Theorem rerecapb 8887

Description: A real number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 18-Jan-2025.)

Assertion

Ref	Expression
rerecapb	|- ( A e. RR -> ( A # 0 <-> E. x e. RR ( A x. x ) = 1 ) )

Distinct variable group:   x, A

Proof of Theorem rerecapb

Step	Hyp	Ref	Expression
1		rerecclap 8774	. . . 4 |- ( ( A e. RR /\ A # 0 ) -> ( 1 / A ) e. RR )
2		recn 8029	. . . . 5 |- ( A e. RR -> A e. CC )
3		recidap 8730	. . . . 5 |- ( ( A e. CC /\ A # 0 ) -> ( A x. ( 1 / A ) ) = 1 )
4	2, 3	sylan 283	. . . 4 |- ( ( A e. RR /\ A # 0 ) -> ( A x. ( 1 / A ) ) = 1 )
5		oveq2 5933	. . . . . 6 |- ( x = ( 1 / A ) -> ( A x. x ) = ( A x. ( 1 / A ) ) )
6	5	eqeq1d 2205	. . . . 5 |- ( x = ( 1 / A ) -> ( ( A x. x ) = 1 <-> ( A x. ( 1 / A ) ) = 1 ) )
7	6	rspcev 2868	. . . 4 |- ( ( ( 1 / A ) e. RR /\ ( A x. ( 1 / A ) ) = 1 ) -> E. x e. RR ( A x. x ) = 1 )
8	1, 4, 7	syl2anc 411	. . 3 |- ( ( A e. RR /\ A # 0 ) -> E. x e. RR ( A x. x ) = 1 )
9	8	ex 115	. 2 |- ( A e. RR -> ( A # 0 -> E. x e. RR ( A x. x ) = 1 ) )
10		ax-resscn 7988	. . . 4 |- RR C_ CC
11		ssrexv 3249	. . . 4 |- ( RR C_ CC -> ( E. x e. RR ( A x. x ) = 1 -> E. x e. CC ( A x. x ) = 1 ) )
12	10, 11	ax-mp 5	. . 3 |- ( E. x e. RR ( A x. x ) = 1 -> E. x e. CC ( A x. x ) = 1 )
13		recapb 8715	. . . 4 |- ( A e. CC -> ( A # 0 <-> E. x e. CC ( A x. x ) = 1 ) )
14	13	biimprd 158	. . 3 |- ( A e. CC -> ( E. x e. CC ( A x. x ) = 1 -> A # 0 ) )
15	2, 12, 14	syl2im 38	. 2 |- ( A e. RR -> ( E. x e. RR ( A x. x ) = 1 -> A # 0 ) )
16	9, 15	impbid 129	1 |- ( A e. RR -> ( A # 0 <-> E. x e. RR ( A x. x ) = 1 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   E.wrex 2476   C_ wss 3157   class class class wbr 4034  (class class class)co 5925   CCcc 7894   RRcr 7895   0cc0 7896   1c1 7897   x. cmul 7901   # cap 8625   / cdiv 8716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717

This theorem is referenced by: (None)