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Theorem tngngp3 24593
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 = (Baseβ€˜πΊ)
tngngp3.z 0 = (0gβ€˜πΊ)
tngngp3.p + = (+gβ€˜πΊ)
tngngp3.i 𝐼 = (invgβ€˜πΊ)
Assertion
Ref Expression
tngngp3 (𝑁:π‘‹βŸΆβ„ β†’ (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))))
Distinct variable groups:   π‘₯,𝐺,𝑦   π‘₯,𝑁,𝑦   π‘₯,𝑇,𝑦   π‘₯,𝑋,𝑦   π‘₯,𝐼,𝑦   π‘₯, + ,𝑦   π‘₯, 0 ,𝑦

Proof of Theorem tngngp3
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tngngp3.x . . . . 5 𝑋 = (Baseβ€˜πΊ)
21fvexi 6916 . . . 4 𝑋 ∈ V
3 fex 7244 . . . 4 ((𝑁:π‘‹βŸΆβ„ ∧ 𝑋 ∈ V) β†’ 𝑁 ∈ V)
42, 3mpan2 689 . . 3 (𝑁:π‘‹βŸΆβ„ β†’ 𝑁 ∈ V)
5 tngngp3.t . . . . . . 7 𝑇 = (𝐺 toNrmGrp 𝑁)
65tnggrpr 24592 . . . . . 6 ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) β†’ 𝐺 ∈ Grp)
7 simp2 1134 . . . . . . . 8 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:π‘‹βŸΆβ„) β†’ 𝐺 ∈ Grp)
8 eqid 2728 . . . . . . . . . . . . . 14 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
9 eqid 2728 . . . . . . . . . . . . . 14 (normβ€˜π‘‡) = (normβ€˜π‘‡)
10 eqid 2728 . . . . . . . . . . . . . 14 (0gβ€˜π‘‡) = (0gβ€˜π‘‡)
118, 9, 10nmeq0 24547 . . . . . . . . . . . . 13 ((𝑇 ∈ NrmGrp ∧ π‘₯ ∈ (Baseβ€˜π‘‡)) β†’ (((normβ€˜π‘‡)β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘‡)))
12 eqid 2728 . . . . . . . . . . . . . 14 (invgβ€˜π‘‡) = (invgβ€˜π‘‡)
138, 9, 12nminv 24550 . . . . . . . . . . . . 13 ((𝑇 ∈ NrmGrp ∧ π‘₯ ∈ (Baseβ€˜π‘‡)) β†’ ((normβ€˜π‘‡)β€˜((invgβ€˜π‘‡)β€˜π‘₯)) = ((normβ€˜π‘‡)β€˜π‘₯))
14 eqid 2728 . . . . . . . . . . . . . . . 16 (+gβ€˜π‘‡) = (+gβ€˜π‘‡)
158, 9, 14nmtri 24555 . . . . . . . . . . . . . . 15 ((𝑇 ∈ NrmGrp ∧ π‘₯ ∈ (Baseβ€˜π‘‡) ∧ 𝑦 ∈ (Baseβ€˜π‘‡)) β†’ ((normβ€˜π‘‡)β€˜(π‘₯(+gβ€˜π‘‡)𝑦)) ≀ (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦)))
16153expa 1115 . . . . . . . . . . . . . 14 (((𝑇 ∈ NrmGrp ∧ π‘₯ ∈ (Baseβ€˜π‘‡)) ∧ 𝑦 ∈ (Baseβ€˜π‘‡)) β†’ ((normβ€˜π‘‡)β€˜(π‘₯(+gβ€˜π‘‡)𝑦)) ≀ (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦)))
1716ralrimiva 3143 . . . . . . . . . . . . 13 ((𝑇 ∈ NrmGrp ∧ π‘₯ ∈ (Baseβ€˜π‘‡)) β†’ βˆ€π‘¦ ∈ (Baseβ€˜π‘‡)((normβ€˜π‘‡)β€˜(π‘₯(+gβ€˜π‘‡)𝑦)) ≀ (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦)))
1811, 13, 173jca 1125 . . . . . . . . . . . 12 ((𝑇 ∈ NrmGrp ∧ π‘₯ ∈ (Baseβ€˜π‘‡)) β†’ ((((normβ€˜π‘‡)β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘‡)) ∧ ((normβ€˜π‘‡)β€˜((invgβ€˜π‘‡)β€˜π‘₯)) = ((normβ€˜π‘‡)β€˜π‘₯) ∧ βˆ€π‘¦ ∈ (Baseβ€˜π‘‡)((normβ€˜π‘‡)β€˜(π‘₯(+gβ€˜π‘‡)𝑦)) ≀ (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦))))
1918ralrimiva 3143 . . . . . . . . . . 11 (𝑇 ∈ NrmGrp β†’ βˆ€π‘₯ ∈ (Baseβ€˜π‘‡)((((normβ€˜π‘‡)β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘‡)) ∧ ((normβ€˜π‘‡)β€˜((invgβ€˜π‘‡)β€˜π‘₯)) = ((normβ€˜π‘‡)β€˜π‘₯) ∧ βˆ€π‘¦ ∈ (Baseβ€˜π‘‡)((normβ€˜π‘‡)β€˜(π‘₯(+gβ€˜π‘‡)𝑦)) ≀ (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦))))
2019adantl 480 . . . . . . . . . 10 ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) β†’ βˆ€π‘₯ ∈ (Baseβ€˜π‘‡)((((normβ€˜π‘‡)β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘‡)) ∧ ((normβ€˜π‘‡)β€˜((invgβ€˜π‘‡)β€˜π‘₯)) = ((normβ€˜π‘‡)β€˜π‘₯) ∧ βˆ€π‘¦ ∈ (Baseβ€˜π‘‡)((normβ€˜π‘‡)β€˜(π‘₯(+gβ€˜π‘‡)𝑦)) ≀ (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦))))
21203ad2ant1 1130 . . . . . . . . 9 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:π‘‹βŸΆβ„) β†’ βˆ€π‘₯ ∈ (Baseβ€˜π‘‡)((((normβ€˜π‘‡)β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘‡)) ∧ ((normβ€˜π‘‡)β€˜((invgβ€˜π‘‡)β€˜π‘₯)) = ((normβ€˜π‘‡)β€˜π‘₯) ∧ βˆ€π‘¦ ∈ (Baseβ€˜π‘‡)((normβ€˜π‘‡)β€˜(π‘₯(+gβ€˜π‘‡)𝑦)) ≀ (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦))))
225, 1tngbas 24571 . . . . . . . . . . . . . 14 (𝑁 ∈ V β†’ 𝑋 = (Baseβ€˜π‘‡))
23 tngngp3.p . . . . . . . . . . . . . . 15 + = (+gβ€˜πΊ)
245, 23tngplusg 24573 . . . . . . . . . . . . . 14 (𝑁 ∈ V β†’ + = (+gβ€˜π‘‡))
25 tngngp3.i . . . . . . . . . . . . . . 15 𝐼 = (invgβ€˜πΊ)
26 eqidd 2729 . . . . . . . . . . . . . . . 16 (𝑁 ∈ V β†’ (Baseβ€˜πΊ) = (Baseβ€˜πΊ))
27 eqid 2728 . . . . . . . . . . . . . . . . 17 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
285, 27tngbas 24571 . . . . . . . . . . . . . . . 16 (𝑁 ∈ V β†’ (Baseβ€˜πΊ) = (Baseβ€˜π‘‡))
29 eqid 2728 . . . . . . . . . . . . . . . . . . 19 (+gβ€˜πΊ) = (+gβ€˜πΊ)
305, 29tngplusg 24573 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ V β†’ (+gβ€˜πΊ) = (+gβ€˜π‘‡))
3130oveqd 7443 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ V β†’ (π‘₯(+gβ€˜πΊ)𝑦) = (π‘₯(+gβ€˜π‘‡)𝑦))
3231adantr 479 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ V ∧ (π‘₯ ∈ (Baseβ€˜πΊ) ∧ 𝑦 ∈ (Baseβ€˜πΊ))) β†’ (π‘₯(+gβ€˜πΊ)𝑦) = (π‘₯(+gβ€˜π‘‡)𝑦))
3326, 28, 32grpinvpropd 18978 . . . . . . . . . . . . . . 15 (𝑁 ∈ V β†’ (invgβ€˜πΊ) = (invgβ€˜π‘‡))
3425, 33eqtrid 2780 . . . . . . . . . . . . . 14 (𝑁 ∈ V β†’ 𝐼 = (invgβ€˜π‘‡))
3522, 24, 343jca 1125 . . . . . . . . . . . . 13 (𝑁 ∈ V β†’ (𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)))
3635adantr 479 . . . . . . . . . . . 12 ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) β†’ (𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)))
37363ad2ant1 1130 . . . . . . . . . . 11 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:π‘‹βŸΆβ„) β†’ (𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)))
38 reex 11237 . . . . . . . . . . . . 13 ℝ ∈ V
395, 1, 38tngnm 24588 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑁:π‘‹βŸΆβ„) β†’ 𝑁 = (normβ€˜π‘‡))
40393adant1 1127 . . . . . . . . . . 11 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:π‘‹βŸΆβ„) β†’ 𝑁 = (normβ€˜π‘‡))
41 tngngp3.z . . . . . . . . . . . . . 14 0 = (0gβ€˜πΊ)
425, 41tng0 24575 . . . . . . . . . . . . 13 (𝑁 ∈ V β†’ 0 = (0gβ€˜π‘‡))
4342adantr 479 . . . . . . . . . . . 12 ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) β†’ 0 = (0gβ€˜π‘‡))
44433ad2ant1 1130 . . . . . . . . . . 11 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:π‘‹βŸΆβ„) β†’ 0 = (0gβ€˜π‘‡))
4537, 40, 443jca 1125 . . . . . . . . . 10 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:π‘‹βŸΆβ„) β†’ ((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)))
46 simp1 1133 . . . . . . . . . . . 12 ((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) β†’ 𝑋 = (Baseβ€˜π‘‡))
47463ad2ant1 1130 . . . . . . . . . . 11 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ 𝑋 = (Baseβ€˜π‘‡))
48 simp2 1134 . . . . . . . . . . . . . . 15 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ 𝑁 = (normβ€˜π‘‡))
4948fveq1d 6904 . . . . . . . . . . . . . 14 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ (π‘β€˜π‘₯) = ((normβ€˜π‘‡)β€˜π‘₯))
5049eqeq1d 2730 . . . . . . . . . . . . 13 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ ((π‘β€˜π‘₯) = 0 ↔ ((normβ€˜π‘‡)β€˜π‘₯) = 0))
51 simp3 1135 . . . . . . . . . . . . . 14 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ 0 = (0gβ€˜π‘‡))
5251eqeq2d 2739 . . . . . . . . . . . . 13 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ (π‘₯ = 0 ↔ π‘₯ = (0gβ€˜π‘‡)))
5350, 52bibi12d 344 . . . . . . . . . . . 12 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ↔ (((normβ€˜π‘‡)β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘‡))))
54 simp3 1135 . . . . . . . . . . . . . . . 16 ((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) β†’ 𝐼 = (invgβ€˜π‘‡))
55543ad2ant1 1130 . . . . . . . . . . . . . . 15 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ 𝐼 = (invgβ€˜π‘‡))
5655fveq1d 6904 . . . . . . . . . . . . . 14 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ (πΌβ€˜π‘₯) = ((invgβ€˜π‘‡)β€˜π‘₯))
5748, 56fveq12d 6909 . . . . . . . . . . . . 13 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ (π‘β€˜(πΌβ€˜π‘₯)) = ((normβ€˜π‘‡)β€˜((invgβ€˜π‘‡)β€˜π‘₯)))
5857, 49eqeq12d 2744 . . . . . . . . . . . 12 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ↔ ((normβ€˜π‘‡)β€˜((invgβ€˜π‘‡)β€˜π‘₯)) = ((normβ€˜π‘‡)β€˜π‘₯)))
59 simp2 1134 . . . . . . . . . . . . . . . . 17 ((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) β†’ + = (+gβ€˜π‘‡))
60593ad2ant1 1130 . . . . . . . . . . . . . . . 16 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ + = (+gβ€˜π‘‡))
6160oveqd 7443 . . . . . . . . . . . . . . 15 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ (π‘₯ + 𝑦) = (π‘₯(+gβ€˜π‘‡)𝑦))
6248, 61fveq12d 6909 . . . . . . . . . . . . . 14 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ (π‘β€˜(π‘₯ + 𝑦)) = ((normβ€˜π‘‡)β€˜(π‘₯(+gβ€˜π‘‡)𝑦)))
63 fveq1 6901 . . . . . . . . . . . . . . . 16 (𝑁 = (normβ€˜π‘‡) β†’ (π‘β€˜π‘₯) = ((normβ€˜π‘‡)β€˜π‘₯))
64 fveq1 6901 . . . . . . . . . . . . . . . 16 (𝑁 = (normβ€˜π‘‡) β†’ (π‘β€˜π‘¦) = ((normβ€˜π‘‡)β€˜π‘¦))
6563, 64oveq12d 7444 . . . . . . . . . . . . . . 15 (𝑁 = (normβ€˜π‘‡) β†’ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) = (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦)))
66653ad2ant2 1131 . . . . . . . . . . . . . 14 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) = (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦)))
6762, 66breq12d 5165 . . . . . . . . . . . . 13 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ ((π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) ↔ ((normβ€˜π‘‡)β€˜(π‘₯(+gβ€˜π‘‡)𝑦)) ≀ (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦))))
6847, 67raleqbidv 3340 . . . . . . . . . . . 12 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ (βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ (Baseβ€˜π‘‡)((normβ€˜π‘‡)β€˜(π‘₯(+gβ€˜π‘‡)𝑦)) ≀ (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦))))
6953, 58, 683anbi123d 1432 . . . . . . . . . . 11 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ ((((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) ↔ ((((normβ€˜π‘‡)β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘‡)) ∧ ((normβ€˜π‘‡)β€˜((invgβ€˜π‘‡)β€˜π‘₯)) = ((normβ€˜π‘‡)β€˜π‘₯) ∧ βˆ€π‘¦ ∈ (Baseβ€˜π‘‡)((normβ€˜π‘‡)β€˜(π‘₯(+gβ€˜π‘‡)𝑦)) ≀ (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦)))))
7047, 69raleqbidv 3340 . . . . . . . . . 10 (((𝑋 = (Baseβ€˜π‘‡) ∧ + = (+gβ€˜π‘‡) ∧ 𝐼 = (invgβ€˜π‘‡)) ∧ 𝑁 = (normβ€˜π‘‡) ∧ 0 = (0gβ€˜π‘‡)) β†’ (βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) ↔ βˆ€π‘₯ ∈ (Baseβ€˜π‘‡)((((normβ€˜π‘‡)β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘‡)) ∧ ((normβ€˜π‘‡)β€˜((invgβ€˜π‘‡)β€˜π‘₯)) = ((normβ€˜π‘‡)β€˜π‘₯) ∧ βˆ€π‘¦ ∈ (Baseβ€˜π‘‡)((normβ€˜π‘‡)β€˜(π‘₯(+gβ€˜π‘‡)𝑦)) ≀ (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦)))))
7145, 70syl 17 . . . . . . . . 9 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:π‘‹βŸΆβ„) β†’ (βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) ↔ βˆ€π‘₯ ∈ (Baseβ€˜π‘‡)((((normβ€˜π‘‡)β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘‡)) ∧ ((normβ€˜π‘‡)β€˜((invgβ€˜π‘‡)β€˜π‘₯)) = ((normβ€˜π‘‡)β€˜π‘₯) ∧ βˆ€π‘¦ ∈ (Baseβ€˜π‘‡)((normβ€˜π‘‡)β€˜(π‘₯(+gβ€˜π‘‡)𝑦)) ≀ (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦)))))
7221, 71mpbird 256 . . . . . . . 8 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:π‘‹βŸΆβ„) β†’ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))
737, 72jca 510 . . . . . . 7 (((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) ∧ 𝐺 ∈ Grp ∧ 𝑁:π‘‹βŸΆβ„) β†’ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))))
74733exp 1116 . . . . . 6 ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) β†’ (𝐺 ∈ Grp β†’ (𝑁:π‘‹βŸΆβ„ β†’ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))))))
756, 74mpd 15 . . . . 5 ((𝑁 ∈ V ∧ 𝑇 ∈ NrmGrp) β†’ (𝑁:π‘‹βŸΆβ„ β†’ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))))
7675expcom 412 . . . 4 (𝑇 ∈ NrmGrp β†’ (𝑁 ∈ V β†’ (𝑁:π‘‹βŸΆβ„ β†’ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))))))
7776com13 88 . . 3 (𝑁:π‘‹βŸΆβ„ β†’ (𝑁 ∈ V β†’ (𝑇 ∈ NrmGrp β†’ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))))))
784, 77mpd 15 . 2 (𝑁:π‘‹βŸΆβ„ β†’ (𝑇 ∈ NrmGrp β†’ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))))
79 eqid 2728 . . . 4 (-gβ€˜πΊ) = (-gβ€˜πΊ)
80 simpl 481 . . . . 5 ((𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))) β†’ 𝐺 ∈ Grp)
8180adantl 480 . . . 4 ((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) β†’ 𝐺 ∈ Grp)
82 simpl 481 . . . 4 ((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) β†’ 𝑁:π‘‹βŸΆβ„)
83 fveq2 6902 . . . . . . . . . . . . 13 (π‘₯ = π‘Ž β†’ (π‘β€˜π‘₯) = (π‘β€˜π‘Ž))
8483eqeq1d 2730 . . . . . . . . . . . 12 (π‘₯ = π‘Ž β†’ ((π‘β€˜π‘₯) = 0 ↔ (π‘β€˜π‘Ž) = 0))
85 eqeq1 2732 . . . . . . . . . . . 12 (π‘₯ = π‘Ž β†’ (π‘₯ = 0 ↔ π‘Ž = 0 ))
8684, 85bibi12d 344 . . . . . . . . . . 11 (π‘₯ = π‘Ž β†’ (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ↔ ((π‘β€˜π‘Ž) = 0 ↔ π‘Ž = 0 )))
87 fveq2 6902 . . . . . . . . . . . . 13 (π‘₯ = π‘Ž β†’ (πΌβ€˜π‘₯) = (πΌβ€˜π‘Ž))
8887fveq2d 6906 . . . . . . . . . . . 12 (π‘₯ = π‘Ž β†’ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜(πΌβ€˜π‘Ž)))
8988, 83eqeq12d 2744 . . . . . . . . . . 11 (π‘₯ = π‘Ž β†’ ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ↔ (π‘β€˜(πΌβ€˜π‘Ž)) = (π‘β€˜π‘Ž)))
90 fvoveq1 7449 . . . . . . . . . . . . 13 (π‘₯ = π‘Ž β†’ (π‘β€˜(π‘₯ + 𝑦)) = (π‘β€˜(π‘Ž + 𝑦)))
9183oveq1d 7441 . . . . . . . . . . . . 13 (π‘₯ = π‘Ž β†’ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) = ((π‘β€˜π‘Ž) + (π‘β€˜π‘¦)))
9290, 91breq12d 5165 . . . . . . . . . . . 12 (π‘₯ = π‘Ž β†’ ((π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) ↔ (π‘β€˜(π‘Ž + 𝑦)) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘¦))))
9392ralbidv 3175 . . . . . . . . . . 11 (π‘₯ = π‘Ž β†’ (βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘Ž + 𝑦)) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘¦))))
9486, 89, 933anbi123d 1432 . . . . . . . . . 10 (π‘₯ = π‘Ž β†’ ((((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) ↔ (((π‘β€˜π‘Ž) = 0 ↔ π‘Ž = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘Ž)) = (π‘β€˜π‘Ž) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘Ž + 𝑦)) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘¦)))))
9594rspccva 3610 . . . . . . . . 9 ((βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) ∧ π‘Ž ∈ 𝑋) β†’ (((π‘β€˜π‘Ž) = 0 ↔ π‘Ž = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘Ž)) = (π‘β€˜π‘Ž) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘Ž + 𝑦)) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘¦))))
96 simp1 1133 . . . . . . . . 9 ((((π‘β€˜π‘Ž) = 0 ↔ π‘Ž = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘Ž)) = (π‘β€˜π‘Ž) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘Ž + 𝑦)) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘¦))) β†’ ((π‘β€˜π‘Ž) = 0 ↔ π‘Ž = 0 ))
9795, 96syl 17 . . . . . . . 8 ((βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) ∧ π‘Ž ∈ 𝑋) β†’ ((π‘β€˜π‘Ž) = 0 ↔ π‘Ž = 0 ))
9897ex 411 . . . . . . 7 (βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ (π‘Ž ∈ 𝑋 β†’ ((π‘β€˜π‘Ž) = 0 ↔ π‘Ž = 0 )))
9998adantl 480 . . . . . 6 ((𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))) β†’ (π‘Ž ∈ 𝑋 β†’ ((π‘β€˜π‘Ž) = 0 ↔ π‘Ž = 0 )))
10099adantl 480 . . . . 5 ((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) β†’ (π‘Ž ∈ 𝑋 β†’ ((π‘β€˜π‘Ž) = 0 ↔ π‘Ž = 0 )))
101100imp 405 . . . 4 (((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) ∧ π‘Ž ∈ 𝑋) β†’ ((π‘β€˜π‘Ž) = 0 ↔ π‘Ž = 0 ))
1021, 23, 25, 79grpsubval 18949 . . . . . . 7 ((π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) β†’ (π‘Ž(-gβ€˜πΊ)𝑏) = (π‘Ž + (πΌβ€˜π‘)))
103102adantl 480 . . . . . 6 (((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (π‘Ž(-gβ€˜πΊ)𝑏) = (π‘Ž + (πΌβ€˜π‘)))
104103fveq2d 6906 . . . . 5 (((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (π‘β€˜(π‘Ž(-gβ€˜πΊ)𝑏)) = (π‘β€˜(π‘Ž + (πΌβ€˜π‘))))
105 3simpc 1147 . . . . . . . . . 10 ((((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))
106105ralimi 3080 . . . . . . . . 9 (βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))
107 simpr 483 . . . . . . . . . . . . . . . 16 (((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
108107ralimi 3080 . . . . . . . . . . . . . . 15 (βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
109 oveq2 7434 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (πΌβ€˜π‘) β†’ (π‘Ž + 𝑦) = (π‘Ž + (πΌβ€˜π‘)))
110109fveq2d 6906 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (πΌβ€˜π‘) β†’ (π‘β€˜(π‘Ž + 𝑦)) = (π‘β€˜(π‘Ž + (πΌβ€˜π‘))))
111 fveq2 6902 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (πΌβ€˜π‘) β†’ (π‘β€˜π‘¦) = (π‘β€˜(πΌβ€˜π‘)))
112111oveq2d 7442 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (πΌβ€˜π‘) β†’ ((π‘β€˜π‘Ž) + (π‘β€˜π‘¦)) = ((π‘β€˜π‘Ž) + (π‘β€˜(πΌβ€˜π‘))))
113110, 112breq12d 5165 . . . . . . . . . . . . . . . . . 18 (𝑦 = (πΌβ€˜π‘) β†’ ((π‘β€˜(π‘Ž + 𝑦)) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘¦)) ↔ (π‘β€˜(π‘Ž + (πΌβ€˜π‘))) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜(πΌβ€˜π‘)))))
11492, 113rspc2v 3622 . . . . . . . . . . . . . . . . 17 ((π‘Ž ∈ 𝑋 ∧ (πΌβ€˜π‘) ∈ 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) β†’ (π‘β€˜(π‘Ž + (πΌβ€˜π‘))) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜(πΌβ€˜π‘)))))
1151, 25grpinvcl 18951 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ Grp ∧ 𝑏 ∈ 𝑋) β†’ (πΌβ€˜π‘) ∈ 𝑋)
116115ex 411 . . . . . . . . . . . . . . . . . . 19 (𝐺 ∈ Grp β†’ (𝑏 ∈ 𝑋 β†’ (πΌβ€˜π‘) ∈ 𝑋))
117116anim2d 610 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ Grp β†’ ((π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) β†’ (π‘Ž ∈ 𝑋 ∧ (πΌβ€˜π‘) ∈ 𝑋)))
118117imp 405 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Grp ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (π‘Ž ∈ 𝑋 ∧ (πΌβ€˜π‘) ∈ 𝑋))
119114, 118syl11 33 . . . . . . . . . . . . . . . 16 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) β†’ ((𝐺 ∈ Grp ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (π‘β€˜(π‘Ž + (πΌβ€˜π‘))) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜(πΌβ€˜π‘)))))
120119expd 414 . . . . . . . . . . . . . . 15 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) β†’ (𝐺 ∈ Grp β†’ ((π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) β†’ (π‘β€˜(π‘Ž + (πΌβ€˜π‘))) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜(πΌβ€˜π‘))))))
121108, 120syl 17 . . . . . . . . . . . . . 14 (βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ (𝐺 ∈ Grp β†’ ((π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) β†’ (π‘β€˜(π‘Ž + (πΌβ€˜π‘))) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜(πΌβ€˜π‘))))))
122121imp 405 . . . . . . . . . . . . 13 ((βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) ∧ 𝐺 ∈ Grp) β†’ ((π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) β†’ (π‘β€˜(π‘Ž + (πΌβ€˜π‘))) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜(πΌβ€˜π‘)))))
123122imp 405 . . . . . . . . . . . 12 (((βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) ∧ 𝐺 ∈ Grp) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (π‘β€˜(π‘Ž + (πΌβ€˜π‘))) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜(πΌβ€˜π‘))))
124 simpl 481 . . . . . . . . . . . . . . . . . 18 (((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯))
125124ralimi 3080 . . . . . . . . . . . . . . . . 17 (βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ βˆ€π‘₯ ∈ 𝑋 (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯))
126 fveq2 6902 . . . . . . . . . . . . . . . . . . . . . 22 (π‘₯ = 𝑏 β†’ (πΌβ€˜π‘₯) = (πΌβ€˜π‘))
127126fveq2d 6906 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ = 𝑏 β†’ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜(πΌβ€˜π‘)))
128 fveq2 6902 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ = 𝑏 β†’ (π‘β€˜π‘₯) = (π‘β€˜π‘))
129127, 128eqeq12d 2744 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ = 𝑏 β†’ ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ↔ (π‘β€˜(πΌβ€˜π‘)) = (π‘β€˜π‘)))
130129rspccva 3610 . . . . . . . . . . . . . . . . . . 19 ((βˆ€π‘₯ ∈ 𝑋 (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ 𝑏 ∈ 𝑋) β†’ (π‘β€˜(πΌβ€˜π‘)) = (π‘β€˜π‘))
131130eqcomd 2734 . . . . . . . . . . . . . . . . . 18 ((βˆ€π‘₯ ∈ 𝑋 (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ 𝑏 ∈ 𝑋) β†’ (π‘β€˜π‘) = (π‘β€˜(πΌβ€˜π‘)))
132131ex 411 . . . . . . . . . . . . . . . . 17 (βˆ€π‘₯ ∈ 𝑋 (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) β†’ (𝑏 ∈ 𝑋 β†’ (π‘β€˜π‘) = (π‘β€˜(πΌβ€˜π‘))))
133125, 132syl 17 . . . . . . . . . . . . . . . 16 (βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ (𝑏 ∈ 𝑋 β†’ (π‘β€˜π‘) = (π‘β€˜(πΌβ€˜π‘))))
134133adantr 479 . . . . . . . . . . . . . . 15 ((βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) ∧ 𝐺 ∈ Grp) β†’ (𝑏 ∈ 𝑋 β†’ (π‘β€˜π‘) = (π‘β€˜(πΌβ€˜π‘))))
135134adantld 489 . . . . . . . . . . . . . 14 ((βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) ∧ 𝐺 ∈ Grp) β†’ ((π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) β†’ (π‘β€˜π‘) = (π‘β€˜(πΌβ€˜π‘))))
136135imp 405 . . . . . . . . . . . . 13 (((βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) ∧ 𝐺 ∈ Grp) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (π‘β€˜π‘) = (π‘β€˜(πΌβ€˜π‘)))
137136oveq2d 7442 . . . . . . . . . . . 12 (((βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) ∧ 𝐺 ∈ Grp) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ ((π‘β€˜π‘Ž) + (π‘β€˜π‘)) = ((π‘β€˜π‘Ž) + (π‘β€˜(πΌβ€˜π‘))))
138123, 137breqtrrd 5180 . . . . . . . . . . 11 (((βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) ∧ 𝐺 ∈ Grp) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (π‘β€˜(π‘Ž + (πΌβ€˜π‘))) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘)))
139138ex 411 . . . . . . . . . 10 ((βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) ∧ 𝐺 ∈ Grp) β†’ ((π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) β†’ (π‘β€˜(π‘Ž + (πΌβ€˜π‘))) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘))))
140139ex 411 . . . . . . . . 9 (βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ (𝐺 ∈ Grp β†’ ((π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) β†’ (π‘β€˜(π‘Ž + (πΌβ€˜π‘))) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘)))))
141106, 140syl 17 . . . . . . . 8 (βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ (𝐺 ∈ Grp β†’ ((π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) β†’ (π‘β€˜(π‘Ž + (πΌβ€˜π‘))) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘)))))
142141impcom 406 . . . . . . 7 ((𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))) β†’ ((π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) β†’ (π‘β€˜(π‘Ž + (πΌβ€˜π‘))) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘))))
143142adantl 480 . . . . . 6 ((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) β†’ ((π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) β†’ (π‘β€˜(π‘Ž + (πΌβ€˜π‘))) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘))))
144143imp 405 . . . . 5 (((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (π‘β€˜(π‘Ž + (πΌβ€˜π‘))) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘)))
145104, 144eqbrtrd 5174 . . . 4 (((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (π‘β€˜(π‘Ž(-gβ€˜πΊ)𝑏)) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘)))
1465, 1, 79, 41, 81, 82, 101, 145tngngpd 24590 . . 3 ((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) β†’ 𝑇 ∈ NrmGrp)
147146ex 411 . 2 (𝑁:π‘‹βŸΆβ„ β†’ ((𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))) β†’ 𝑇 ∈ NrmGrp))
14878, 147impbid 211 1 (𝑁:π‘‹βŸΆβ„ β†’ (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  Vcvv 3473   class class class wbr 5152  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426  β„cr 11145  0cc0 11146   + caddc 11149   ≀ cle 11287  Basecbs 17187  +gcplusg 17240  0gc0g 17428  Grpcgrp 18897  invgcminusg 18898  -gcsg 18899  normcnm 24505  NrmGrpcngp 24506   toNrmGrp ctng 24507
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-sup 9473  df-inf 9474  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-q 12971  df-rp 13015  df-xneg 13132  df-xadd 13133  df-xmul 13134  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-plusg 17253  df-tset 17259  df-ds 17262  df-rest 17411  df-topn 17412  df-0g 17430  df-topgen 17432  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-grp 18900  df-minusg 18901  df-sbg 18902  df-psmet 21278  df-xmet 21279  df-met 21280  df-bl 21281  df-mopn 21282  df-top 22816  df-topon 22833  df-topsp 22855  df-bases 22869  df-xms 24246  df-ms 24247  df-nm 24511  df-ngp 24512  df-tng 24513
This theorem is referenced by: (None)
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