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Theorem nrmmetd 23118
Description: Show that a group norm generates a metric. Part of Definition 2.2-1 of [Kreyszig] p. 58. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
nrmmetd.x 𝑋 = (Base‘𝐺)
nrmmetd.m = (-g𝐺)
nrmmetd.z 0 = (0g𝐺)
nrmmetd.g (𝜑𝐺 ∈ Grp)
nrmmetd.f (𝜑𝐹:𝑋⟶ℝ)
nrmmetd.1 ((𝜑𝑥𝑋) → ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ))
nrmmetd.2 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
Assertion
Ref Expression
nrmmetd (𝜑 → (𝐹 ) ∈ (Met‘𝑋))
Distinct variable groups:   𝑥,𝑦,   𝑥, 0 ,𝑦   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐺(𝑥,𝑦)

Proof of Theorem nrmmetd
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nrmmetd.f . . 3 (𝜑𝐹:𝑋⟶ℝ)
2 nrmmetd.g . . . 4 (𝜑𝐺 ∈ Grp)
3 nrmmetd.x . . . . 5 𝑋 = (Base‘𝐺)
4 nrmmetd.m . . . . 5 = (-g𝐺)
53, 4grpsubf 18123 . . . 4 (𝐺 ∈ Grp → :(𝑋 × 𝑋)⟶𝑋)
62, 5syl 17 . . 3 (𝜑 :(𝑋 × 𝑋)⟶𝑋)
7 fco 6530 . . 3 ((𝐹:𝑋⟶ℝ ∧ :(𝑋 × 𝑋)⟶𝑋) → (𝐹 ):(𝑋 × 𝑋)⟶ℝ)
81, 6, 7syl2anc 584 . 2 (𝜑 → (𝐹 ):(𝑋 × 𝑋)⟶ℝ)
9 opelxpi 5591 . . . . . . . 8 ((𝑎𝑋𝑏𝑋) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋))
10 fvco3 6759 . . . . . . . 8 (( :(𝑋 × 𝑋)⟶𝑋 ∧ ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋)) → ((𝐹 )‘⟨𝑎, 𝑏⟩) = (𝐹‘( ‘⟨𝑎, 𝑏⟩)))
116, 9, 10syl2an 595 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹 )‘⟨𝑎, 𝑏⟩) = (𝐹‘( ‘⟨𝑎, 𝑏⟩)))
12 df-ov 7153 . . . . . . 7 (𝑎(𝐹 )𝑏) = ((𝐹 )‘⟨𝑎, 𝑏⟩)
13 df-ov 7153 . . . . . . . 8 (𝑎 𝑏) = ( ‘⟨𝑎, 𝑏⟩)
1413fveq2i 6672 . . . . . . 7 (𝐹‘(𝑎 𝑏)) = (𝐹‘( ‘⟨𝑎, 𝑏⟩))
1511, 12, 143eqtr4g 2886 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(𝐹 )𝑏) = (𝐹‘(𝑎 𝑏)))
1615eqeq1d 2828 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎(𝐹 )𝑏) = 0 ↔ (𝐹‘(𝑎 𝑏)) = 0))
17 nrmmetd.1 . . . . . . 7 ((𝜑𝑥𝑋) → ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ))
1817ralrimiva 3187 . . . . . 6 (𝜑 → ∀𝑥𝑋 ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ))
193, 4grpsubcl 18124 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑎𝑋𝑏𝑋) → (𝑎 𝑏) ∈ 𝑋)
20193expb 1114 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑎𝑋𝑏𝑋)) → (𝑎 𝑏) ∈ 𝑋)
212, 20sylan 580 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝑎 𝑏) ∈ 𝑋)
22 fveq2 6669 . . . . . . . . 9 (𝑥 = (𝑎 𝑏) → (𝐹𝑥) = (𝐹‘(𝑎 𝑏)))
2322eqeq1d 2828 . . . . . . . 8 (𝑥 = (𝑎 𝑏) → ((𝐹𝑥) = 0 ↔ (𝐹‘(𝑎 𝑏)) = 0))
24 eqeq1 2830 . . . . . . . 8 (𝑥 = (𝑎 𝑏) → (𝑥 = 0 ↔ (𝑎 𝑏) = 0 ))
2523, 24bibi12d 347 . . . . . . 7 (𝑥 = (𝑎 𝑏) → (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝐹‘(𝑎 𝑏)) = 0 ↔ (𝑎 𝑏) = 0 )))
2625rspccva 3626 . . . . . 6 ((∀𝑥𝑋 ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑎 𝑏) ∈ 𝑋) → ((𝐹‘(𝑎 𝑏)) = 0 ↔ (𝑎 𝑏) = 0 ))
2718, 21, 26syl2an2r 681 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹‘(𝑎 𝑏)) = 0 ↔ (𝑎 𝑏) = 0 ))
28 nrmmetd.z . . . . . . . 8 0 = (0g𝐺)
293, 28, 4grpsubeq0 18130 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑎𝑋𝑏𝑋) → ((𝑎 𝑏) = 0𝑎 = 𝑏))
30293expb 1114 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎 𝑏) = 0𝑎 = 𝑏))
312, 30sylan 580 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎 𝑏) = 0𝑎 = 𝑏))
3216, 27, 313bitrd 306 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎(𝐹 )𝑏) = 0 ↔ 𝑎 = 𝑏))
331adantr 481 . . . . . . . . 9 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → 𝐹:𝑋⟶ℝ)
3421adantrr 713 . . . . . . . . 9 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝑎 𝑏) ∈ 𝑋)
3533, 34ffvelrnd 6850 . . . . . . . 8 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝐹‘(𝑎 𝑏)) ∈ ℝ)
362adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → 𝐺 ∈ Grp)
37 simprll 775 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → 𝑎𝑋)
38 simprr 769 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → 𝑐𝑋)
393, 4grpsubcl 18124 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑎𝑋𝑐𝑋) → (𝑎 𝑐) ∈ 𝑋)
4036, 37, 38, 39syl3anc 1365 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝑎 𝑐) ∈ 𝑋)
4133, 40ffvelrnd 6850 . . . . . . . . 9 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝐹‘(𝑎 𝑐)) ∈ ℝ)
42 simprlr 776 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → 𝑏𝑋)
433, 4grpsubcl 18124 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑏𝑋𝑐𝑋) → (𝑏 𝑐) ∈ 𝑋)
4436, 42, 38, 43syl3anc 1365 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝑏 𝑐) ∈ 𝑋)
4533, 44ffvelrnd 6850 . . . . . . . . 9 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝐹‘(𝑏 𝑐)) ∈ ℝ)
4641, 45readdcld 10664 . . . . . . . 8 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → ((𝐹‘(𝑎 𝑐)) + (𝐹‘(𝑏 𝑐))) ∈ ℝ)
473, 4grpsubcl 18124 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑐𝑋𝑎𝑋) → (𝑐 𝑎) ∈ 𝑋)
4836, 38, 37, 47syl3anc 1365 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝑐 𝑎) ∈ 𝑋)
4933, 48ffvelrnd 6850 . . . . . . . . 9 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝐹‘(𝑐 𝑎)) ∈ ℝ)
503, 4grpsubcl 18124 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑐𝑋𝑏𝑋) → (𝑐 𝑏) ∈ 𝑋)
5136, 38, 42, 50syl3anc 1365 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝑐 𝑏) ∈ 𝑋)
5233, 51ffvelrnd 6850 . . . . . . . . 9 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝐹‘(𝑐 𝑏)) ∈ ℝ)
5349, 52readdcld 10664 . . . . . . . 8 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → ((𝐹‘(𝑐 𝑎)) + (𝐹‘(𝑐 𝑏))) ∈ ℝ)
543, 4grpnnncan2 18141 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝑎 𝑐) (𝑏 𝑐)) = (𝑎 𝑏))
5536, 37, 42, 38, 54syl13anc 1366 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → ((𝑎 𝑐) (𝑏 𝑐)) = (𝑎 𝑏))
5655fveq2d 6673 . . . . . . . . 9 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝐹‘((𝑎 𝑐) (𝑏 𝑐))) = (𝐹‘(𝑎 𝑏)))
57 nrmmetd.2 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
5857ralrimivva 3196 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
5958adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
60 fvoveq1 7173 . . . . . . . . . . . 12 (𝑥 = (𝑎 𝑐) → (𝐹‘(𝑥 𝑦)) = (𝐹‘((𝑎 𝑐) 𝑦)))
61 fveq2 6669 . . . . . . . . . . . . 13 (𝑥 = (𝑎 𝑐) → (𝐹𝑥) = (𝐹‘(𝑎 𝑐)))
6261oveq1d 7165 . . . . . . . . . . . 12 (𝑥 = (𝑎 𝑐) → ((𝐹𝑥) + (𝐹𝑦)) = ((𝐹‘(𝑎 𝑐)) + (𝐹𝑦)))
6360, 62breq12d 5076 . . . . . . . . . . 11 (𝑥 = (𝑎 𝑐) → ((𝐹‘(𝑥 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)) ↔ (𝐹‘((𝑎 𝑐) 𝑦)) ≤ ((𝐹‘(𝑎 𝑐)) + (𝐹𝑦))))
64 oveq2 7158 . . . . . . . . . . . . 13 (𝑦 = (𝑏 𝑐) → ((𝑎 𝑐) 𝑦) = ((𝑎 𝑐) (𝑏 𝑐)))
6564fveq2d 6673 . . . . . . . . . . . 12 (𝑦 = (𝑏 𝑐) → (𝐹‘((𝑎 𝑐) 𝑦)) = (𝐹‘((𝑎 𝑐) (𝑏 𝑐))))
66 fveq2 6669 . . . . . . . . . . . . 13 (𝑦 = (𝑏 𝑐) → (𝐹𝑦) = (𝐹‘(𝑏 𝑐)))
6766oveq2d 7166 . . . . . . . . . . . 12 (𝑦 = (𝑏 𝑐) → ((𝐹‘(𝑎 𝑐)) + (𝐹𝑦)) = ((𝐹‘(𝑎 𝑐)) + (𝐹‘(𝑏 𝑐))))
6865, 67breq12d 5076 . . . . . . . . . . 11 (𝑦 = (𝑏 𝑐) → ((𝐹‘((𝑎 𝑐) 𝑦)) ≤ ((𝐹‘(𝑎 𝑐)) + (𝐹𝑦)) ↔ (𝐹‘((𝑎 𝑐) (𝑏 𝑐))) ≤ ((𝐹‘(𝑎 𝑐)) + (𝐹‘(𝑏 𝑐)))))
6963, 68rspc2va 3638 . . . . . . . . . 10 ((((𝑎 𝑐) ∈ 𝑋 ∧ (𝑏 𝑐) ∈ 𝑋) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))) → (𝐹‘((𝑎 𝑐) (𝑏 𝑐))) ≤ ((𝐹‘(𝑎 𝑐)) + (𝐹‘(𝑏 𝑐))))
7040, 44, 59, 69syl21anc 835 . . . . . . . . 9 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝐹‘((𝑎 𝑐) (𝑏 𝑐))) ≤ ((𝐹‘(𝑎 𝑐)) + (𝐹‘(𝑏 𝑐))))
7156, 70eqbrtrrd 5087 . . . . . . . 8 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝐹‘(𝑎 𝑏)) ≤ ((𝐹‘(𝑎 𝑐)) + (𝐹‘(𝑏 𝑐))))
72 eleq1w 2900 . . . . . . . . . . . . . 14 (𝑏 = 𝑐 → (𝑏𝑋𝑐𝑋))
7372anbi2d 628 . . . . . . . . . . . . 13 (𝑏 = 𝑐 → ((𝑎𝑋𝑏𝑋) ↔ (𝑎𝑋𝑐𝑋)))
7473anbi2d 628 . . . . . . . . . . . 12 (𝑏 = 𝑐 → ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) ↔ (𝜑 ∧ (𝑎𝑋𝑐𝑋))))
75 oveq2 7158 . . . . . . . . . . . . . 14 (𝑏 = 𝑐 → (𝑎 𝑏) = (𝑎 𝑐))
7675fveq2d 6673 . . . . . . . . . . . . 13 (𝑏 = 𝑐 → (𝐹‘(𝑎 𝑏)) = (𝐹‘(𝑎 𝑐)))
77 fvoveq1 7173 . . . . . . . . . . . . 13 (𝑏 = 𝑐 → (𝐹‘(𝑏 𝑎)) = (𝐹‘(𝑐 𝑎)))
7876, 77breq12d 5076 . . . . . . . . . . . 12 (𝑏 = 𝑐 → ((𝐹‘(𝑎 𝑏)) ≤ (𝐹‘(𝑏 𝑎)) ↔ (𝐹‘(𝑎 𝑐)) ≤ (𝐹‘(𝑐 𝑎))))
7974, 78imbi12d 346 . . . . . . . . . . 11 (𝑏 = 𝑐 → (((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝐹‘(𝑎 𝑏)) ≤ (𝐹‘(𝑏 𝑎))) ↔ ((𝜑 ∧ (𝑎𝑋𝑐𝑋)) → (𝐹‘(𝑎 𝑐)) ≤ (𝐹‘(𝑐 𝑎)))))
802adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → 𝐺 ∈ Grp)
813, 28grpidcl 18076 . . . . . . . . . . . . . 14 (𝐺 ∈ Grp → 0𝑋)
8280, 81syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → 0𝑋)
83 simprr 769 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → 𝑏𝑋)
84 simprl 767 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → 𝑎𝑋)
853, 4grpsubcl 18124 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑏𝑋𝑎𝑋) → (𝑏 𝑎) ∈ 𝑋)
8680, 83, 84, 85syl3anc 1365 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝑏 𝑎) ∈ 𝑋)
8758adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
88 fvoveq1 7173 . . . . . . . . . . . . . . 15 (𝑥 = 0 → (𝐹‘(𝑥 𝑦)) = (𝐹‘( 0 𝑦)))
89 fveq2 6669 . . . . . . . . . . . . . . . 16 (𝑥 = 0 → (𝐹𝑥) = (𝐹0 ))
9089oveq1d 7165 . . . . . . . . . . . . . . 15 (𝑥 = 0 → ((𝐹𝑥) + (𝐹𝑦)) = ((𝐹0 ) + (𝐹𝑦)))
9188, 90breq12d 5076 . . . . . . . . . . . . . 14 (𝑥 = 0 → ((𝐹‘(𝑥 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)) ↔ (𝐹‘( 0 𝑦)) ≤ ((𝐹0 ) + (𝐹𝑦))))
92 oveq2 7158 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑏 𝑎) → ( 0 𝑦) = ( 0 (𝑏 𝑎)))
9392fveq2d 6673 . . . . . . . . . . . . . . 15 (𝑦 = (𝑏 𝑎) → (𝐹‘( 0 𝑦)) = (𝐹‘( 0 (𝑏 𝑎))))
94 fveq2 6669 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑏 𝑎) → (𝐹𝑦) = (𝐹‘(𝑏 𝑎)))
9594oveq2d 7166 . . . . . . . . . . . . . . 15 (𝑦 = (𝑏 𝑎) → ((𝐹0 ) + (𝐹𝑦)) = ((𝐹0 ) + (𝐹‘(𝑏 𝑎))))
9693, 95breq12d 5076 . . . . . . . . . . . . . 14 (𝑦 = (𝑏 𝑎) → ((𝐹‘( 0 𝑦)) ≤ ((𝐹0 ) + (𝐹𝑦)) ↔ (𝐹‘( 0 (𝑏 𝑎))) ≤ ((𝐹0 ) + (𝐹‘(𝑏 𝑎)))))
9791, 96rspc2va 3638 . . . . . . . . . . . . 13 ((( 0𝑋 ∧ (𝑏 𝑎) ∈ 𝑋) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))) → (𝐹‘( 0 (𝑏 𝑎))) ≤ ((𝐹0 ) + (𝐹‘(𝑏 𝑎))))
9882, 86, 87, 97syl21anc 835 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝐹‘( 0 (𝑏 𝑎))) ≤ ((𝐹0 ) + (𝐹‘(𝑏 𝑎))))
99 eqid 2826 . . . . . . . . . . . . . . . 16 (invg𝐺) = (invg𝐺)
1003, 4, 99, 28grpinvval2 18127 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ (𝑏 𝑎) ∈ 𝑋) → ((invg𝐺)‘(𝑏 𝑎)) = ( 0 (𝑏 𝑎)))
1012, 86, 100syl2an2r 681 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((invg𝐺)‘(𝑏 𝑎)) = ( 0 (𝑏 𝑎)))
1023, 4, 99grpinvsub 18126 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝑏𝑋𝑎𝑋) → ((invg𝐺)‘(𝑏 𝑎)) = (𝑎 𝑏))
10380, 83, 84, 102syl3anc 1365 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((invg𝐺)‘(𝑏 𝑎)) = (𝑎 𝑏))
104101, 103eqtr3d 2863 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ( 0 (𝑏 𝑎)) = (𝑎 𝑏))
105104fveq2d 6673 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝐹‘( 0 (𝑏 𝑎))) = (𝐹‘(𝑎 𝑏)))
1062, 81syl 17 . . . . . . . . . . . . . . . 16 (𝜑0𝑋)
107 pm5.501 368 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 0 → ((𝐹𝑥) = 0 ↔ (𝑥 = 0 ↔ (𝐹𝑥) = 0)))
108 bicom 223 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = 0 ↔ (𝐹𝑥) = 0) ↔ ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ))
109107, 108syl6bb 288 . . . . . . . . . . . . . . . . . 18 (𝑥 = 0 → ((𝐹𝑥) = 0 ↔ ((𝐹𝑥) = 0 ↔ 𝑥 = 0 )))
11089eqeq1d 2828 . . . . . . . . . . . . . . . . . 18 (𝑥 = 0 → ((𝐹𝑥) = 0 ↔ (𝐹0 ) = 0))
111109, 110bitr3d 282 . . . . . . . . . . . . . . . . 17 (𝑥 = 0 → (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ↔ (𝐹0 ) = 0))
112111rspccva 3626 . . . . . . . . . . . . . . . 16 ((∀𝑥𝑋 ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ 0𝑋) → (𝐹0 ) = 0)
11318, 106, 112syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹0 ) = 0)
114113adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝐹0 ) = 0)
115114oveq1d 7165 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹0 ) + (𝐹‘(𝑏 𝑎))) = (0 + (𝐹‘(𝑏 𝑎))))
1161adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → 𝐹:𝑋⟶ℝ)
117116, 86ffvelrnd 6850 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝐹‘(𝑏 𝑎)) ∈ ℝ)
118117recnd 10663 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝐹‘(𝑏 𝑎)) ∈ ℂ)
119118addid2d 10835 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (0 + (𝐹‘(𝑏 𝑎))) = (𝐹‘(𝑏 𝑎)))
120115, 119eqtrd 2861 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹0 ) + (𝐹‘(𝑏 𝑎))) = (𝐹‘(𝑏 𝑎)))
12198, 105, 1203brtr3d 5094 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝐹‘(𝑎 𝑏)) ≤ (𝐹‘(𝑏 𝑎)))
12279, 121chvarvv 1998 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝑋𝑐𝑋)) → (𝐹‘(𝑎 𝑐)) ≤ (𝐹‘(𝑐 𝑎)))
123122adantrlr 719 . . . . . . . . 9 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝐹‘(𝑎 𝑐)) ≤ (𝐹‘(𝑐 𝑎)))
124 eleq1w 2900 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝑎𝑋𝑏𝑋))
125124anbi1d 629 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → ((𝑎𝑋𝑐𝑋) ↔ (𝑏𝑋𝑐𝑋)))
126125anbi2d 628 . . . . . . . . . . . 12 (𝑎 = 𝑏 → ((𝜑 ∧ (𝑎𝑋𝑐𝑋)) ↔ (𝜑 ∧ (𝑏𝑋𝑐𝑋))))
127 fvoveq1 7173 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝐹‘(𝑎 𝑐)) = (𝐹‘(𝑏 𝑐)))
128 oveq2 7158 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝑐 𝑎) = (𝑐 𝑏))
129128fveq2d 6673 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝐹‘(𝑐 𝑎)) = (𝐹‘(𝑐 𝑏)))
130127, 129breq12d 5076 . . . . . . . . . . . 12 (𝑎 = 𝑏 → ((𝐹‘(𝑎 𝑐)) ≤ (𝐹‘(𝑐 𝑎)) ↔ (𝐹‘(𝑏 𝑐)) ≤ (𝐹‘(𝑐 𝑏))))
131126, 130imbi12d 346 . . . . . . . . . . 11 (𝑎 = 𝑏 → (((𝜑 ∧ (𝑎𝑋𝑐𝑋)) → (𝐹‘(𝑎 𝑐)) ≤ (𝐹‘(𝑐 𝑎))) ↔ ((𝜑 ∧ (𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑏 𝑐)) ≤ (𝐹‘(𝑐 𝑏)))))
132131, 122chvarvv 1998 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑏 𝑐)) ≤ (𝐹‘(𝑐 𝑏)))
133132adantrll 718 . . . . . . . . 9 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝐹‘(𝑏 𝑐)) ≤ (𝐹‘(𝑐 𝑏)))
13441, 45, 49, 52, 123, 133le2addd 11253 . . . . . . . 8 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → ((𝐹‘(𝑎 𝑐)) + (𝐹‘(𝑏 𝑐))) ≤ ((𝐹‘(𝑐 𝑎)) + (𝐹‘(𝑐 𝑏))))
13535, 46, 53, 71, 134letrd 10791 . . . . . . 7 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝐹‘(𝑎 𝑏)) ≤ ((𝐹‘(𝑐 𝑎)) + (𝐹‘(𝑐 𝑏))))
13615adantrr 713 . . . . . . 7 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝑎(𝐹 )𝑏) = (𝐹‘(𝑎 𝑏)))
137 opelxpi 5591 . . . . . . . . . . 11 ((𝑐𝑋𝑎𝑋) → ⟨𝑐, 𝑎⟩ ∈ (𝑋 × 𝑋))
13838, 37, 137syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → ⟨𝑐, 𝑎⟩ ∈ (𝑋 × 𝑋))
139 fvco3 6759 . . . . . . . . . 10 (( :(𝑋 × 𝑋)⟶𝑋 ∧ ⟨𝑐, 𝑎⟩ ∈ (𝑋 × 𝑋)) → ((𝐹 )‘⟨𝑐, 𝑎⟩) = (𝐹‘( ‘⟨𝑐, 𝑎⟩)))
1406, 138, 139syl2an2r 681 . . . . . . . . 9 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → ((𝐹 )‘⟨𝑐, 𝑎⟩) = (𝐹‘( ‘⟨𝑐, 𝑎⟩)))
141 df-ov 7153 . . . . . . . . 9 (𝑐(𝐹 )𝑎) = ((𝐹 )‘⟨𝑐, 𝑎⟩)
142 df-ov 7153 . . . . . . . . . 10 (𝑐 𝑎) = ( ‘⟨𝑐, 𝑎⟩)
143142fveq2i 6672 . . . . . . . . 9 (𝐹‘(𝑐 𝑎)) = (𝐹‘( ‘⟨𝑐, 𝑎⟩))
144140, 141, 1433eqtr4g 2886 . . . . . . . 8 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝑐(𝐹 )𝑎) = (𝐹‘(𝑐 𝑎)))
145 opelxpi 5591 . . . . . . . . . . 11 ((𝑐𝑋𝑏𝑋) → ⟨𝑐, 𝑏⟩ ∈ (𝑋 × 𝑋))
14638, 42, 145syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → ⟨𝑐, 𝑏⟩ ∈ (𝑋 × 𝑋))
147 fvco3 6759 . . . . . . . . . 10 (( :(𝑋 × 𝑋)⟶𝑋 ∧ ⟨𝑐, 𝑏⟩ ∈ (𝑋 × 𝑋)) → ((𝐹 )‘⟨𝑐, 𝑏⟩) = (𝐹‘( ‘⟨𝑐, 𝑏⟩)))
1486, 146, 147syl2an2r 681 . . . . . . . . 9 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → ((𝐹 )‘⟨𝑐, 𝑏⟩) = (𝐹‘( ‘⟨𝑐, 𝑏⟩)))
149 df-ov 7153 . . . . . . . . 9 (𝑐(𝐹 )𝑏) = ((𝐹 )‘⟨𝑐, 𝑏⟩)
150 df-ov 7153 . . . . . . . . . 10 (𝑐 𝑏) = ( ‘⟨𝑐, 𝑏⟩)
151150fveq2i 6672 . . . . . . . . 9 (𝐹‘(𝑐 𝑏)) = (𝐹‘( ‘⟨𝑐, 𝑏⟩))
152148, 149, 1513eqtr4g 2886 . . . . . . . 8 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝑐(𝐹 )𝑏) = (𝐹‘(𝑐 𝑏)))
153144, 152oveq12d 7168 . . . . . . 7 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → ((𝑐(𝐹 )𝑎) + (𝑐(𝐹 )𝑏)) = ((𝐹‘(𝑐 𝑎)) + (𝐹‘(𝑐 𝑏))))
154135, 136, 1533brtr4d 5095 . . . . . 6 ((𝜑 ∧ ((𝑎𝑋𝑏𝑋) ∧ 𝑐𝑋)) → (𝑎(𝐹 )𝑏) ≤ ((𝑐(𝐹 )𝑎) + (𝑐(𝐹 )𝑏)))
155154expr 457 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝑐𝑋 → (𝑎(𝐹 )𝑏) ≤ ((𝑐(𝐹 )𝑎) + (𝑐(𝐹 )𝑏))))
156155ralrimiv 3186 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ∀𝑐𝑋 (𝑎(𝐹 )𝑏) ≤ ((𝑐(𝐹 )𝑎) + (𝑐(𝐹 )𝑏)))
15732, 156jca 512 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (((𝑎(𝐹 )𝑏) = 0 ↔ 𝑎 = 𝑏) ∧ ∀𝑐𝑋 (𝑎(𝐹 )𝑏) ≤ ((𝑐(𝐹 )𝑎) + (𝑐(𝐹 )𝑏))))
158157ralrimivva 3196 . 2 (𝜑 → ∀𝑎𝑋𝑏𝑋 (((𝑎(𝐹 )𝑏) = 0 ↔ 𝑎 = 𝑏) ∧ ∀𝑐𝑋 (𝑎(𝐹 )𝑏) ≤ ((𝑐(𝐹 )𝑎) + (𝑐(𝐹 )𝑏))))
1593fvexi 6683 . . 3 𝑋 ∈ V
160 ismet 22867 . . 3 (𝑋 ∈ V → ((𝐹 ) ∈ (Met‘𝑋) ↔ ((𝐹 ):(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑎𝑋𝑏𝑋 (((𝑎(𝐹 )𝑏) = 0 ↔ 𝑎 = 𝑏) ∧ ∀𝑐𝑋 (𝑎(𝐹 )𝑏) ≤ ((𝑐(𝐹 )𝑎) + (𝑐(𝐹 )𝑏))))))
161159, 160ax-mp 5 . 2 ((𝐹 ) ∈ (Met‘𝑋) ↔ ((𝐹 ):(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑎𝑋𝑏𝑋 (((𝑎(𝐹 )𝑏) = 0 ↔ 𝑎 = 𝑏) ∧ ∀𝑐𝑋 (𝑎(𝐹 )𝑏) ≤ ((𝑐(𝐹 )𝑎) + (𝑐(𝐹 )𝑏)))))
1628, 158, 161sylanbrc 583 1 (𝜑 → (𝐹 ) ∈ (Met‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3143  Vcvv 3500  cop 4570   class class class wbr 5063   × cxp 5552  ccom 5558  wf 6350  cfv 6354  (class class class)co 7150  cr 10530  0cc0 10531   + caddc 10534  cle 10670  Basecbs 16478  0gc0g 16708  Grpcgrp 18048  invgcminusg 18049  -gcsg 18050  Metcmet 20466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-po 5473  df-so 5474  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-er 8284  df-map 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18051  df-minusg 18052  df-sbg 18053  df-met 20474
This theorem is referenced by:  abvmet  23119  tngngpd  23196
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