Metamath Proof Explorer


Theorem normlem7

Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of Beran p. 97. (Contributed by NM, 11-Aug-1999) (New usage is discouraged.)

Ref Expression
Hypotheses normlem1.1 S
normlem1.2 F
normlem1.3 G
normlem7.4 S = 1
Assertion normlem7 S F ih G + S G ih F 2 G ih G F ih F

Proof

Step Hyp Ref Expression
1 normlem1.1 S
2 normlem1.2 F
3 normlem1.3 G
4 normlem7.4 S = 1
5 eqid S F ih G + S G ih F = S F ih G + S G ih F
6 1 2 3 5 normlem2 S F ih G + S G ih F
7 1 cjcli S
8 2 3 hicli F ih G
9 7 8 mulcli S F ih G
10 3 2 hicli G ih F
11 1 10 mulcli S G ih F
12 9 11 addcli S F ih G + S G ih F
13 12 negrebi S F ih G + S G ih F S F ih G + S G ih F
14 6 13 mpbi S F ih G + S G ih F
15 14 leabsi S F ih G + S G ih F S F ih G + S G ih F
16 12 absnegi S F ih G + S G ih F = S F ih G + S G ih F
17 15 16 breqtrri S F ih G + S G ih F S F ih G + S G ih F
18 eqid G ih G = G ih G
19 eqid F ih F = F ih F
20 1 2 3 5 18 19 4 normlem6 S F ih G + S G ih F 2 G ih G F ih F
21 12 negcli S F ih G + S G ih F
22 21 abscli S F ih G + S G ih F
23 2re 2
24 hiidge0 G 0 G ih G
25 hiidrcl G G ih G
26 3 25 ax-mp G ih G
27 26 sqrtcli 0 G ih G G ih G
28 3 24 27 mp2b G ih G
29 hiidge0 F 0 F ih F
30 hiidrcl F F ih F
31 2 30 ax-mp F ih F
32 31 sqrtcli 0 F ih F F ih F
33 2 29 32 mp2b F ih F
34 28 33 remulcli G ih G F ih F
35 23 34 remulcli 2 G ih G F ih F
36 14 22 35 letri S F ih G + S G ih F S F ih G + S G ih F S F ih G + S G ih F 2 G ih G F ih F S F ih G + S G ih F 2 G ih G F ih F
37 17 20 36 mp2an S F ih G + S G ih F 2 G ih G F ih F