Theorem gicsubgen 19526

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	⊢ (𝑅 ≃𝑔 𝑆 → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))

Proof of Theorem gicsubgen

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brgic 19517	. . 3 ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2		n0 4303	. . 3 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
3	1, 2	bitri 278	. 2 ⊢ (𝑅 ≃𝑔 𝑆 ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
4		fvexd 6926	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ∈ V)
5		fvexd 6926	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑆) ∈ V)
6		vex 3458	. . . . . 6 ⊢ 𝑎 ∈ V
7	6	imaex 7953	. . . . 5 ⊢ (𝑎 “ 𝑏) ∈ V
8	7	2a1i 12	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑏 ∈ (SubGrp‘𝑅) → (𝑎 “ 𝑏) ∈ V))
9	6	cnvex 7964	. . . . . 6 ⊢ ◡𝑎 ∈ V
10	9	imaex 7953	. . . . 5 ⊢ (◡𝑎 “ 𝑐) ∈ V
11	10	2a1i 12	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑐 ∈ (SubGrp‘𝑆) → (◡𝑎 “ 𝑐) ∈ V))
12		gimghm 19510	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎 ∈ (𝑅 GrpHom 𝑆))
13		ghmima 19483	. . . . . . . . 9 ⊢ ((𝑎 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑎 “ 𝑏) ∈ (SubGrp‘𝑆))
14	12, 13	sylan 592	. . . . . . . 8 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑎 “ 𝑏) ∈ (SubGrp‘𝑆))
15		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
16		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
17	15, 16	gimf1o 19509	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎:(Base‘𝑅)⟶1-1onto(Base‘𝑆))
18		f1of1 6848	. . . . . . . . . . 11 ⊢ (𝑎:(Base‘𝑅)⟶1-1onto(Base‘𝑆) → 𝑎:(Base‘𝑅)⟶1-1(Base‘𝑆))
19	17, 18	syl 18	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎:(Base‘𝑅)⟶1-1(Base‘𝑆))
20	15	subgss 19369	. . . . . . . . . 10 ⊢ (𝑏 ∈ (SubGrp‘𝑅) → 𝑏 ⊆ (Base‘𝑅))
21		f1imacnv 6866	. . . . . . . . . 10 ⊢ ((𝑎:(Base‘𝑅)⟶1-1(Base‘𝑆) ∧ 𝑏 ⊆ (Base‘𝑅)) → (◡𝑎 “ (𝑎 “ 𝑏)) = 𝑏)
22	19, 20, 21	syl2an 608	. . . . . . . . 9 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (◡𝑎 “ (𝑎 “ 𝑏)) = 𝑏)
23	22	eqcomd 2769	. . . . . . . 8 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → 𝑏 = (◡𝑎 “ (𝑎 “ 𝑏)))
24	14, 23	jca 521	. . . . . . 7 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → ((𝑎 “ 𝑏) ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ (𝑎 “ 𝑏))))
25		eleq1 2852	. . . . . . . 8 ⊢ (𝑐 = (𝑎 “ 𝑏) → (𝑐 ∈ (SubGrp‘𝑆) ↔ (𝑎 “ 𝑏) ∈ (SubGrp‘𝑆)))
26		imaeq2 6061	. . . . . . . . 9 ⊢ (𝑐 = (𝑎 “ 𝑏) → (◡𝑎 “ 𝑐) = (◡𝑎 “ (𝑎 “ 𝑏)))
27	26	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑐 = (𝑎 “ 𝑏) → (𝑏 = (◡𝑎 “ 𝑐) ↔ 𝑏 = (◡𝑎 “ (𝑎 “ 𝑏))))
28	25, 27	anbi12d 644	. . . . . . 7 ⊢ (𝑐 = (𝑎 “ 𝑏) → ((𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ 𝑐)) ↔ ((𝑎 “ 𝑏) ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ (𝑎 “ 𝑏)))))
29	24, 28	syl5ibrcom 250	. . . . . 6 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑐 = (𝑎 “ 𝑏) → (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ 𝑐))))
30	29	impr 460	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ 𝑏))) → (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ 𝑐)))
31		ghmpreima 19484	. . . . . . . . 9 ⊢ ((𝑎 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (◡𝑎 “ 𝑐) ∈ (SubGrp‘𝑅))
32	12, 31	sylan 592	. . . . . . . 8 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (◡𝑎 “ 𝑐) ∈ (SubGrp‘𝑅))
33		f1ofo 6857	. . . . . . . . . . 11 ⊢ (𝑎:(Base‘𝑅)⟶1-1onto(Base‘𝑆) → 𝑎:(Base‘𝑅)⟶onto(Base‘𝑆))
34	17, 33	syl 18	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎:(Base‘𝑅)⟶onto(Base‘𝑆))
35	16	subgss 19369	. . . . . . . . . 10 ⊢ (𝑐 ∈ (SubGrp‘𝑆) → 𝑐 ⊆ (Base‘𝑆))
36		foimacnv 6867	. . . . . . . . . 10 ⊢ ((𝑎:(Base‘𝑅)⟶onto(Base‘𝑆) ∧ 𝑐 ⊆ (Base‘𝑆)) → (𝑎 “ (◡𝑎 “ 𝑐)) = 𝑐)
37	34, 35, 36	syl2an 608	. . . . . . . . 9 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑎 “ (◡𝑎 “ 𝑐)) = 𝑐)
38	37	eqcomd 2769	. . . . . . . 8 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → 𝑐 = (𝑎 “ (◡𝑎 “ 𝑐)))
39	32, 38	jca 521	. . . . . . 7 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → ((◡𝑎 “ 𝑐) ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ (◡𝑎 “ 𝑐))))
40		eleq1 2852	. . . . . . . 8 ⊢ (𝑏 = (◡𝑎 “ 𝑐) → (𝑏 ∈ (SubGrp‘𝑅) ↔ (◡𝑎 “ 𝑐) ∈ (SubGrp‘𝑅)))
41		imaeq2 6061	. . . . . . . . 9 ⊢ (𝑏 = (◡𝑎 “ 𝑐) → (𝑎 “ 𝑏) = (𝑎 “ (◡𝑎 “ 𝑐)))
42	41	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑏 = (◡𝑎 “ 𝑐) → (𝑐 = (𝑎 “ 𝑏) ↔ 𝑐 = (𝑎 “ (◡𝑎 “ 𝑐))))
43	40, 42	anbi12d 644	. . . . . . 7 ⊢ (𝑏 = (◡𝑎 “ 𝑐) → ((𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ 𝑏)) ↔ ((◡𝑎 “ 𝑐) ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ (◡𝑎 “ 𝑐)))))
44	39, 43	syl5ibrcom 250	. . . . . 6 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑏 = (◡𝑎 “ 𝑐) → (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ 𝑏))))
45	44	impr 460	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ 𝑐))) → (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ 𝑏)))
46	30, 45	impbida 813	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → ((𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ 𝑏)) ↔ (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ 𝑐))))
47	4, 5, 8, 11, 46	en2d 9046	. . 3 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
48	47	exlimiv 1965	. 2 ⊢ (∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
49	3, 48	sylbi 220	1 ⊢ (𝑅 ≃𝑔 𝑆 → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454   ⊆ wss 3900  ∅c0 4281   class class class wbr 5106  ^◡ccnv 5655   “ cima 5659  ⟶1-1wf1 6558  ⟶ontowfo 6559  ⟶1-1ontowf1o 6560  ‘cfv 6561  (class class class)co 7446   ≈ cen 9001  Basecbs 17419  SubGrpcsubg 19362   GrpHom cghm 19459   GrpIso cgim 19503   ≃_𝑔 cgic 19504

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-er 8748  df-map 8880  df-en 9005  df-dom 9006  df-sdom 9007  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-nn 12366  df-2 12435  df-sets 17374  df-slot 17392  df-ndx 17404  df-base 17420  df-ress 17441  df-plusg 17473  df-0g 17644  df-mgm 18848  df-sgrp 18940  df-mnd 18956  df-grp 19179  df-minusg 19180  df-subg 19365  df-ghm 19460  df-gim 19505  df-gic 19506

This theorem is used by: (None)