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Theorem tngngp3 22370
 Description: Alternate definition of a normed group (i.e. a group equipped with a norm) without using the properties of a metric space. This corresponds to the definition in N. H. Bingham, A. J. Ostaszewski: "Normed versus topological groups: dichotomy and duality", 2010, Dissertationes Mathematicae 472, pp. 1-138 and E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006. (Contributed by AV, 16-Oct-2021.)
Hypotheses
Ref Expression
tngngp3.t 𝑇 = (𝐺 toNrmGrp 𝑁)
tngngp3.x 𝑋 = (Base‘𝐺)
tngngp3.z 0 = (0g𝐺)
tngngp3.p + = (+g𝐺)
tngngp3.i 𝐼 = (invg𝐺)
Assertion
Ref Expression
tngngp3 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))
Distinct variable groups:   𝑥,𝐺,𝑦   𝑥,𝑁,𝑦   𝑥,𝑇,𝑦   𝑥,𝑋,𝑦   𝑥,𝐼,𝑦   𝑥, + ,𝑦   𝑥, 0 ,𝑦

Proof of Theorem tngngp3
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tngngp3.x . . . . 5 𝑋 = (Base‘𝐺)
2 fvex 6158 . . . . 5 (Base‘𝐺) ∈ V
31, 2eqeltri 2694 . . . 4 𝑋 ∈ V
4 fex 6444 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V) → 𝑁 ∈ V)
53, 4mpan2 706 . . 3 (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
6 tngngp3.t . . . . . . 7 𝑇 = (𝐺 toNrmGrp 𝑁)
76tnggrpr 22369 . . . . . 6 ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
8 simp2 1060 . . . . . . . 8 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝐺 ∈ Grp)
9 eqid 2621 . . . . . . . . . . . . . 14 (Base‘𝑇) = (Base‘𝑇)
10 eqid 2621 . . . . . . . . . . . . . 14 (norm‘𝑇) = (norm‘𝑇)
11 eqid 2621 . . . . . . . . . . . . . 14 (0g𝑇) = (0g𝑇)
129, 10, 11nmeq0 22332 . . . . . . . . . . . . 13 ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)))
13 eqid 2621 . . . . . . . . . . . . . 14 (invg𝑇) = (invg𝑇)
149, 10, 13nminv 22335 . . . . . . . . . . . . 13 ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘((invg𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥))
15 eqid 2621 . . . . . . . . . . . . . . . 16 (+g𝑇) = (+g𝑇)
169, 10, 15nmtri 22340 . . . . . . . . . . . . . . 15 ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝑥(+g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
17163expa 1262 . . . . . . . . . . . . . 14 (((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) ∧ 𝑦 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝑥(+g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
1817ralrimiva 2960 . . . . . . . . . . . . 13 ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
1912, 14, 183jca 1240 . . . . . . . . . . . 12 ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → ((((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)) ∧ ((norm‘𝑇)‘((invg𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥) ∧ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))))
2019ralrimiva 2960 . . . . . . . . . . 11 (𝑇 ∈ NrmGrp → ∀𝑥 ∈ (Base‘𝑇)((((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)) ∧ ((norm‘𝑇)‘((invg𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥) ∧ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))))
2120adantl 482 . . . . . . . . . 10 ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → ∀𝑥 ∈ (Base‘𝑇)((((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)) ∧ ((norm‘𝑇)‘((invg𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥) ∧ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))))
22213ad2ant1 1080 . . . . . . . . 9 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → ∀𝑥 ∈ (Base‘𝑇)((((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)) ∧ ((norm‘𝑇)‘((invg𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥) ∧ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))))
236, 1tngbas 22355 . . . . . . . . . . . . . 14 (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
24 tngngp3.p . . . . . . . . . . . . . . 15 + = (+g𝐺)
256, 24tngplusg 22356 . . . . . . . . . . . . . 14 (𝑁 ∈ V → + = (+g𝑇))
26 tngngp3.i . . . . . . . . . . . . . . 15 𝐼 = (invg𝐺)
27 eqidd 2622 . . . . . . . . . . . . . . . 16 (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝐺))
28 eqid 2621 . . . . . . . . . . . . . . . . 17 (Base‘𝐺) = (Base‘𝐺)
296, 28tngbas 22355 . . . . . . . . . . . . . . . 16 (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝑇))
30 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (+g𝐺) = (+g𝐺)
316, 30tngplusg 22356 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ V → (+g𝐺) = (+g𝑇))
3231oveqd 6621 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ V → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝑇)𝑦))
3332adantr 481 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ V ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝑇)𝑦))
3427, 29, 33grpinvpropd 17411 . . . . . . . . . . . . . . 15 (𝑁 ∈ V → (invg𝐺) = (invg𝑇))
3526, 34syl5eq 2667 . . . . . . . . . . . . . 14 (𝑁 ∈ V → 𝐼 = (invg𝑇))
3623, 25, 353jca 1240 . . . . . . . . . . . . 13 (𝑁 ∈ V → (𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)))
3736adantr 481 . . . . . . . . . . . 12 ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → (𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)))
38373ad2ant1 1080 . . . . . . . . . . 11 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → (𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)))
39 reex 9971 . . . . . . . . . . . . 13 ℝ ∈ V
406, 1, 39tngnm 22365 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
41403adant1 1077 . . . . . . . . . . 11 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
42 tngngp3.z . . . . . . . . . . . . . 14 0 = (0g𝐺)
436, 42tng0 22357 . . . . . . . . . . . . 13 (𝑁 ∈ V → 0 = (0g𝑇))
4443adantr 481 . . . . . . . . . . . 12 ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → 0 = (0g𝑇))
45443ad2ant1 1080 . . . . . . . . . . 11 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 0 = (0g𝑇))
4638, 41, 453jca 1240 . . . . . . . . . 10 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → ((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)))
47 simp1 1059 . . . . . . . . . . . 12 ((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) → 𝑋 = (Base‘𝑇))
48473ad2ant1 1080 . . . . . . . . . . 11 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → 𝑋 = (Base‘𝑇))
49 simp2 1060 . . . . . . . . . . . . . . 15 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → 𝑁 = (norm‘𝑇))
5049fveq1d 6150 . . . . . . . . . . . . . 14 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → (𝑁𝑥) = ((norm‘𝑇)‘𝑥))
5150eqeq1d 2623 . . . . . . . . . . . . 13 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → ((𝑁𝑥) = 0 ↔ ((norm‘𝑇)‘𝑥) = 0))
52 simp3 1061 . . . . . . . . . . . . . 14 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → 0 = (0g𝑇))
5352eqeq2d 2631 . . . . . . . . . . . . 13 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → (𝑥 = 0𝑥 = (0g𝑇)))
5451, 53bibi12d 335 . . . . . . . . . . . 12 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ↔ (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇))))
55 simp3 1061 . . . . . . . . . . . . . . . 16 ((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) → 𝐼 = (invg𝑇))
56553ad2ant1 1080 . . . . . . . . . . . . . . 15 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → 𝐼 = (invg𝑇))
5756fveq1d 6150 . . . . . . . . . . . . . 14 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → (𝐼𝑥) = ((invg𝑇)‘𝑥))
5849, 57fveq12d 6154 . . . . . . . . . . . . 13 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → (𝑁‘(𝐼𝑥)) = ((norm‘𝑇)‘((invg𝑇)‘𝑥)))
5958, 50eqeq12d 2636 . . . . . . . . . . . 12 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ↔ ((norm‘𝑇)‘((invg𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥)))
60 simp2 1060 . . . . . . . . . . . . . . . . 17 ((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) → + = (+g𝑇))
61603ad2ant1 1080 . . . . . . . . . . . . . . . 16 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → + = (+g𝑇))
6261oveqd 6621 . . . . . . . . . . . . . . 15 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → (𝑥 + 𝑦) = (𝑥(+g𝑇)𝑦))
6349, 62fveq12d 6154 . . . . . . . . . . . . . 14 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → (𝑁‘(𝑥 + 𝑦)) = ((norm‘𝑇)‘(𝑥(+g𝑇)𝑦)))
64 fveq1 6147 . . . . . . . . . . . . . . . 16 (𝑁 = (norm‘𝑇) → (𝑁𝑥) = ((norm‘𝑇)‘𝑥))
65 fveq1 6147 . . . . . . . . . . . . . . . 16 (𝑁 = (norm‘𝑇) → (𝑁𝑦) = ((norm‘𝑇)‘𝑦))
6664, 65oveq12d 6622 . . . . . . . . . . . . . . 15 (𝑁 = (norm‘𝑇) → ((𝑁𝑥) + (𝑁𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
67663ad2ant2 1081 . . . . . . . . . . . . . 14 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → ((𝑁𝑥) + (𝑁𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
6863, 67breq12d 4626 . . . . . . . . . . . . 13 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → ((𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ↔ ((norm‘𝑇)‘(𝑥(+g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))))
6948, 68raleqbidv 3141 . . . . . . . . . . . 12 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → (∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ↔ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))))
7054, 59, 693anbi123d 1396 . . . . . . . . . . 11 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → ((((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ↔ ((((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)) ∧ ((norm‘𝑇)‘((invg𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥) ∧ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))))
7148, 70raleqbidv 3141 . . . . . . . . . 10 (((𝑋 = (Base‘𝑇) ∧ + = (+g𝑇) ∧ 𝐼 = (invg𝑇)) ∧ 𝑁 = (norm‘𝑇) ∧ 0 = (0g𝑇)) → (∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ↔ ∀𝑥 ∈ (Base‘𝑇)((((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)) ∧ ((norm‘𝑇)‘((invg𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥) ∧ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))))
7246, 71syl 17 . . . . . . . . 9 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → (∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ↔ ∀𝑥 ∈ (Base‘𝑇)((((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g𝑇)) ∧ ((norm‘𝑇)‘((invg𝑇)‘𝑥)) = ((norm‘𝑇)‘𝑥) ∧ ∀𝑦 ∈ (Base‘𝑇)((norm‘𝑇)‘(𝑥(+g𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))))
7322, 72mpbird 247 . . . . . . . 8 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
748, 73jca 554 . . . . . . 7 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
75743exp 1261 . . . . . 6 ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → (𝐺 ∈ Grp → (𝑁:𝑋⟶ℝ → (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))))
767, 75mpd 15 . . . . 5 ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) → (𝑁:𝑋⟶ℝ → (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))
7776expcom 451 . . . 4 (𝑇 ∈ NrmGrp → (𝑁 ∈ V → (𝑁:𝑋⟶ℝ → (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))))
7877com13 88 . . 3 (𝑁:𝑋⟶ℝ → (𝑁 ∈ V → (𝑇 ∈ NrmGrp → (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))))
795, 78mpd 15 . 2 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp → (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))
80 eqid 2621 . . . 4 (-g𝐺) = (-g𝐺)
81 simpl 473 . . . . 5 ((𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))) → 𝐺 ∈ Grp)
8281adantl 482 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝐺 ∈ Grp)
83 simpl 473 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝑁:𝑋⟶ℝ)
84 fveq2 6148 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑁𝑥) = (𝑁𝑎))
8584eqeq1d 2623 . . . . . . . . . . . 12 (𝑥 = 𝑎 → ((𝑁𝑥) = 0 ↔ (𝑁𝑎) = 0))
86 eqeq1 2625 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑥 = 0𝑎 = 0 ))
8785, 86bibi12d 335 . . . . . . . . . . 11 (𝑥 = 𝑎 → (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝑁𝑎) = 0 ↔ 𝑎 = 0 )))
88 fveq2 6148 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝐼𝑥) = (𝐼𝑎))
8988fveq2d 6152 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑁‘(𝐼𝑥)) = (𝑁‘(𝐼𝑎)))
9089, 84eqeq12d 2636 . . . . . . . . . . 11 (𝑥 = 𝑎 → ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ↔ (𝑁‘(𝐼𝑎)) = (𝑁𝑎)))
91 oveq1 6611 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑥 + 𝑦) = (𝑎 + 𝑦))
9291fveq2d 6152 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑁‘(𝑥 + 𝑦)) = (𝑁‘(𝑎 + 𝑦)))
9384oveq1d 6619 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → ((𝑁𝑥) + (𝑁𝑦)) = ((𝑁𝑎) + (𝑁𝑦)))
9492, 93breq12d 4626 . . . . . . . . . . . 12 (𝑥 = 𝑎 → ((𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ↔ (𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦))))
9594ralbidv 2980 . . . . . . . . . . 11 (𝑥 = 𝑎 → (∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ↔ ∀𝑦𝑋 (𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦))))
9687, 90, 953anbi123d 1396 . . . . . . . . . 10 (𝑥 = 𝑎 → ((((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ↔ (((𝑁𝑎) = 0 ↔ 𝑎 = 0 ) ∧ (𝑁‘(𝐼𝑎)) = (𝑁𝑎) ∧ ∀𝑦𝑋 (𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦)))))
9796rspccva 3294 . . . . . . . . 9 ((∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ∧ 𝑎𝑋) → (((𝑁𝑎) = 0 ↔ 𝑎 = 0 ) ∧ (𝑁‘(𝐼𝑎)) = (𝑁𝑎) ∧ ∀𝑦𝑋 (𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦))))
98 simp1 1059 . . . . . . . . 9 ((((𝑁𝑎) = 0 ↔ 𝑎 = 0 ) ∧ (𝑁‘(𝐼𝑎)) = (𝑁𝑎) ∧ ∀𝑦𝑋 (𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦))) → ((𝑁𝑎) = 0 ↔ 𝑎 = 0 ))
9997, 98syl 17 . . . . . . . 8 ((∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ∧ 𝑎𝑋) → ((𝑁𝑎) = 0 ↔ 𝑎 = 0 ))
10099ex 450 . . . . . . 7 (∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → (𝑎𝑋 → ((𝑁𝑎) = 0 ↔ 𝑎 = 0 )))
101100adantl 482 . . . . . 6 ((𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))) → (𝑎𝑋 → ((𝑁𝑎) = 0 ↔ 𝑎 = 0 )))
102101adantl 482 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → (𝑎𝑋 → ((𝑁𝑎) = 0 ↔ 𝑎 = 0 )))
103102imp 445 . . . 4 (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) ∧ 𝑎𝑋) → ((𝑁𝑎) = 0 ↔ 𝑎 = 0 ))
1041, 24, 26, 80grpsubval 17386 . . . . . . 7 ((𝑎𝑋𝑏𝑋) → (𝑎(-g𝐺)𝑏) = (𝑎 + (𝐼𝑏)))
105104adantl 482 . . . . . 6 (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(-g𝐺)𝑏) = (𝑎 + (𝐼𝑏)))
106105fveq2d 6152 . . . . 5 (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) ∧ (𝑎𝑋𝑏𝑋)) → (𝑁‘(𝑎(-g𝐺)𝑏)) = (𝑁‘(𝑎 + (𝐼𝑏))))
107 3simpc 1058 . . . . . . . . . 10 ((((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
108107ralimi 2947 . . . . . . . . 9 (∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑥𝑋 ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
109 simpr 477 . . . . . . . . . . . . . . . 16 (((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
110109ralimi 2947 . . . . . . . . . . . . . . 15 (∀𝑥𝑋 ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
111 oveq2 6612 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝐼𝑏) → (𝑎 + 𝑦) = (𝑎 + (𝐼𝑏)))
112111fveq2d 6152 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝐼𝑏) → (𝑁‘(𝑎 + 𝑦)) = (𝑁‘(𝑎 + (𝐼𝑏))))
113 fveq2 6148 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝐼𝑏) → (𝑁𝑦) = (𝑁‘(𝐼𝑏)))
114113oveq2d 6620 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝐼𝑏) → ((𝑁𝑎) + (𝑁𝑦)) = ((𝑁𝑎) + (𝑁‘(𝐼𝑏))))
115112, 114breq12d 4626 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝐼𝑏) → ((𝑁‘(𝑎 + 𝑦)) ≤ ((𝑁𝑎) + (𝑁𝑦)) ↔ (𝑁‘(𝑎 + (𝐼𝑏))) ≤ ((𝑁𝑎) + (𝑁‘(𝐼𝑏)))))
11694, 115rspc2v 3306 . . . . . . . . . . . . . . . . 17 ((𝑎𝑋 ∧ (𝐼𝑏) ∈ 𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) → (𝑁‘(𝑎 + (𝐼𝑏))) ≤ ((𝑁𝑎) + (𝑁‘(𝐼𝑏)))))
1171, 26grpinvcl 17388 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ Grp ∧ 𝑏𝑋) → (𝐼𝑏) ∈ 𝑋)
118117ex 450 . . . . . . . . . . . . . . . . . . 19 (𝐺 ∈ Grp → (𝑏𝑋 → (𝐼𝑏) ∈ 𝑋))
119118anim2d 588 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ Grp → ((𝑎𝑋𝑏𝑋) → (𝑎𝑋 ∧ (𝐼𝑏) ∈ 𝑋)))
120119imp 445 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Grp ∧ (𝑎𝑋𝑏𝑋)) → (𝑎𝑋 ∧ (𝐼𝑏) ∈ 𝑋))
121116, 120syl11 33 . . . . . . . . . . . . . . . 16 (∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) → ((𝐺 ∈ Grp ∧ (𝑎𝑋𝑏𝑋)) → (𝑁‘(𝑎 + (𝐼𝑏))) ≤ ((𝑁𝑎) + (𝑁‘(𝐼𝑏)))))
122121expd 452 . . . . . . . . . . . . . . 15 (∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) → (𝐺 ∈ Grp → ((𝑎𝑋𝑏𝑋) → (𝑁‘(𝑎 + (𝐼𝑏))) ≤ ((𝑁𝑎) + (𝑁‘(𝐼𝑏))))))
123110, 122syl 17 . . . . . . . . . . . . . 14 (∀𝑥𝑋 ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → (𝐺 ∈ Grp → ((𝑎𝑋𝑏𝑋) → (𝑁‘(𝑎 + (𝐼𝑏))) ≤ ((𝑁𝑎) + (𝑁‘(𝐼𝑏))))))
124123imp 445 . . . . . . . . . . . . 13 ((∀𝑥𝑋 ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ∧ 𝐺 ∈ Grp) → ((𝑎𝑋𝑏𝑋) → (𝑁‘(𝑎 + (𝐼𝑏))) ≤ ((𝑁𝑎) + (𝑁‘(𝐼𝑏)))))
125124imp 445 . . . . . . . . . . . 12 (((∀𝑥𝑋 ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ∧ 𝐺 ∈ Grp) ∧ (𝑎𝑋𝑏𝑋)) → (𝑁‘(𝑎 + (𝐼𝑏))) ≤ ((𝑁𝑎) + (𝑁‘(𝐼𝑏))))
126 simpl 473 . . . . . . . . . . . . . . . . . 18 (((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → (𝑁‘(𝐼𝑥)) = (𝑁𝑥))
127126ralimi 2947 . . . . . . . . . . . . . . . . 17 (∀𝑥𝑋 ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑥𝑋 (𝑁‘(𝐼𝑥)) = (𝑁𝑥))
128 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑏 → (𝐼𝑥) = (𝐼𝑏))
129128fveq2d 6152 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑏 → (𝑁‘(𝐼𝑥)) = (𝑁‘(𝐼𝑏)))
130 fveq2 6148 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑏 → (𝑁𝑥) = (𝑁𝑏))
131129, 130eqeq12d 2636 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑏 → ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ↔ (𝑁‘(𝐼𝑏)) = (𝑁𝑏)))
132131rspccva 3294 . . . . . . . . . . . . . . . . . . 19 ((∀𝑥𝑋 (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ 𝑏𝑋) → (𝑁‘(𝐼𝑏)) = (𝑁𝑏))
133132eqcomd 2627 . . . . . . . . . . . . . . . . . 18 ((∀𝑥𝑋 (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ 𝑏𝑋) → (𝑁𝑏) = (𝑁‘(𝐼𝑏)))
134133ex 450 . . . . . . . . . . . . . . . . 17 (∀𝑥𝑋 (𝑁‘(𝐼𝑥)) = (𝑁𝑥) → (𝑏𝑋 → (𝑁𝑏) = (𝑁‘(𝐼𝑏))))
135127, 134syl 17 . . . . . . . . . . . . . . . 16 (∀𝑥𝑋 ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → (𝑏𝑋 → (𝑁𝑏) = (𝑁‘(𝐼𝑏))))
136135adantr 481 . . . . . . . . . . . . . . 15 ((∀𝑥𝑋 ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ∧ 𝐺 ∈ Grp) → (𝑏𝑋 → (𝑁𝑏) = (𝑁‘(𝐼𝑏))))
137136adantld 483 . . . . . . . . . . . . . 14 ((∀𝑥𝑋 ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ∧ 𝐺 ∈ Grp) → ((𝑎𝑋𝑏𝑋) → (𝑁𝑏) = (𝑁‘(𝐼𝑏))))
138137imp 445 . . . . . . . . . . . . 13 (((∀𝑥𝑋 ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ∧ 𝐺 ∈ Grp) ∧ (𝑎𝑋𝑏𝑋)) → (𝑁𝑏) = (𝑁‘(𝐼𝑏)))
139138oveq2d 6620 . . . . . . . . . . . 12 (((∀𝑥𝑋 ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ∧ 𝐺 ∈ Grp) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑁𝑎) + (𝑁𝑏)) = ((𝑁𝑎) + (𝑁‘(𝐼𝑏))))
140125, 139breqtrrd 4641 . . . . . . . . . . 11 (((∀𝑥𝑋 ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ∧ 𝐺 ∈ Grp) ∧ (𝑎𝑋𝑏𝑋)) → (𝑁‘(𝑎 + (𝐼𝑏))) ≤ ((𝑁𝑎) + (𝑁𝑏)))
141140ex 450 . . . . . . . . . 10 ((∀𝑥𝑋 ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ∧ 𝐺 ∈ Grp) → ((𝑎𝑋𝑏𝑋) → (𝑁‘(𝑎 + (𝐼𝑏))) ≤ ((𝑁𝑎) + (𝑁𝑏))))
142141ex 450 . . . . . . . . 9 (∀𝑥𝑋 ((𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → (𝐺 ∈ Grp → ((𝑎𝑋𝑏𝑋) → (𝑁‘(𝑎 + (𝐼𝑏))) ≤ ((𝑁𝑎) + (𝑁𝑏)))))
143108, 142syl 17 . . . . . . . 8 (∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → (𝐺 ∈ Grp → ((𝑎𝑋𝑏𝑋) → (𝑁‘(𝑎 + (𝐼𝑏))) ≤ ((𝑁𝑎) + (𝑁𝑏)))))
144143impcom 446 . . . . . . 7 ((𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))) → ((𝑎𝑋𝑏𝑋) → (𝑁‘(𝑎 + (𝐼𝑏))) ≤ ((𝑁𝑎) + (𝑁𝑏))))
145144adantl 482 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → ((𝑎𝑋𝑏𝑋) → (𝑁‘(𝑎 + (𝐼𝑏))) ≤ ((𝑁𝑎) + (𝑁𝑏))))
146145imp 445 . . . . 5 (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) ∧ (𝑎𝑋𝑏𝑋)) → (𝑁‘(𝑎 + (𝐼𝑏))) ≤ ((𝑁𝑎) + (𝑁𝑏)))
147106, 146eqbrtrd 4635 . . . 4 (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) ∧ (𝑎𝑋𝑏𝑋)) → (𝑁‘(𝑎(-g𝐺)𝑏)) ≤ ((𝑁𝑎) + (𝑁𝑏)))
1486, 1, 80, 42, 82, 83, 103, 147tngngpd 22367 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))) → 𝑇 ∈ NrmGrp)
149148ex 450 . 2 (𝑁:𝑋⟶ℝ → ((𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))) → 𝑇 ∈ NrmGrp))
15079, 149impbid 202 1 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907  Vcvv 3186   class class class wbr 4613  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  ℝcr 9879  0cc0 9880   + caddc 9883   ≤ cle 10019  Basecbs 15781  +gcplusg 15862  0gc0g 16021  Grpcgrp 17343  invgcminusg 17344  -gcsg 17345  normcnm 22291  NrmGrpcngp 22292   toNrmGrp ctng 22293 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-plusg 15875  df-tset 15881  df-ds 15885  df-rest 16004  df-topn 16005  df-0g 16023  df-topgen 16025  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-sbg 17348  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-xms 22035  df-ms 22036  df-nm 22297  df-ngp 22298  df-tng 22299 This theorem is referenced by: (None)
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