Theorem isnumbasgrplem2 39702

Description: If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)

Assertion

Ref	Expression
isnumbasgrplem2	⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

Proof of Theorem isnumbasgrplem2

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

Step	Hyp	Ref	Expression
1		basfn 16502	. . 3 ⊢ Base Fn V
2		ssv 3990	. . 3 ⊢ Grp ⊆ V
3		fvelimab 6736	. . 3 ⊢ ((Base Fn V ∧ Grp ⊆ V) → ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))))
4	1, 2, 3	mp2an 690	. 2 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
5		harcl 9024	. . . . . 6 ⊢ (har‘𝑆) ∈ On
6		onenon 9377	. . . . . 6 ⊢ ((har‘𝑆) ∈ On → (har‘𝑆) ∈ dom card)
7	5, 6	ax-mp 5	. . . . 5 ⊢ (har‘𝑆) ∈ dom card
8		xpnum 9379	. . . . 5 ⊢ (((har‘𝑆) ∈ dom card ∧ (har‘𝑆) ∈ dom card) → ((har‘𝑆) × (har‘𝑆)) ∈ dom card)
9	7, 7, 8	mp2an 690	. . . 4 ⊢ ((har‘𝑆) × (har‘𝑆)) ∈ dom card
10		ssun1 4147	. . . . . . . 8 ⊢ 𝑆 ⊆ (𝑆 ∪ (har‘𝑆))
11		simpr 487	. . . . . . . 8 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
12	10, 11	sseqtrrid 4019	. . . . . . 7 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
13		fvex 6682	. . . . . . . 8 ⊢ (Base‘𝑥) ∈ V
14	13	ssex 5224	. . . . . . 7 ⊢ (𝑆 ⊆ (Base‘𝑥) → 𝑆 ∈ V)
15	12, 14	syl 17	. . . . . 6 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ V)
16	7	a1i 11	. . . . . 6 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ∈ dom card)
17		simp1l 1193	. . . . . . . 8 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑥 ∈ Grp)
18	12	3ad2ant1 1129	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑆 ⊆ (Base‘𝑥))
19		simp2 1133	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑎 ∈ 𝑆)
20	18, 19	sseldd 3967	. . . . . . . 8 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑎 ∈ (Base‘𝑥))
21		ssun2 4148	. . . . . . . . . . 11 ⊢ (har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆))
22	21, 11	sseqtrrid 4019	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
23	22	3ad2ant1 1129	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
24		simp3 1134	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (har‘𝑆))
25	23, 24	sseldd 3967	. . . . . . . 8 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (Base‘𝑥))
26		eqid 2821	. . . . . . . . 9 ⊢ (Base‘𝑥) = (Base‘𝑥)
27		eqid 2821	. . . . . . . . 9 ⊢ (+g‘𝑥) = (+g‘𝑥)
28	26, 27	grpcl 18110	. . . . . . . 8 ⊢ ((𝑥 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑐 ∈ (Base‘𝑥)) → (𝑎(+g‘𝑥)𝑐) ∈ (Base‘𝑥))
29	17, 20, 25, 28	syl3anc 1367	. . . . . . 7 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (𝑎(+g‘𝑥)𝑐) ∈ (Base‘𝑥))
30		simp1r 1194	. . . . . . 7 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
31	29, 30	eleqtrd 2915	. . . . . 6 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (𝑎(+g‘𝑥)𝑐) ∈ (𝑆 ∪ (har‘𝑆)))
32		simplll 773	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑥 ∈ Grp)
33	22	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
34		simprl 769	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (har‘𝑆))
35	33, 34	sseldd 3967	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (Base‘𝑥))
36		simprr 771	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (har‘𝑆))
37	33, 36	sseldd 3967	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (Base‘𝑥))
38	12	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
39		simplr 767	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎 ∈ 𝑆)
40	38, 39	sseldd 3967	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎 ∈ (Base‘𝑥))
41	26, 27	grplcan 18160	. . . . . . 7 ⊢ ((𝑥 ∈ Grp ∧ (𝑐 ∈ (Base‘𝑥) ∧ 𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥))) → ((𝑎(+g‘𝑥)𝑐) = (𝑎(+g‘𝑥)𝑑) ↔ 𝑐 = 𝑑))
42	32, 35, 37, 40, 41	syl13anc 1368	. . . . . 6 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → ((𝑎(+g‘𝑥)𝑐) = (𝑎(+g‘𝑥)𝑑) ↔ 𝑐 = 𝑑))
43		simplll 773	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑥 ∈ Grp)
44	12	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑆 ⊆ (Base‘𝑥))
45		simprr 771	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑑 ∈ 𝑆)
46	44, 45	sseldd 3967	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑑 ∈ (Base‘𝑥))
47		simprl 769	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑎 ∈ 𝑆)
48	44, 47	sseldd 3967	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑎 ∈ (Base‘𝑥))
49	22	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
50		simplr 767	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑏 ∈ (har‘𝑆))
51	49, 50	sseldd 3967	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑏 ∈ (Base‘𝑥))
52	26, 27	grprcan 18136	. . . . . . 7 ⊢ ((𝑥 ∈ Grp ∧ (𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑏 ∈ (Base‘𝑥))) → ((𝑑(+g‘𝑥)𝑏) = (𝑎(+g‘𝑥)𝑏) ↔ 𝑑 = 𝑎))
53	43, 46, 48, 51, 52	syl13anc 1368	. . . . . 6 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → ((𝑑(+g‘𝑥)𝑏) = (𝑎(+g‘𝑥)𝑏) ↔ 𝑑 = 𝑎))
54		harndom 9027	. . . . . . 7 ⊢ ¬ (har‘𝑆) ≼ 𝑆
55	54	a1i 11	. . . . . 6 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → ¬ (har‘𝑆) ≼ 𝑆)
56	15, 16, 16, 31, 42, 53, 55	unxpwdom3 39693	. . . . 5 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ≼* ((har‘𝑆) × (har‘𝑆)))
57		wdomnumr 9489	. . . . . 6 ⊢ (((har‘𝑆) × (har‘𝑆)) ∈ dom card → (𝑆 ≼* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))))
58	9, 57	ax-mp 5	. . . . 5 ⊢ (𝑆 ≼* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
59	56, 58	sylib 220	. . . 4 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
60		numdom 9463	. . . 4 ⊢ ((((har‘𝑆) × (har‘𝑆)) ∈ dom card ∧ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))) → 𝑆 ∈ dom card)
61	9, 59, 60	sylancr 589	. . 3 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ dom card)
62	61	rexlimiva 3281	. 2 ⊢ (∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)) → 𝑆 ∈ dom card)
63	4, 62	sylbi 219	1 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  Vcvv 3494   ∪ cun 3933   ⊆ wss 3935   class class class wbr 5065   × cxp 5552  dom cdm 5554   “ cima 5557  Oncon0 6190   Fn wfn 6349  ‘cfv 6354  (class class class)co 7155   ≼ cdom 8506  harchar 9019   ≼^* cwdom 9020  cardccrd 9363  Basecbs 16482  +_gcplusg 16564  Grpcgrp 18102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-omul 8106  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-oi 8973  df-har 9021  df-wdom 9022  df-card 9367  df-acn 9370  df-slot 16486  df-base 16488  df-0g 16714  df-mgm 17851  df-sgrp 17900  df-mnd 17911  df-grp 18105  df-minusg 18106

This theorem is referenced by:  isnumbasabl  39704  isnumbasgrp  39705