readdcan - Intuitionistic Logic Explorer

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Theorem readdcan 8361

Description: Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)

**Assertion**
Ref	Expression
readdcan	$/\$ $/\$

Proof of Theorem readdcan Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-rnegex 8184	. . . 4
2	1	3ad2ant3 1047	. . 3 $/\$ $/\$
3		oveq2 6036	. . . . . . 7
4	3	adantl 277	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
5		simprl 531	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
6	5	recnd 8250	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
7		simpl3 1029	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
8	7	recnd 8250	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
9		simpl1 1027	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
10	9	recnd 8250	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
11	6, 8, 10	addassd 8244	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
12		simpl2 1028	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
13	12	recnd 8250	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
14	6, 8, 13	addassd 8244	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
15	11, 14	eqeq12d 2246	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
16	15	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
17	4, 16	mpbird 167	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
18	8	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
19	6	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
20		addcom 8358	. . . . . . . . 9 $/\$
21	18, 19, 20	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
22		simplrr 538	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
23	21, 22	eqtr3d 2266	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
24	23	oveq1d 6043	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
25	10	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
26		addlid 8360	. . . . . . 7
27	25, 26	syl 14	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
28	24, 27	eqtrd 2264	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
29	23	oveq1d 6043	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
30	13	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
31		addlid 8360	. . . . . . 7
32	30, 31	syl 14	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
33	29, 32	eqtrd 2264	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
34	17, 28, 33	3eqtr3d 2272	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
35	34	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
36	2, 35	rexlimddv 2656	. 2 $/\$ $/\$
37		oveq2 6036	. 2
38	36, 37	impbid1 142	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1005

wceq 1398

wcel 2202

wrex 2512 (class class class)co 6028

cc 8073

cr 8074

cc0 8075

caddc 8078

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-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-ext 2213 ax-resscn 8167 ax-1cn 8168 ax-icn 8170 ax-addcl 8171 ax-mulcl 8173 ax-addcom 8175 ax-addass 8177 ax-i2m1 8180 ax-0id 8183 ax-rnegex 8184

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ral 2516 df-rex 2517 df-v 2805 df-un 3205 df-in 3207 df-ss 3214 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-iota 5293 df-fv 5341 df-ov 6031

This theorem is referenced by: (None)

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