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Theorem tngngp 24034
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 2737 . . . . 5 (distβ€˜π‘‡) = (distβ€˜π‘‡)
41, 2, 3tngngp2 24032 . . . 4 (𝑁:π‘‹βŸΆβ„ β†’ (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (distβ€˜π‘‡) ∈ (Metβ€˜π‘‹))))
54simprbda 500 . . 3 ((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) β†’ 𝐺 ∈ Grp)
6 simplr 768 . . . . . . 7 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ 𝑇 ∈ NrmGrp)
7 simpr 486 . . . . . . . 8 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
82fvexi 6861 . . . . . . . . . . 11 𝑋 ∈ V
9 reex 11149 . . . . . . . . . . 11 ℝ ∈ V
10 fex2 7875 . . . . . . . . . . 11 ((𝑁:π‘‹βŸΆβ„ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) β†’ 𝑁 ∈ V)
118, 9, 10mp3an23 1454 . . . . . . . . . 10 (𝑁:π‘‹βŸΆβ„ β†’ 𝑁 ∈ V)
1211ad2antrr 725 . . . . . . . . 9 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ 𝑁 ∈ V)
131, 2tngbas 24014 . . . . . . . . 9 (𝑁 ∈ V β†’ 𝑋 = (Baseβ€˜π‘‡))
1412, 13syl 17 . . . . . . . 8 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ 𝑋 = (Baseβ€˜π‘‡))
157, 14eleqtrd 2840 . . . . . . 7 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ (Baseβ€˜π‘‡))
16 eqid 2737 . . . . . . . 8 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
17 eqid 2737 . . . . . . . 8 (normβ€˜π‘‡) = (normβ€˜π‘‡)
18 eqid 2737 . . . . . . . 8 (0gβ€˜π‘‡) = (0gβ€˜π‘‡)
1916, 17, 18nmeq0 23990 . . . . . . 7 ((𝑇 ∈ NrmGrp ∧ π‘₯ ∈ (Baseβ€˜π‘‡)) β†’ (((normβ€˜π‘‡)β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘‡)))
206, 15, 19syl2anc 585 . . . . . 6 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (((normβ€˜π‘‡)β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘‡)))
215adantr 482 . . . . . . . . 9 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ 𝐺 ∈ Grp)
22 simpll 766 . . . . . . . . 9 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ 𝑁:π‘‹βŸΆβ„)
231, 2, 9tngnm 24031 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑁:π‘‹βŸΆβ„) β†’ 𝑁 = (normβ€˜π‘‡))
2421, 22, 23syl2anc 585 . . . . . . . 8 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ 𝑁 = (normβ€˜π‘‡))
2524fveq1d 6849 . . . . . . 7 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (π‘β€˜π‘₯) = ((normβ€˜π‘‡)β€˜π‘₯))
2625eqeq1d 2739 . . . . . 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 24018 . . . . . . . 8 (𝑁 ∈ V β†’ 0 = (0gβ€˜π‘‡))
2912, 28syl 17 . . . . . . 7 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ 0 = (0gβ€˜π‘‡))
3029eqeq2d 2748 . . . . . 6 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯ = 0 ↔ π‘₯ = (0gβ€˜π‘‡)))
3120, 26, 303bitr4d 311 . . . . 5 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ ((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ))
32 simpllr 775 . . . . . . . 8 ((((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ 𝑇 ∈ NrmGrp)
3315adantr 482 . . . . . . . 8 ((((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ π‘₯ ∈ (Baseβ€˜π‘‡))
3414eleq2d 2824 . . . . . . . . 9 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ (Baseβ€˜π‘‡)))
3534biimpa 478 . . . . . . . 8 ((((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ 𝑦 ∈ (Baseβ€˜π‘‡))
36 eqid 2737 . . . . . . . . 9 (-gβ€˜π‘‡) = (-gβ€˜π‘‡)
3716, 17, 36nmmtri 23994 . . . . . . . 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 1372 . . . . . . 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 2791 . . . . . . . . . . . 12 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (Baseβ€˜πΊ) = (Baseβ€˜π‘‡))
41 eqid 2737 . . . . . . . . . . . . . 14 (+gβ€˜πΊ) = (+gβ€˜πΊ)
421, 41tngplusg 24016 . . . . . . . . . . . . 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 18859 . . . . . . . . . . 11 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (-gβ€˜πΊ) = (-gβ€˜π‘‡))
4539, 44eqtrid 2789 . . . . . . . . . 10 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ βˆ’ = (-gβ€˜π‘‡))
4645oveqd 7379 . . . . . . . . 9 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯ βˆ’ 𝑦) = (π‘₯(-gβ€˜π‘‡)𝑦))
4724, 46fveq12d 6854 . . . . . . . 8 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (π‘β€˜(π‘₯ βˆ’ 𝑦)) = ((normβ€˜π‘‡)β€˜(π‘₯(-gβ€˜π‘‡)𝑦)))
4847adantr 482 . . . . . . 7 ((((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ (π‘β€˜(π‘₯ βˆ’ 𝑦)) = ((normβ€˜π‘‡)β€˜(π‘₯(-gβ€˜π‘‡)𝑦)))
4924fveq1d 6849 . . . . . . . . 9 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ (π‘β€˜π‘¦) = ((normβ€˜π‘‡)β€˜π‘¦))
5025, 49oveq12d 7380 . . . . . . . 8 (((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) β†’ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) = (((normβ€˜π‘‡)β€˜π‘₯) + ((normβ€˜π‘‡)β€˜π‘¦)))
5150adantr 482 . . . . . . 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 5142 . . . . . 6 ((((𝑁:π‘‹βŸΆβ„ ∧ 𝑇 ∈ NrmGrp) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
5352ralrimiva 3144 . . . . 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 513 . . . 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 3144 . . 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 513 . 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 770 . . 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 484 . . 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 484 . . . . . 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 3087 . . . . 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 728 . . . 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 6847 . . . . . . 7 (π‘₯ = π‘Ž β†’ (π‘β€˜π‘₯) = (π‘β€˜π‘Ž))
6362eqeq1d 2739 . . . . . 6 (π‘₯ = π‘Ž β†’ ((π‘β€˜π‘₯) = 0 ↔ (π‘β€˜π‘Ž) = 0))
64 eqeq1 2741 . . . . . 6 (π‘₯ = π‘Ž β†’ (π‘₯ = 0 ↔ π‘Ž = 0 ))
6563, 64bibi12d 346 . . . . 5 (π‘₯ = π‘Ž β†’ (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ↔ ((π‘β€˜π‘Ž) = 0 ↔ π‘Ž = 0 )))
6665rspccva 3583 . . . 4 ((βˆ€π‘₯ ∈ 𝑋 ((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ π‘Ž ∈ 𝑋) β†’ ((π‘β€˜π‘Ž) = 0 ↔ π‘Ž = 0 ))
6761, 66sylan 581 . . 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 486 . . . . . 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 3087 . . . . 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 728 . . . 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 7385 . . . . . . 7 (π‘₯ = π‘Ž β†’ (π‘β€˜(π‘₯ βˆ’ 𝑦)) = (π‘β€˜(π‘Ž βˆ’ 𝑦)))
7262oveq1d 7377 . . . . . . 7 (π‘₯ = π‘Ž β†’ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) = ((π‘β€˜π‘Ž) + (π‘β€˜π‘¦)))
7371, 72breq12d 5123 . . . . . 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 7370 . . . . . . . 8 (𝑦 = 𝑏 β†’ (π‘Ž βˆ’ 𝑦) = (π‘Ž βˆ’ 𝑏))
7574fveq2d 6851 . . . . . . 7 (𝑦 = 𝑏 β†’ (π‘β€˜(π‘Ž βˆ’ 𝑦)) = (π‘β€˜(π‘Ž βˆ’ 𝑏)))
76 fveq2 6847 . . . . . . . 8 (𝑦 = 𝑏 β†’ (π‘β€˜π‘¦) = (π‘β€˜π‘))
7776oveq2d 7378 . . . . . . 7 (𝑦 = 𝑏 β†’ ((π‘β€˜π‘Ž) + (π‘β€˜π‘¦)) = ((π‘β€˜π‘Ž) + (π‘β€˜π‘)))
7875, 77breq12d 5123 . . . . . 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 3594 . . . . 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 460 . . . 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 581 . . 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 24033 . 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 800 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  Vcvv 3448   class class class wbr 5110  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  β„cr 11057  0cc0 11058   + caddc 11061   ≀ cle 11197  Basecbs 17090  +gcplusg 17140  distcds 17149  0gc0g 17328  Grpcgrp 18755  -gcsg 18757  Metcmet 20798  normcnm 23948  NrmGrpcngp 23949   toNrmGrp ctng 23950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9385  df-inf 9386  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-z 12507  df-dec 12626  df-uz 12771  df-q 12881  df-rp 12923  df-xneg 13040  df-xadd 13041  df-xmul 13042  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-plusg 17153  df-tset 17159  df-ds 17162  df-rest 17311  df-topn 17312  df-0g 17330  df-topgen 17332  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-grp 18758  df-minusg 18759  df-sbg 18760  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807  df-mopn 20808  df-top 22259  df-topon 22276  df-topsp 22298  df-bases 22312  df-xms 23689  df-ms 23690  df-nm 23954  df-ngp 23955  df-tng 23956
This theorem is referenced by: (None)
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