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Theorem cjap 11672

Description: Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)

**Assertion**
Ref	Expression
cjap	$/\$ # #

Proof of Theorem cjap Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 8322	. . 3
2	1	adantr 276	. 2 $/\$
3		cnre 8322	. . . . . 6
4	3	ad3antlr 497	. . . . 5 $/\$ $/\$ $/\$ $/\$
5		simplrr 542	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
6	5	ad2antrr 492	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7	6	recnd 8354	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simplrr 542	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9	8	recnd 8354	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		apneg 8939	. . . . . . . . . 10 $/\$ # #
11	7, 9, 10	syl2anc 415	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
12	11	orbi2d 802	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $\/$ # # $\/$ #
13		simpllr 540	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		simpr 110	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15	13, 14	breq12d 4143	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
16		simplrl 541	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
17	16	ad2antrr 492	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18		simplrl 541	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19		apreim 8931	. . . . . . . . . 10 $/\$ $/\$ $/\$ # # $\/$ #
20	17, 6, 18, 8, 19	syl22anc 1279	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
21	15, 20	bitrd 188	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
22	13	fveq2d 5699	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		cjreim 11669	. . . . . . . . . . . 12 $/\$
24	17, 6, 23	syl2anc 415	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	22, 24	eqtrd 2271	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	14	fveq2d 5699	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		cjreim 11669	. . . . . . . . . . . 12 $/\$
28	18, 8, 27	syl2anc 415	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	26, 28	eqtrd 2271	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	25, 29	breq12d 4143	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
31	17	recnd 8354	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		ax-icn 8274	. . . . . . . . . . . 12
33	32	a1i 9	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34		submul2 8726	. . . . . . . . . . 11 $/\$ $/\$
35	31, 33, 7, 34	syl3anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	18	recnd 8354	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		submul2 8726	. . . . . . . . . . 11 $/\$ $/\$
38	36, 33, 9, 37	syl3anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	35, 38	breq12d 4143	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
40	6	renegcld 8707	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	8	renegcld 8707	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		apreim 8931	. . . . . . . . . 10 $/\$ $/\$ $/\$ # # $\/$ #
43	17, 40, 18, 41, 42	syl22anc 1279	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
44	30, 39, 43	3bitrd 214	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
45	12, 21, 44	3bitr4rd 221	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
46	45	ex 115	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
47	46	rexlimdvva 2676	. . . . 5 $/\$ $/\$ $/\$ $/\$ # #
48	4, 47	mpd 13	. . . 4 $/\$ $/\$ $/\$ $/\$ # #
49	48	ex 115	. . 3 $/\$ $/\$ $/\$ # #
50	49	rexlimdvva 2676	. 2 $/\$ # #
51	2, 50	mpd 13	1 $/\$ # #

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720

wceq 1402

wcel 2209

wrex 2529 class class class wbr 4130

cfv 5377 (class class class)co 6085

cc 8177

cr 8178

ci 8181

caddc 8182

cmul 8184

cmin 8497

cneg 8498 # cap 8909

ccj 11604

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-mpt 4194 df-id 4438 df-po 4441 df-iso 4442 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-2 9363 df-cj 11607 df-re 11608 df-im 11609

This theorem is used by: cjap0 11673

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