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Theorem gicsubgen 19297
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 19288 . . 3 (𝑅𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2 n0 4353 . . 3 ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
31, 2bitri 275 . 2 (𝑅𝑔 𝑆 ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
4 fvexd 6921 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ∈ V)
5 fvexd 6921 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑆) ∈ V)
6 vex 3484 . . . . . 6 𝑎 ∈ V
76imaex 7936 . . . . 5 (𝑎𝑏) ∈ V
872a1i 12 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑏 ∈ (SubGrp‘𝑅) → (𝑎𝑏) ∈ V))
96cnvex 7947 . . . . . 6 𝑎 ∈ V
109imaex 7936 . . . . 5 (𝑎𝑐) ∈ V
11102a1i 12 . . . 4 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑐 ∈ (SubGrp‘𝑆) → (𝑎𝑐) ∈ V))
12 gimghm 19282 . . . . . . . . 9 (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎 ∈ (𝑅 GrpHom 𝑆))
13 ghmima 19255 . . . . . . . . 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 2737 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
16 eqid 2737 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
1715, 16gimf1o 19281 . . . . . . . . . . 11 (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
18 f1of1 6847 . . . . . . . . . . 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 19145 . . . . . . . . . 10 (𝑏 ∈ (SubGrp‘𝑅) → 𝑏 ⊆ (Base‘𝑅))
21 f1imacnv 6864 . . . . . . . . . 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 2743 . . . . . . . 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 2829 . . . . . . . 8 (𝑐 = (𝑎𝑏) → (𝑐 ∈ (SubGrp‘𝑆) ↔ (𝑎𝑏) ∈ (SubGrp‘𝑆)))
26 imaeq2 6074 . . . . . . . . 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 454 . . . . 5 ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎𝑏))) → (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (𝑎𝑐)))
31 ghmpreima 19256 . . . . . . . . 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 6855 . . . . . . . . . . 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 19145 . . . . . . . . . 10 (𝑐 ∈ (SubGrp‘𝑆) → 𝑐 ⊆ (Base‘𝑆))
36 foimacnv 6865 . . . . . . . . . 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 2743 . . . . . . . 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 2829 . . . . . . . 8 (𝑏 = (𝑎𝑐) → (𝑏 ∈ (SubGrp‘𝑅) ↔ (𝑎𝑐) ∈ (SubGrp‘𝑅)))
41 imaeq2 6074 . . . . . . . . 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 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 9028 . . 3 (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
4847exlimiv 1930 . 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 1540  wex 1779  wcel 2108  wne 2940  Vcvv 3480  wss 3951  c0 4333   class class class wbr 5143  ccnv 5684  cima 5688  1-1wf1 6558  ontowfo 6559  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cen 8982  Basecbs 17247  SubGrpcsubg 19138   GrpHom cghm 19230   GrpIso cgim 19275  𝑔 cgic 19276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-subg 19141  df-ghm 19231  df-gim 19277  df-gic 19278
This theorem is referenced by: (None)
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