (𝑎𝑐)) = 𝑐)
3734, 35, 36syl2an 597 . . . . . . . . 9 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑎 “ (𝑎𝑐)) = 𝑐)
3837eqcomd 2743 . . . . . . . 8 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → 𝑐 = (𝑎 “ (𝑎𝑐)))
3932, 38jca 513 . . . . . . 7 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → ((𝑎𝑐) ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ (𝑎𝑐))))
40 eleq1 2826 . . . . . . . 8 (𝑏 = (𝑎𝑐) → (𝑏 ∈ (SubGrp‘𝑅) ↔ (𝑎𝑐) ∈ (SubGrp‘𝑅)))
41 imaeq2 6010 . . . . . . . . 9 (𝑏 = (𝑎𝑐) → (𝑎𝑏) = (𝑎 “ (𝑎𝑐)))
4241eqeq2d 2748 . . . . . . . 8 (𝑏 = (𝑎𝑐) → (𝑐 = (𝑎𝑏) ↔ 𝑐 = (𝑎 “ (𝑎𝑐))))
4340, 42anbi12d 632 . . . . . . 7 (𝑏 = (𝑎𝑐) → ((𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏)) ↔ ((𝑎𝑐) ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ (𝑎𝑐)))))
4439, 43syl5ibrcom 247 . . . . . 6 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑏 = (𝑎𝑐) → (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏))))
4544impr 456 . . . . 5 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐))) → (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏)))
4630, 45impbida 800 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → ((𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏)) ↔ (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐))))
474, 5, 8, 11, 46en2d 8929 . . 3 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
4847exlimiv 1934 . 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 397   = wceq 1542  wex 1782  wcel 2107  wne 2944  Vcvv 3446  wss 3911  c0 4283   class class class wbr 5106  ccnv 5633  cima 5637  1-1wf1 6494  ontowfo 6495  1-1-ontowf1o 6496  cfv 6497  (class class class)co 7358  cen 8881  Basecbs 17084  SubGrpcsubg 18923   GrpHom cghm 19006   GrpIso cgim 19048  𝑔 cgic 19049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8649  df-en 8885  df-dom 8886  df-sdom 8887  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-nn 12155  df-2 12217  df-sets 17037  df-slot 17055  df-ndx 17067  df-base 17085  df-ress 17114  df-plusg 17147  df-0g 17324  df-mgm 18498  df-sgrp 18547  df-mnd 18558  df-grp 18752  df-minusg 18753  df-subg 18926  df-ghm 19007  df-gim 19050  df-gic 19051
This theorem is referenced by: (None)
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