Theorem gicsubgen 19380

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 19371	. . 3 ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2		n0 4310	. . 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 6903	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ∈ V)
5		fvexd 6903	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑆) ∈ V)
6		vex 3462	. . . . . 6 ⊢ 𝑎 ∈ V
7	6	imaex 7920	. . . . 5 ⊢ (𝑎 “ 𝑏) ∈ V
8	7	2a1i 12	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑏 ∈ (SubGrp‘𝑅) → (𝑎 “ 𝑏) ∈ V))
9	6	cnvex 7931	. . . . . 6 ⊢ ◡𝑎 ∈ V
10	9	imaex 7920	. . . . 5 ⊢ (◡𝑎 “ 𝑐) ∈ V
11	10	2a1i 12	. . . 4 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (𝑐 ∈ (SubGrp‘𝑆) → (◡𝑎 “ 𝑐) ∈ V))
12		gimghm 19365	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎 ∈ (𝑅 GrpHom 𝑆))
13		ghmima 19338	. . . . . . . . 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 2766	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
16		eqid 2766	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
17	15, 16	gimf1o 19364	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → 𝑎:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
18		f1of1 6826	. . . . . . . . . . 11 ⊢ (𝑎:(Base‘𝑅)–1-1-onto→(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 19224	. . . . . . . . . 10 ⊢ (𝑏 ∈ (SubGrp‘𝑅) → 𝑏 ⊆ (Base‘𝑅))
21		f1imacnv 6844	. . . . . . . . . 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 2772	. . . . . . . 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 2854	. . . . . . . 8 ⊢ (𝑐 = (𝑎 “ 𝑏) → (𝑐 ∈ (SubGrp‘𝑆) ↔ (𝑎 “ 𝑏) ∈ (SubGrp‘𝑆)))
26		imaeq2 6063	. . . . . . . . 9 ⊢ (𝑐 = (𝑎 “ 𝑏) → (◡𝑎 “ 𝑐) = (◡𝑎 “ (𝑎 “ 𝑏)))
27	26	eqeq2d 2777	. . . . . . . 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 19339	. . . . . . . . 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 6835	. . . . . . . . . . 11 ⊢ (𝑎:(Base‘𝑅)–1-1-onto→(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 19224	. . . . . . . . . 10 ⊢ (𝑐 ∈ (SubGrp‘𝑆) → 𝑐 ⊆ (Base‘𝑆))
36		foimacnv 6845	. . . . . . . . . 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 2772	. . . . . . . 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 2854	. . . . . . . 8 ⊢ (𝑏 = (◡𝑎 “ 𝑐) → (𝑏 ∈ (SubGrp‘𝑅) ↔ (◡𝑎 “ 𝑐) ∈ (SubGrp‘𝑅)))
41		imaeq2 6063	. . . . . . . . 9 ⊢ (𝑏 = (◡𝑎 “ 𝑐) → (𝑎 “ 𝑏) = (𝑎 “ (◡𝑎 “ 𝑐)))
42	41	eqeq2d 2777	. . . . . . . 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 8994	. . 3 ⊢ (𝑎 ∈ (𝑅 GrpIso 𝑆) → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
48	47	exlimiv 1963	. 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 2146   ≠ wne 2961  Vcvv 3458   ⊆ wss 3908  ∅c0 4289   class class class wbr 5114  ^◡ccnv 5665   “ cima 5669  –1-1→wf1 6540  –onto→wfo 6541  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7423   ≈ cen 8949  Basecbs 17294  SubGrpcsubg 19217   GrpHom cghm 19314   GrpIso cgim 19358   ≃_𝑔 cgic 19359

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-map 8835  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-ress 17316  df-plusg 17348  df-0g 17519  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-grp 19034  df-minusg 19035  df-subg 19220  df-ghm 19315  df-gim 19360  df-gic 19361

This theorem is used by: (None)