Theorem rerecapb 9022

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 8909	. . . 4 |- ( ( A e. RR /\ A # 0 ) -> ( 1 / A ) e. RR )
2		recn 8164	. . . . 5 |- ( A e. RR -> A e. CC )
3		recidap 8865	. . . . 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 6025	. . . . . 6 |- ( x = ( 1 / A ) -> ( A x. x ) = ( A x. ( 1 / A ) ) )
6	5	eqeq1d 2240	. . . . 5 |- ( x = ( 1 / A ) -> ( ( A x. x ) = 1 <-> ( A x. ( 1 / A ) ) = 1 ) )
7	6	rspcev 2910	. . . 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 8123	. . . 4 |- RR C_ CC
11		ssrexv 3292	. . . 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 8850	. . . 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 1397   e. wcel 2202   E.wrex 2511   C_ wss 3200   class class class wbr 4088  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   1c1 8032   x. cmul 8036   # cap 8760   / cdiv 8851

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852

This theorem is referenced by: (None)