Theorem gicsubgen 19068

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 19059	. . 3 ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2		n0 4306	. . 3 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
3	1, 2	bitri 274	. 2 ⊢ (𝑅 ≃𝑔 𝑆 ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
4		fvexd 6857	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ∈ V)
5		fvexd 6857	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑆) ∈ V)
6		vex 3449	. . . . . 6 ⊢ 𝑎 ∈ V
7	6	imaex 7853	. . . . 5 ⊢ (𝑎 “ 𝑏) ∈ V
8	7	2a1i 12	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑏 ∈ (SubGrp‘𝑅) → (𝑎 “ 𝑏) ∈ V))
9	6	cnvex 7862	. . . . . 6 ⊢ ◡𝑎 ∈ V
10	9	imaex 7853	. . . . 5 ⊢ (◡𝑎 “ 𝑐) ∈ V
11	10	2a1i 12	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑐 ∈ (SubGrp‘𝑆) → (◡𝑎 “ 𝑐) ∈ V))
12		gimghm 19054	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎 ∈ (𝑅 GrpHom 𝑆))
13		ghmima 19029	. . . . . . . . 9 ⊢ ((𝑎 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑎 “ 𝑏) ∈ (SubGrp‘𝑆))
14	12, 13	sylan 580	. . . . . . . 8 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑎 “ 𝑏) ∈ (SubGrp‘𝑆))
15		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
16		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
17	15, 16	gimf1o 19053	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
18		f1of1 6783	. . . . . . . . . . 11 ⊢ (𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆) → 𝑎:(Base‘𝑅)–1-1→(Base‘𝑆))
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎:(Base‘𝑅)–1-1→(Base‘𝑆))
20	15	subgss 18929	. . . . . . . . . 10 ⊢ (𝑏 ∈ (SubGrp‘𝑅) → 𝑏 ⊆ (Base‘𝑅))
21		f1imacnv 6800	. . . . . . . . . 10 ⊢ ((𝑎:(Base‘𝑅)–1-1→(Base‘𝑆) ∧ 𝑏 ⊆ (Base‘𝑅)) → (◡𝑎 “ (𝑎 “ 𝑏)) = 𝑏)
22	19, 20, 21	syl2an 596	. . . . . . . . 9 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (◡𝑎 “ (𝑎 “ 𝑏)) = 𝑏)
23	22	eqcomd 2742	. . . . . . . 8 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → 𝑏 = (◡𝑎 “ (𝑎 “ 𝑏)))
24	14, 23	jca 512	. . . . . . 7 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → ((𝑎 “ 𝑏) ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ (𝑎 “ 𝑏))))
25		eleq1 2825	. . . . . . . 8 ⊢ (𝑐 = (𝑎 “ 𝑏) → (𝑐 ∈ (SubGrp‘𝑆) ↔ (𝑎 “ 𝑏) ∈ (SubGrp‘𝑆)))
26		imaeq2 6009	. . . . . . . . 9 ⊢ (𝑐 = (𝑎 “ 𝑏) → (◡𝑎 “ 𝑐) = (◡𝑎 “ (𝑎 “ 𝑏)))
27	26	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑐 = (𝑎 “ 𝑏) → (𝑏 = (◡𝑎 “ 𝑐) ↔ 𝑏 = (◡𝑎 “ (𝑎 “ 𝑏))))
28	25, 27	anbi12d 631	. . . . . . 7 ⊢ (𝑐 = (𝑎 “ 𝑏) → ((𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ 𝑐)) ↔ ((𝑎 “ 𝑏) ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ (𝑎 “ 𝑏)))))
29	24, 28	syl5ibrcom 246	. . . . . 6 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑐 = (𝑎 “ 𝑏) → (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ 𝑐))))
30	29	impr 455	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ 𝑏))) → (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ 𝑐)))
31		ghmpreima 19030	. . . . . . . . 9 ⊢ ((𝑎 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (◡𝑎 “ 𝑐) ∈ (SubGrp‘𝑅))
32	12, 31	sylan 580	. . . . . . . 8 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (◡𝑎 “ 𝑐) ∈ (SubGrp‘𝑅))
33		f1ofo 6791	. . . . . . . . . . 11 ⊢ (𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆) → 𝑎:(Base‘𝑅)–onto→(Base‘𝑆))
34	17, 33	syl 17	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎:(Base‘𝑅)–onto→(Base‘𝑆))
35	16	subgss 18929	. . . . . . . . . 10 ⊢ (𝑐 ∈ (SubGrp‘𝑆) → 𝑐 ⊆ (Base‘𝑆))
36		foimacnv 6801	. . . . . . . . . 10 ⊢ ((𝑎:(Base‘𝑅)–onto→(Base‘𝑆) ∧ 𝑐 ⊆ (Base‘𝑆)) → (𝑎 “ (◡𝑎 “ 𝑐)) = 𝑐)
37	34, 35, 36	syl2an 596	. . . . . . . . 9 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑎 “ (◡𝑎 “ 𝑐)) = 𝑐)
38	37	eqcomd 2742	. . . . . . . 8 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → 𝑐 = (𝑎 “ (◡𝑎 “ 𝑐)))
39	32, 38	jca 512	. . . . . . 7 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → ((◡𝑎 “ 𝑐) ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ (◡𝑎 “ 𝑐))))
40		eleq1 2825	. . . . . . . 8 ⊢ (𝑏 = (◡𝑎 “ 𝑐) → (𝑏 ∈ (SubGrp‘𝑅) ↔ (◡𝑎 “ 𝑐) ∈ (SubGrp‘𝑅)))
41		imaeq2 6009	. . . . . . . . 9 ⊢ (𝑏 = (◡𝑎 “ 𝑐) → (𝑎 “ 𝑏) = (𝑎 “ (◡𝑎 “ 𝑐)))
42	41	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑏 = (◡𝑎 “ 𝑐) → (𝑐 = (𝑎 “ 𝑏) ↔ 𝑐 = (𝑎 “ (◡𝑎 “ 𝑐))))
43	40, 42	anbi12d 631	. . . . . . 7 ⊢ (𝑏 = (◡𝑎 “ 𝑐) → ((𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ 𝑏)) ↔ ((◡𝑎 “ 𝑐) ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ (◡𝑎 “ 𝑐)))))
44	39, 43	syl5ibrcom 246	. . . . . 6 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑏 = (◡𝑎 “ 𝑐) → (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ 𝑏))))
45	44	impr 455	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ 𝑐))) → (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ 𝑏)))
46	30, 45	impbida 799	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → ((𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ 𝑏)) ↔ (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ 𝑐))))
47	4, 5, 8, 11, 46	en2d 8928	. . 3 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
48	47	exlimiv 1933	. 2 ⊢ (∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
49	3, 48	sylbi 216	1 ⊢ (𝑅 ≃𝑔 𝑆 → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2943  Vcvv 3445   ⊆ wss 3910  ∅c0 4282   class class class wbr 5105  ^◡ccnv 5632   “ cima 5636  –1-1→wf1 6493  –onto→wfo 6494  –1-1-onto→wf1o 6495  ‘cfv 6496  (class class class)co 7357   ≈ cen 8880  Basecbs 17083  SubGrpcsubg 18922   GrpHom cghm 19005   GrpIso cgim 19047   ≃_𝑔 cgic 19048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-subg 18925  df-ghm 19006  df-gim 19049  df-gic 19050

This theorem is referenced by: (None)