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

Theorem gicsubgen 19310
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 19301 . . 3 (𝑅𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2 n0 4359 . . 3 ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
31, 2bitri 275 . 2 (𝑅𝑔 𝑆 ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
4 fvexd 6922 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ∈ V)
5 fvexd 6922 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑆) ∈ V)
6 vex 3482 . . . . . 6 𝑎 ∈ V
76imaex 7937 . . . . 5 (𝑎𝑏) ∈ V
872a1i 12 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑏 ∈ (SubGrp‘𝑅) → (𝑎𝑏) ∈ V))
96cnvex 7948 . . . . . 6 𝑎 ∈ V
109imaex 7937 . . . . 5 (𝑎𝑐) ∈ V
11102a1i 12 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑐 ∈ (SubGrp‘𝑆) → (𝑎𝑐) ∈ V))
12 gimghm 19295 . . . . . . . . 9 (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎 ∈ (𝑅 GrpHom 𝑆))
13 ghmima 19268 . . . . . . . . 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 2735 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
16 eqid 2735 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
1715, 16gimf1o 19294 . . . . . . . . . . 11 (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
18 f1of1 6848 . . . . . . . . . . 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 19158 . . . . . . . . . 10 (𝑏 ∈ (SubGrp‘𝑅) → 𝑏 ⊆ (Base‘𝑅))
21 f1imacnv 6865 . . . . . . . . . 10 ((𝑎:(Base‘𝑅)–1-1→(Base‘𝑆) ∧ 𝑏 ⊆ (Base‘𝑅)) → (𝑎 “ (𝑎𝑏)) = 𝑏)
2219, 20, 21syl2an 596 . . . . . . . . 9 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑎 “ (𝑎𝑏)) = 𝑏)
2322eqcomd 2741 . . . . . . . 8 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → 𝑏 = (𝑎 “ (𝑎𝑏)))
2414, 23jca 511 . . . . . . 7 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → ((𝑎𝑏) ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎 “ (𝑎𝑏))))
25 eleq1 2827 . . . . . . . 8 (𝑐 = (𝑎𝑏) → (𝑐 ∈ (SubGrp‘𝑆) ↔ (𝑎𝑏) ∈ (SubGrp‘𝑆)))
26 imaeq2 6076 . . . . . . . . 9 (𝑐 = (𝑎𝑏) → (𝑎𝑐) = (𝑎 “ (𝑎𝑏)))
2726eqeq2d 2746 . . . . . . . 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 454 . . . . 5 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏))) → (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐)))
31 ghmpreima 19269 . . . . . . . . 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 6856 . . . . . . . . . . 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 19158 . . . . . . . . . 10 (𝑐 ∈ (SubGrp‘𝑆) → 𝑐 ⊆ (Base‘𝑆))
36 foimacnv 6866 . . . . . . . . . 10 ((𝑎:(Base‘𝑅)–onto→(Base‘𝑆) ∧ 𝑐 ⊆ (Base‘𝑆)) → (𝑎 “ (𝑎𝑐)) = 𝑐)
3734, 35, 36syl2an 596 . . . . . . . . 9 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑎 “ (𝑎𝑐)) = 𝑐)
3837eqcomd 2741 . . . . . . . 8 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → 𝑐 = (𝑎 “ (𝑎𝑐)))
3932, 38jca 511 . . . . . . 7 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → ((𝑎𝑐) ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ (𝑎𝑐))))
40 eleq1 2827 . . . . . . . 8 (𝑏 = (𝑎𝑐) → (𝑏 ∈ (SubGrp‘𝑅) ↔ (𝑎𝑐) ∈ (SubGrp‘𝑅)))
41 imaeq2 6076 . . . . . . . . 9 (𝑏 = (𝑎𝑐) → (𝑎𝑏) = (𝑎 “ (𝑎𝑐)))
4241eqeq2d 2746 . . . . . . . 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 454 . . . . 5 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐))) → (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏)))
4630, 45impbida 801 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → ((𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏)) ↔ (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐))))
474, 5, 8, 11, 46en2d 9027 . . 3 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
4847exlimiv 1928 . 2 (∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
493, 48sylbi 217 1 (𝑅𝑔 𝑆 → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  wne 2938  Vcvv 3478  wss 3963  c0 4339   class class class wbr 5148  ccnv 5688  cima 5692  1-1wf1 6560  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  cen 8981  Basecbs 17245  SubGrpcsubg 19151   GrpHom cghm 19243   GrpIso cgim 19288  𝑔 cgic 19289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-subg 19154  df-ghm 19244  df-gim 19290  df-gic 19291
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator