Theorem gicsubgen 19208

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 19199	. . 3 ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2		n0 4305	. . 3 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
3	1, 2	bitri 275	. 2 ⊢ (𝑅 ≃𝑔 𝑆 ↔ ∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆))
4		fvexd 6849	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ∈ V)
5		fvexd 6849	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑆) ∈ V)
6		vex 3444	. . . . . 6 ⊢ 𝑎 ∈ V
7	6	imaex 7856	. . . . 5 ⊢ (𝑎 “ 𝑏) ∈ V
8	7	2a1i 12	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑏 ∈ (SubGrp‘𝑅) → (𝑎 “ 𝑏) ∈ V))
9	6	cnvex 7867	. . . . . 6 ⊢ ◡𝑎 ∈ V
10	9	imaex 7856	. . . . 5 ⊢ (◡𝑎 “ 𝑐) ∈ V
11	10	2a1i 12	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑐 ∈ (SubGrp‘𝑆) → (◡𝑎 “ 𝑐) ∈ V))
12		gimghm 19193	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎 ∈ (𝑅 GrpHom 𝑆))
13		ghmima 19166	. . . . . . . . 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 19192	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
18		f1of1 6773	. . . . . . . . . . 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 19057	. . . . . . . . . 10 ⊢ (𝑏 ∈ (SubGrp‘𝑅) → 𝑏 ⊆ (Base‘𝑅))
21		f1imacnv 6790	. . . . . . . . . 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 511	. . . . . . 7 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → ((𝑎 “ 𝑏) ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ (𝑎 “ 𝑏))))
25		eleq1 2824	. . . . . . . 8 ⊢ (𝑐 = (𝑎 “ 𝑏) → (𝑐 ∈ (SubGrp‘𝑆) ↔ (𝑎 “ 𝑏) ∈ (SubGrp‘𝑆)))
26		imaeq2 6015	. . . . . . . . 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 632	. . . . . . 7 ⊢ (𝑐 = (𝑎 “ 𝑏) → ((𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ 𝑐)) ↔ ((𝑎 “ 𝑏) ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ (𝑎 “ 𝑏)))))
29	24, 28	syl5ibrcom 247	. . . . . 6 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑏 ∈ (SubGrp‘𝑅)) → (𝑐 = (𝑎 “ 𝑏) → (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ 𝑐))))
30	29	impr 454	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ 𝑏))) → (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ 𝑐)))
31		ghmpreima 19167	. . . . . . . . 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 6781	. . . . . . . . . . 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 19057	. . . . . . . . . 10 ⊢ (𝑐 ∈ (SubGrp‘𝑆) → 𝑐 ⊆ (Base‘𝑆))
36		foimacnv 6791	. . . . . . . . . 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 511	. . . . . . 7 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → ((◡𝑎 “ 𝑐) ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ (◡𝑎 “ 𝑐))))
40		eleq1 2824	. . . . . . . 8 ⊢ (𝑏 = (◡𝑎 “ 𝑐) → (𝑏 ∈ (SubGrp‘𝑅) ↔ (◡𝑎 “ 𝑐) ∈ (SubGrp‘𝑅)))
41		imaeq2 6015	. . . . . . . . 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 632	. . . . . . 7 ⊢ (𝑏 = (◡𝑎 “ 𝑐) → ((𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ 𝑏)) ↔ ((◡𝑎 “ 𝑐) ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ (◡𝑎 “ 𝑐)))))
44	39, 43	syl5ibrcom 247	. . . . . 6 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑐 ∈ (SubGrp‘𝑆)) → (𝑏 = (◡𝑎 “ 𝑐) → (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ 𝑏))))
45	44	impr 454	. . . . 5 ⊢ ((𝑎 ∈ (𝑅 GrpIso 𝑆) ∧ (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ 𝑐))) → (𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ 𝑏)))
46	30, 45	impbida 800	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → ((𝑏 ∈ (SubGrp‘𝑅) ∧ 𝑐 = (𝑎 “ 𝑏)) ↔ (𝑐 ∈ (SubGrp‘𝑆) ∧ 𝑏 = (◡𝑎 “ 𝑐))))
47	4, 5, 8, 11, 46	en2d 8925	. . 3 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
48	47	exlimiv 1931	. 2 ⊢ (∃𝑎 𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
49	3, 48	sylbi 217	1 ⊢ (𝑅 ≃𝑔 𝑆 → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  Vcvv 3440   ⊆ wss 3901  ∅c0 4285   class class class wbr 5098  ^◡ccnv 5623   “ cima 5627  –1-1→wf1 6489  –onto→wfo 6490  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358   ≈ cen 8880  Basecbs 17136  SubGrpcsubg 19050   GrpHom cghm 19141   GrpIso cgim 19186   ≃_𝑔 cgic 19187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-0g 17361  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18866  df-minusg 18867  df-subg 19053  df-ghm 19142  df-gim 19188  df-gic 19189

This theorem is referenced by: (None)