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Theorem isnvlem 29863
Description: Lemma for isnv 29865. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
isnvlem.1 𝑋 = ran 𝐺
isnvlem.2 𝑍 = (GIdβ€˜πΊ)
Assertion
Ref Expression
isnvlem ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) β†’ (⟨⟨𝐺, π‘†βŸ©, π‘βŸ© ∈ NrmCVec ↔ (⟨𝐺, π‘†βŸ© ∈ CVecOLD ∧ 𝑁:π‘‹βŸΆβ„ ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))))
Distinct variable groups:   π‘₯,𝑦,𝐺   π‘₯,𝑁,𝑦   π‘₯,𝑆,𝑦   π‘₯,𝑋,𝑦
Allowed substitution hints:   𝑍(π‘₯,𝑦)

Proof of Theorem isnvlem
Dummy variables 𝑔 𝑛 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nv 29845 . . 3 NrmCVec = {βŸ¨βŸ¨π‘”, π‘ βŸ©, π‘›βŸ© ∣ (βŸ¨π‘”, π‘ βŸ© ∈ CVecOLD ∧ 𝑛:ran π‘”βŸΆβ„ ∧ βˆ€π‘₯ ∈ ran 𝑔(((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = (GIdβ€˜π‘”)) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑠π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ ran 𝑔(π‘›β€˜(π‘₯𝑔𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦))))}
21eleq2i 2826 . 2 (⟨⟨𝐺, π‘†βŸ©, π‘βŸ© ∈ NrmCVec ↔ ⟨⟨𝐺, π‘†βŸ©, π‘βŸ© ∈ {βŸ¨βŸ¨π‘”, π‘ βŸ©, π‘›βŸ© ∣ (βŸ¨π‘”, π‘ βŸ© ∈ CVecOLD ∧ 𝑛:ran π‘”βŸΆβ„ ∧ βˆ€π‘₯ ∈ ran 𝑔(((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = (GIdβ€˜π‘”)) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑠π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ ran 𝑔(π‘›β€˜(π‘₯𝑔𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦))))})
3 opeq1 4874 . . . . 5 (𝑔 = 𝐺 β†’ βŸ¨π‘”, π‘ βŸ© = ⟨𝐺, π‘ βŸ©)
43eleq1d 2819 . . . 4 (𝑔 = 𝐺 β†’ (βŸ¨π‘”, π‘ βŸ© ∈ CVecOLD ↔ ⟨𝐺, π‘ βŸ© ∈ CVecOLD))
5 rneq 5936 . . . . . 6 (𝑔 = 𝐺 β†’ ran 𝑔 = ran 𝐺)
6 isnvlem.1 . . . . . 6 𝑋 = ran 𝐺
75, 6eqtr4di 2791 . . . . 5 (𝑔 = 𝐺 β†’ ran 𝑔 = 𝑋)
87feq2d 6704 . . . 4 (𝑔 = 𝐺 β†’ (𝑛:ran π‘”βŸΆβ„ ↔ 𝑛:π‘‹βŸΆβ„))
9 fveq2 6892 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (GIdβ€˜π‘”) = (GIdβ€˜πΊ))
10 isnvlem.2 . . . . . . . . 9 𝑍 = (GIdβ€˜πΊ)
119, 10eqtr4di 2791 . . . . . . . 8 (𝑔 = 𝐺 β†’ (GIdβ€˜π‘”) = 𝑍)
1211eqeq2d 2744 . . . . . . 7 (𝑔 = 𝐺 β†’ (π‘₯ = (GIdβ€˜π‘”) ↔ π‘₯ = 𝑍))
1312imbi2d 341 . . . . . 6 (𝑔 = 𝐺 β†’ (((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = (GIdβ€˜π‘”)) ↔ ((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍)))
14 oveq 7415 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (π‘₯𝑔𝑦) = (π‘₯𝐺𝑦))
1514fveq2d 6896 . . . . . . . 8 (𝑔 = 𝐺 β†’ (π‘›β€˜(π‘₯𝑔𝑦)) = (π‘›β€˜(π‘₯𝐺𝑦)))
1615breq1d 5159 . . . . . . 7 (𝑔 = 𝐺 β†’ ((π‘›β€˜(π‘₯𝑔𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦)) ↔ (π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦))))
177, 16raleqbidv 3343 . . . . . 6 (𝑔 = 𝐺 β†’ (βˆ€π‘¦ ∈ ran 𝑔(π‘›β€˜(π‘₯𝑔𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ 𝑋 (π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦))))
1813, 173anbi13d 1439 . . . . 5 (𝑔 = 𝐺 β†’ ((((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = (GIdβ€˜π‘”)) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑠π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ ran 𝑔(π‘›β€˜(π‘₯𝑔𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦))) ↔ (((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑠π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦)))))
197, 18raleqbidv 3343 . . . 4 (𝑔 = 𝐺 β†’ (βˆ€π‘₯ ∈ ran 𝑔(((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = (GIdβ€˜π‘”)) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑠π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ ran 𝑔(π‘›β€˜(π‘₯𝑔𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦))) ↔ βˆ€π‘₯ ∈ 𝑋 (((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑠π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦)))))
204, 8, 193anbi123d 1437 . . 3 (𝑔 = 𝐺 β†’ ((βŸ¨π‘”, π‘ βŸ© ∈ CVecOLD ∧ 𝑛:ran π‘”βŸΆβ„ ∧ βˆ€π‘₯ ∈ ran 𝑔(((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = (GIdβ€˜π‘”)) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑠π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ ran 𝑔(π‘›β€˜(π‘₯𝑔𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦)))) ↔ (⟨𝐺, π‘ βŸ© ∈ CVecOLD ∧ 𝑛:π‘‹βŸΆβ„ ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑠π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦))))))
21 opeq2 4875 . . . . 5 (𝑠 = 𝑆 β†’ ⟨𝐺, π‘ βŸ© = ⟨𝐺, π‘†βŸ©)
2221eleq1d 2819 . . . 4 (𝑠 = 𝑆 β†’ (⟨𝐺, π‘ βŸ© ∈ CVecOLD ↔ ⟨𝐺, π‘†βŸ© ∈ CVecOLD))
23 oveq 7415 . . . . . . . 8 (𝑠 = 𝑆 β†’ (𝑦𝑠π‘₯) = (𝑦𝑆π‘₯))
2423fveqeq2d 6900 . . . . . . 7 (𝑠 = 𝑆 β†’ ((π‘›β€˜(𝑦𝑠π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ↔ (π‘›β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯))))
2524ralbidv 3178 . . . . . 6 (𝑠 = 𝑆 β†’ (βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑠π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ↔ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯))))
26253anbi2d 1442 . . . . 5 (𝑠 = 𝑆 β†’ ((((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑠π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦))) ↔ (((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦)))))
2726ralbidv 3178 . . . 4 (𝑠 = 𝑆 β†’ (βˆ€π‘₯ ∈ 𝑋 (((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑠π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦))) ↔ βˆ€π‘₯ ∈ 𝑋 (((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦)))))
2822, 273anbi13d 1439 . . 3 (𝑠 = 𝑆 β†’ ((⟨𝐺, π‘ βŸ© ∈ CVecOLD ∧ 𝑛:π‘‹βŸΆβ„ ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑠π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦)))) ↔ (⟨𝐺, π‘†βŸ© ∈ CVecOLD ∧ 𝑛:π‘‹βŸΆβ„ ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦))))))
29 feq1 6699 . . . 4 (𝑛 = 𝑁 β†’ (𝑛:π‘‹βŸΆβ„ ↔ 𝑁:π‘‹βŸΆβ„))
30 fveq1 6891 . . . . . . . 8 (𝑛 = 𝑁 β†’ (π‘›β€˜π‘₯) = (π‘β€˜π‘₯))
3130eqeq1d 2735 . . . . . . 7 (𝑛 = 𝑁 β†’ ((π‘›β€˜π‘₯) = 0 ↔ (π‘β€˜π‘₯) = 0))
3231imbi1d 342 . . . . . 6 (𝑛 = 𝑁 β†’ (((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ↔ ((π‘β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍)))
33 fveq1 6891 . . . . . . . 8 (𝑛 = 𝑁 β†’ (π‘›β€˜(𝑦𝑆π‘₯)) = (π‘β€˜(𝑦𝑆π‘₯)))
3430oveq2d 7425 . . . . . . . 8 (𝑛 = 𝑁 β†’ ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)))
3533, 34eqeq12d 2749 . . . . . . 7 (𝑛 = 𝑁 β†’ ((π‘›β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ↔ (π‘β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯))))
3635ralbidv 3178 . . . . . 6 (𝑛 = 𝑁 β†’ (βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ↔ βˆ€π‘¦ ∈ β„‚ (π‘β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯))))
37 fveq1 6891 . . . . . . . 8 (𝑛 = 𝑁 β†’ (π‘›β€˜(π‘₯𝐺𝑦)) = (π‘β€˜(π‘₯𝐺𝑦)))
38 fveq1 6891 . . . . . . . . 9 (𝑛 = 𝑁 β†’ (π‘›β€˜π‘¦) = (π‘β€˜π‘¦))
3930, 38oveq12d 7427 . . . . . . . 8 (𝑛 = 𝑁 β†’ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦)) = ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
4037, 39breq12d 5162 . . . . . . 7 (𝑛 = 𝑁 β†’ ((π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦)) ↔ (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))
4140ralbidv 3178 . . . . . 6 (𝑛 = 𝑁 β†’ (βˆ€π‘¦ ∈ 𝑋 (π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))
4232, 36, 413anbi123d 1437 . . . . 5 (𝑛 = 𝑁 β†’ ((((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦))) ↔ (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))))
4342ralbidv 3178 . . . 4 (𝑛 = 𝑁 β†’ (βˆ€π‘₯ ∈ 𝑋 (((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦))) ↔ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))))
4429, 433anbi23d 1440 . . 3 (𝑛 = 𝑁 β†’ ((⟨𝐺, π‘†βŸ© ∈ CVecOLD ∧ 𝑛:π‘‹βŸΆβ„ ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘›β€˜(π‘₯𝐺𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦)))) ↔ (⟨𝐺, π‘†βŸ© ∈ CVecOLD ∧ 𝑁:π‘‹βŸΆβ„ ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))))
4520, 28, 44eloprabg 7518 . 2 ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) β†’ (⟨⟨𝐺, π‘†βŸ©, π‘βŸ© ∈ {βŸ¨βŸ¨π‘”, π‘ βŸ©, π‘›βŸ© ∣ (βŸ¨π‘”, π‘ βŸ© ∈ CVecOLD ∧ 𝑛:ran π‘”βŸΆβ„ ∧ βˆ€π‘₯ ∈ ran 𝑔(((π‘›β€˜π‘₯) = 0 β†’ π‘₯ = (GIdβ€˜π‘”)) ∧ βˆ€π‘¦ ∈ β„‚ (π‘›β€˜(𝑦𝑠π‘₯)) = ((absβ€˜π‘¦) Β· (π‘›β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ ran 𝑔(π‘›β€˜(π‘₯𝑔𝑦)) ≀ ((π‘›β€˜π‘₯) + (π‘›β€˜π‘¦))))} ↔ (⟨𝐺, π‘†βŸ© ∈ CVecOLD ∧ 𝑁:π‘‹βŸΆβ„ ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))))
462, 45bitrid 283 1 ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) β†’ (⟨⟨𝐺, π‘†βŸ©, π‘βŸ© ∈ NrmCVec ↔ (⟨𝐺, π‘†βŸ© ∈ CVecOLD ∧ 𝑁:π‘‹βŸΆβ„ ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  Vcvv 3475  βŸ¨cop 4635   class class class wbr 5149  ran crn 5678  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  {coprab 7410  β„‚cc 11108  β„cr 11109  0cc0 11110   + caddc 11113   Β· cmul 11115   ≀ cle 11249  abscabs 15181  GIdcgi 29743  CVecOLDcvc 29811  NrmCVeccnv 29837
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-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-nv 29845
This theorem is referenced by:  isnv  29865
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