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Theorem tngngp 24162
Description: Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
tngngp.t 𝑇 = (𝐺 toNrmGrp 𝑁)
tngngp.x 𝑋 = (Baseβ€˜πΊ)
tngngp.m βˆ’ = (-gβ€˜πΊ)
tngngp.z 0 = (0gβ€˜πΊ)
Assertion
Ref Expression
tngngp (𝑁:π‘‹βŸΆβ„ β†’ (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))))
Distinct variable groups:   π‘₯,𝑦, βˆ’   π‘₯,𝑁,𝑦   π‘₯,𝑇,𝑦   π‘₯,𝑋,𝑦   π‘₯, 0 ,𝑦
Allowed substitution hints:   𝐺(π‘₯,𝑦)

Proof of Theorem tngngp
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tngngp.t . . . . 5 𝑇 = (𝐺 toNrmGrp 𝑁)
2 tngngp.x . . . . 5 𝑋 = (Baseβ€˜πΊ)
3 eqid 2732 . . . . 5 (distβ€˜π‘‡) = (distβ€˜π‘‡)
41, 2, 3tngngp2 24160 . . . 4 (𝑁:π‘‹βŸΆβ„ β†’ (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (distβ€˜π‘‡) ∈ (Metβ€˜π‘‹))))
54simprbda 499 . . 3 ((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) β†’ 𝐺 ∈ Grp)
6 simplr 767 . . . . . . 7 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ 𝑇 ∈ NrmGrp)
7 simpr 485 . . . . . . . 8 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
82fvexi 6902 . . . . . . . . . . 11 𝑋 ∈ V
9 reex 11197 . . . . . . . . . . 11 ℝ ∈ V
10 fex2 7920 . . . . . . . . . . 11 ((𝑁:π‘‹βŸΆβ„ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) β†’ 𝑁 ∈ V)
118, 9, 10mp3an23 1453 . . . . . . . . . 10 (𝑁:π‘‹βŸΆβ„ β†’ 𝑁 ∈ V)
1211ad2antrr 724 . . . . . . . . 9 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ 𝑁 ∈ V)
131, 2tngbas 24142 . . . . . . . . 9 (𝑁 ∈ V β†’ 𝑋 = (Baseβ€˜π‘‡))
1412, 13syl 17 . . . . . . . 8 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ 𝑋 = (Baseβ€˜π‘‡))
157, 14eleqtrd 2835 . . . . . . 7 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ (Baseβ€˜π‘‡))
16 eqid 2732 . . . . . . . 8 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
17 eqid 2732 . . . . . . . 8 (normβ€˜π‘‡) = (normβ€˜π‘‡)
18 eqid 2732 . . . . . . . 8 (0gβ€˜π‘‡) = (0gβ€˜π‘‡)
1916, 17, 18nmeq0 24118 . . . . . . 7 ((𝑇 ∈ NrmGrp ∧ π‘₯ ∈ (Baseβ€˜π‘‡)) β†’ (((normβ€˜π‘‡)β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘‡)))
206, 15, 19syl2anc 584 . . . . . 6 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (((normβ€˜π‘‡)β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘‡)))
215adantr 481 . . . . . . . . 9 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ 𝐺 ∈ Grp)
22 simpll 765 . . . . . . . . 9 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ 𝑁:π‘‹βŸΆβ„)
231, 2, 9tngnm 24159 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑁:π‘‹βŸΆβ„) β†’ 𝑁 = (normβ€˜π‘‡))
2421, 22, 23syl2anc 584 . . . . . . . 8 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ 𝑁 = (normβ€˜π‘‡))
2524fveq1d 6890 . . . . . . 7 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (π‘β€˜π‘₯) = ((normβ€˜π‘‡)β€˜π‘₯))
2625eqeq1d 2734 . . . . . 6 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ ((π‘β€˜π‘₯) = 0 ↔ ((normβ€˜π‘‡)β€˜π‘₯) = 0))
27 tngngp.z . . . . . . . . 9 0 = (0gβ€˜πΊ)
281, 27tng0 24146 . . . . . . . 8 (𝑁 ∈ V β†’ 0 = (0gβ€˜π‘‡))
2912, 28syl 17 . . . . . . 7 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ 0 = (0gβ€˜π‘‡))
3029eqeq2d 2743 . . . . . 6 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯ = 0 ↔ π‘₯ = (0gβ€˜π‘‡)))
3120, 26, 303bitr4d 310 . . . . 5 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ ((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ))
32 simpllr 774 . . . . . . . 8 ((((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ 𝑇 ∈ NrmGrp)
3315adantr 481 . . . . . . . 8 ((((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ π‘₯ ∈ (Baseβ€˜π‘‡))
3414eleq2d 2819 . . . . . . . . 9 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ (Baseβ€˜π‘‡)))
3534biimpa 477 . . . . . . . 8 ((((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ 𝑦 ∈ (Baseβ€˜π‘‡))
36 eqid 2732 . . . . . . . . 9 (-gβ€˜π‘‡) = (-gβ€˜π‘‡)
3716, 17, 36nmmtri 24122 . . . . . . . 8 ((𝑇 ∈ NrmGrp ∧ π‘₯ ∈ (Baseβ€˜π‘‡) ∧ 𝑦 ∈ (Baseβ€˜π‘‡)) β†’ ((normβ€˜π‘‡)β€˜(π‘₯(-gβ€˜π‘‡)𝑦)) ≀ (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦)))
3832, 33, 35, 37syl3anc 1371 . . . . . . 7 ((((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ ((normβ€˜π‘‡)β€˜(π‘₯(-gβ€˜π‘‡)𝑦)) ≀ (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦)))
39 tngngp.m . . . . . . . . . . 11 βˆ’ = (-gβ€˜πΊ)
402, 14eqtr3id 2786 . . . . . . . . . . . 12 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (Baseβ€˜πΊ) = (Baseβ€˜π‘‡))
41 eqid 2732 . . . . . . . . . . . . . 14 (+gβ€˜πΊ) = (+gβ€˜πΊ)
421, 41tngplusg 24144 . . . . . . . . . . . . 13 (𝑁 ∈ V β†’ (+gβ€˜πΊ) = (+gβ€˜π‘‡))
4312, 42syl 17 . . . . . . . . . . . 12 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (+gβ€˜πΊ) = (+gβ€˜π‘‡))
4440, 43grpsubpropd 18924 . . . . . . . . . . 11 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (-gβ€˜πΊ) = (-gβ€˜π‘‡))
4539, 44eqtrid 2784 . . . . . . . . . 10 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ βˆ’ = (-gβ€˜π‘‡))
4645oveqd 7422 . . . . . . . . 9 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯ βˆ’ 𝑦) = (π‘₯(-gβ€˜π‘‡)𝑦))
4724, 46fveq12d 6895 . . . . . . . 8 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (π‘β€˜(π‘₯ βˆ’ 𝑦)) = ((normβ€˜π‘‡)β€˜(π‘₯(-gβ€˜π‘‡)𝑦)))
4847adantr 481 . . . . . . 7 ((((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ (π‘β€˜(π‘₯ βˆ’ 𝑦)) = ((normβ€˜π‘‡)β€˜(π‘₯(-gβ€˜π‘‡)𝑦)))
4924fveq1d 6890 . . . . . . . . 9 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (π‘β€˜π‘¦) = ((normβ€˜π‘‡)β€˜π‘¦))
5025, 49oveq12d 7423 . . . . . . . 8 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) = (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦)))
5150adantr 481 . . . . . . 7 ((((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) = (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦)))
5238, 48, 513brtr4d 5179 . . . . . 6 ((((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
5352ralrimiva 3146 . . . . 5 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
5431, 53jca 512 . . . 4 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))
5554ralrimiva 3146 . . 3 ((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) β†’ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))
565, 55jca 512 . 2 ((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) β†’ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))))
57 simprl 769 . . 3 ((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) β†’ 𝐺 ∈ Grp)
58 simpl 483 . . 3 ((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) β†’ 𝑁:π‘‹βŸΆβ„)
59 simpl 483 . . . . . 6 ((((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ ((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ))
6059ralimi 3083 . . . . 5 (βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ))
6160ad2antll 727 . . . 4 ((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) β†’ βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ))
62 fveq2 6888 . . . . . . 7 (π‘₯ = π‘Ž β†’ (π‘β€˜π‘₯) = (π‘β€˜π‘Ž))
6362eqeq1d 2734 . . . . . 6 (π‘₯ = π‘Ž β†’ ((π‘β€˜π‘₯) = 0 ↔ (π‘β€˜π‘Ž) = 0))
64 eqeq1 2736 . . . . . 6 (π‘₯ = π‘Ž β†’ (π‘₯ = 0 ↔ π‘Ž = 0 ))
6563, 64bibi12d 345 . . . . 5 (π‘₯ = π‘Ž β†’ (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ↔ ((π‘β€˜π‘Ž) = 0 ↔ π‘Ž = 0 )))
6665rspccva 3611 . . . 4 ((βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ π‘Ž ∈ 𝑋) β†’ ((π‘β€˜π‘Ž) = 0 ↔ π‘Ž = 0 ))
6761, 66sylan 580 . . 3 (((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) ∧ π‘Ž ∈ 𝑋) β†’ ((π‘β€˜π‘Ž) = 0 ↔ π‘Ž = 0 ))
68 simpr 485 . . . . . 6 ((((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
6968ralimi 3083 . . . . 5 (βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
7069ad2antll 727 . . . 4 ((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
71 fvoveq1 7428 . . . . . . 7 (π‘₯ = π‘Ž β†’ (π‘β€˜(π‘₯ βˆ’ 𝑦)) = (π‘β€˜(π‘Ž βˆ’ 𝑦)))
7262oveq1d 7420 . . . . . . 7 (π‘₯ = π‘Ž β†’ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) = ((π‘β€˜π‘Ž) + (π‘β€˜π‘¦)))
7371, 72breq12d 5160 . . . . . 6 (π‘₯ = π‘Ž β†’ ((π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) ↔ (π‘β€˜(π‘Ž βˆ’ 𝑦)) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘¦))))
74 oveq2 7413 . . . . . . . 8 (𝑦 = 𝑏 β†’ (π‘Ž βˆ’ 𝑦) = (π‘Ž βˆ’ 𝑏))
7574fveq2d 6892 . . . . . . 7 (𝑦 = 𝑏 β†’ (π‘β€˜(π‘Ž βˆ’ 𝑦)) = (π‘β€˜(π‘Ž βˆ’ 𝑏)))
76 fveq2 6888 . . . . . . . 8 (𝑦 = 𝑏 β†’ (π‘β€˜π‘¦) = (π‘β€˜π‘))
7776oveq2d 7421 . . . . . . 7 (𝑦 = 𝑏 β†’ ((π‘β€˜π‘Ž) + (π‘β€˜π‘¦)) = ((π‘β€˜π‘Ž) + (π‘β€˜π‘)))
7875, 77breq12d 5160 . . . . . 6 (𝑦 = 𝑏 β†’ ((π‘β€˜(π‘Ž βˆ’ 𝑦)) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘¦)) ↔ (π‘β€˜(π‘Ž βˆ’ 𝑏)) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘))))
7973, 78rspc2va 3622 . . . . 5 (((π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ (π‘β€˜(π‘Ž βˆ’ 𝑏)) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘)))
8079ancoms 459 . . . 4 ((βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (π‘β€˜(π‘Ž βˆ’ 𝑏)) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘)))
8170, 80sylan 580 . . 3 (((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (π‘β€˜(π‘Ž βˆ’ 𝑏)) ≀ ((π‘β€˜π‘Ž) + (π‘β€˜π‘)))
821, 2, 39, 27, 57, 58, 67, 81tngngpd 24161 . 2 ((𝑁:π‘‹βŸΆβ„ ∧ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))) β†’ 𝑇 ∈ NrmGrp)
8356, 82impbida 799 1 (𝑁:π‘‹βŸΆβ„ β†’ (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   class class class wbr 5147  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  β„cr 11105  0cc0 11106   + caddc 11109   ≀ cle 11245  Basecbs 17140  +gcplusg 17193  distcds 17202  0gc0g 17381  Grpcgrp 18815  -gcsg 18817  Metcmet 20922  normcnm 24076  NrmGrpcngp 24077   toNrmGrp ctng 24078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-sup 9433  df-inf 9434  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-plusg 17206  df-tset 17212  df-ds 17215  df-rest 17364  df-topn 17365  df-0g 17383  df-topgen 17385  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-sbg 18820  df-psmet 20928  df-xmet 20929  df-met 20930  df-bl 20931  df-mopn 20932  df-top 22387  df-topon 22404  df-topsp 22426  df-bases 22440  df-xms 23817  df-ms 23818  df-nm 24082  df-ngp 24083  df-tng 24084
This theorem is referenced by: (None)
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