Theorem ssgt0sr 4011

Description: The sum of squares of signed reals is positive if one is nonzero.

Hypotheses

Ref	Expression
ssgt0sr.1	⊢ A ∈ V
ssgt0sr.2	⊢ B ∈ V

Assertion

Ref	Expression
ssgt0sr	⊢ ((A ∈ R ∧ B ∈ R) → (¬ (A = 0R ∧ B = 0R) → 0R <R ((A ·R A) +R (B ·R B))))

Proof of Theorem ssgt0sr

Step	Hyp	Ref	Expression
1		ssgt0sr.1	. . . . . . . . . . . 12 ⊢ A ∈ V
2	1	sqgt0sr 4009	. . . . . . . . . . 11 ⊢ (A ∈ R → (¬ A = 0R → 0R <R (A ·R A)))
3	2	imp 277	. . . . . . . . . 10 ⊢ ((A ∈ R ∧ ¬ A = 0R) → 0R <R (A ·R A))
4	3	adantrr 312	. . . . . . . . 9 ⊢ ((A ∈ R ∧ (¬ A = 0R ∧ B = 0R)) → 0R <R (A ·R A))
5		opreq12 3008	. . . . . . . . . . . . . . 15 ⊢ ((B = 0R ∧ B = 0R) → (B ·R B) = (0R ·R 0R))
6	5	anidms 332	. . . . . . . . . . . . . 14 ⊢ (B = 0R → (B ·R B) = (0R ·R 0R))
7		0r 3983	. . . . . . . . . . . . . . 15 ⊢ 0R ∈ R
8		00sr 4002	. . . . . . . . . . . . . . 15 ⊢ (0R ∈ R → (0R ·R 0R) = 0R)
9	7, 8	ax-mp 6	. . . . . . . . . . . . . 14 ⊢ (0R ·R 0R) = 0R
10	6, 9	syl6eq 1140	. . . . . . . . . . . . 13 ⊢ (B = 0R → (B ·R B) = 0R)
11	10	opreq2d 3013	. . . . . . . . . . . 12 ⊢ (B = 0R → ((A ·R A) +R (B ·R B)) = ((A ·R A) +R 0R))
12		mulclsr 3987	. . . . . . . . . . . . . 14 ⊢ ((A ∈ R ∧ A ∈ R) → (A ·R A) ∈ R)
13	12	anidms 332	. . . . . . . . . . . . 13 ⊢ (A ∈ R → (A ·R A) ∈ R)
14		0idsr 4000	. . . . . . . . . . . . 13 ⊢ ((A ·R A) ∈ R → ((A ·R A) +R 0R) = (A ·R A))
15	13, 14	syl 12	. . . . . . . . . . . 12 ⊢ (A ∈ R → ((A ·R A) +R 0R) = (A ·R A))
16	11, 15	sylan9eqr 1145	. . . . . . . . . . 11 ⊢ ((A ∈ R ∧ B = 0R) → ((A ·R A) +R (B ·R B)) = (A ·R A))
17	16	breq2d 2072	. . . . . . . . . 10 ⊢ ((A ∈ R ∧ B = 0R) → (0R <R ((A ·R A) +R (B ·R B)) ↔ 0R <R (A ·R A)))
18	17	adantrl 311	. . . . . . . . 9 ⊢ ((A ∈ R ∧ (¬ A = 0R ∧ B = 0R)) → (0R <R ((A ·R A) +R (B ·R B)) ↔ 0R <R (A ·R A)))
19	4, 18	mpbird 171	. . . . . . . 8 ⊢ ((A ∈ R ∧ (¬ A = 0R ∧ B = 0R)) → 0R <R ((A ·R A) +R (B ·R B)))
20	19	adantlr 310	. . . . . . 7 ⊢ (((A ∈ R ∧ B ∈ R) ∧ (¬ A = 0R ∧ B = 0R)) → 0R <R ((A ·R A) +R (B ·R B)))
21	20	ancoms 334	. . . . . 6 ⊢ (((¬ A = 0R ∧ B = 0R) ∧ (A ∈ R ∧ B ∈ R)) → 0R <R ((A ·R A) +R (B ·R B)))
22	21	exp31 293	. . . . 5 ⊢ (¬ A = 0R → (B = 0R → ((A ∈ R ∧ B ∈ R) → 0R <R ((A ·R A) +R (B ·R B)))))
23		ssgt0sr.2	. . . . . . . . . 10 ⊢ B ∈ V
24	23	sqgt0sr 4009	. . . . . . . . 9 ⊢ (B ∈ R → (¬ B = 0R → 0R <R (B ·R B)))
25	2, 24	im2anan9 434	. . . . . . . 8 ⊢ ((A ∈ R ∧ B ∈ R) → ((¬ A = 0R ∧ ¬ B = 0R) → (0R <R (A ·R A) ∧ 0R <R (B ·R B))))
26		oprex 3018	. . . . . . . . 9 ⊢ (A ·R A) ∈ V
27		oprex 3018	. . . . . . . . 9 ⊢ (B ·R B) ∈ V
28	26, 27	addgt0sr 4007	. . . . . . . 8 ⊢ ((0R <R (A ·R A) ∧ 0R <R (B ·R B)) → 0R <R ((A ·R A) +R (B ·R B)))
29	25, 28	syl6 23	. . . . . . 7 ⊢ ((A ∈ R ∧ B ∈ R) → ((¬ A = 0R ∧ ¬ B = 0R) → 0R <R ((A ·R A) +R (B ·R B))))
30	29	exp3a 292	. . . . . 6 ⊢ ((A ∈ R ∧ B ∈ R) → (¬ A = 0R → (¬ B = 0R → 0R <R ((A ·R A) +R (B ·R B)))))
31	30	com3l 34	. . . . 5 ⊢ (¬ A = 0R → (¬ B = 0R → ((A ∈ R ∧ B ∈ R) → 0R <R ((A ·R A) +R (B ·R B)))))
32	22, 31	pm2.61d 112	. . . 4 ⊢ (¬ A = 0R → ((A ∈ R ∧ B ∈ R) → 0R <R ((A ·R A) +R (B ·R B))))
33	32	com12 13	. . 3 ⊢ ((A ∈ R ∧ B ∈ R) → (¬ A = 0R → 0R <R ((A ·R A) +R (B ·R B))))
34	24	imp 277	. . . . . . . 8 ⊢ ((B ∈ R ∧ ¬ B = 0R) → 0R <R (B ·R B))
35	34	adantrl 311	. . . . . . 7 ⊢ ((B ∈ R ∧ (A = 0R ∧ ¬ B = 0R)) → 0R <R (B ·R B))
36		opreq12 3008	. . . . . . . . . . . . 13 ⊢ ((A = 0R ∧ A = 0R) → (A ·R A) = (0R ·R 0R))
37	36	anidms 332	. . . . . . . . . . . 12 ⊢ (A = 0R → (A ·R A) = (0R ·R 0R))
38	37, 9	syl6eq 1140	. . . . . . . . . . 11 ⊢ (A = 0R → (A ·R A) = 0R)
39	38	opreq1d 3012	. . . . . . . . . 10 ⊢ (A = 0R → ((A ·R A) +R (B ·R B)) = (0R +R (B ·R B)))
40		mulclsr 3987	. . . . . . . . . . . 12 ⊢ ((B ∈ R ∧ B ∈ R) → (B ·R B) ∈ R)
41	40	anidms 332	. . . . . . . . . . 11 ⊢ (B ∈ R → (B ·R B) ∈ R)
42		0idsr 4000	. . . . . . . . . . . 12 ⊢ ((B ·R B) ∈ R → ((B ·R B) +R 0R) = (B ·R B))
43	7	elisseti 1355	. . . . . . . . . . . . 13 ⊢ 0R ∈ V
44	43, 27	addcomsr 3990	. . . . . . . . . . . 12 ⊢ (0R +R (B ·R B)) = ((B ·R B) +R 0R)
45	42, 44	syl5eq 1136	. . . . . . . . . . 11 ⊢ ((B ·R B) ∈ R → (0R +R (B ·R B)) = (B ·R B))
46	41, 45	syl 12	. . . . . . . . . 10 ⊢ (B ∈ R → (0R +R (B ·R B)) = (B ·R B))
47	39, 46	sylan9eqr 1145	. . . . . . . . 9 ⊢ ((B ∈ R ∧ A = 0R) → ((A ·R A) +R (B ·R B)) = (B ·R B))
48	47	breq2d 2072	. . . . . . . 8 ⊢ ((B ∈ R ∧ A = 0R) → (0R <R ((A ·R A) +R (B ·R B)) ↔ 0R <R (B ·R B)))
49	48	adantrr 312	. . . . . . 7 ⊢ ((B ∈ R ∧ (A = 0R ∧ ¬ B = 0R)) → (0R <R ((A ·R A) +R (B ·R B)) ↔ 0R <R (B ·R B)))
50	35, 49	mpbird 171	. . . . . 6 ⊢ ((B ∈ R ∧ (A = 0R ∧ ¬ B = 0R)) → 0R <R ((A ·R A) +R (B ·R B)))
51	50	adantll 309	. . . . 5 ⊢ (((A ∈ R ∧ B ∈ R) ∧ (A = 0R ∧ ¬ B = 0R)) → 0R <R ((A ·R A) +R (B ·R B)))
52	51	exp32 294	. . . 4 ⊢ ((A ∈ R ∧ B ∈ R) → (A = 0R → (¬ B = 0R → 0R <R ((A ·R A) +R (B ·R B)))))
53	52, 30	pm2.61d 112	. . 3 ⊢ ((A ∈ R ∧ B ∈ R) → (¬ B = 0R → 0R <R ((A ·R A) +R (B ·R B))))
54	33, 53	jaod 329	. 2 ⊢ ((A ∈ R ∧ B ∈ R) → ((¬ A = 0R ∨ ¬ B = 0R) → 0R <R ((A ·R A) +R (B ·R B))))
55		ianor 253	. 2 ⊢ (¬ (A = 0R ∧ B = 0R) ↔ (¬ A = 0R ∨ ¬ B = 0R))
56	54, 55	syl5ib 181	1 ⊢ ((A ∈ R ∧ B ∈ R) → (¬ (A = 0R ∧ B = 0R) → 0R <R ((A ·R A) +R (B ·R B))))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054  (class class class)co 3001  Rcnr 3787  0_Rc0r 3788   +_R cplr 3791   ·_R cmr 3792   <_R cltr 3793

This theorem is referenced by:  axrecex 4079

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967

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