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Theorem imsmetlem 29931
Description: Lemma for imsmet 29932. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
imsmetlem.1 𝑋 = (BaseSetβ€˜π‘ˆ)
imsmetlem.2 𝐺 = ( +𝑣 β€˜π‘ˆ)
imsmetlem.7 𝑀 = (invβ€˜πΊ)
imsmetlem.4 𝑆 = ( ·𝑠OLD β€˜π‘ˆ)
imsmetlem.5 𝑍 = (0vecβ€˜π‘ˆ)
imsmetlem.6 𝑁 = (normCVβ€˜π‘ˆ)
imsmetlem.8 𝐷 = (IndMetβ€˜π‘ˆ)
imsmetlem.9 π‘ˆ ∈ NrmCVec
Assertion
Ref Expression
imsmetlem 𝐷 ∈ (Metβ€˜π‘‹)
Proof of Theorem imsmetlem
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imsmetlem.1 . . 3 𝑋 = (BaseSetβ€˜π‘ˆ)
21fvexi 6903 . 2 𝑋 ∈ V
3 imsmetlem.9 . . 3 π‘ˆ ∈ NrmCVec
4 imsmetlem.8 . . . 4 𝐷 = (IndMetβ€˜π‘ˆ)
51, 4imsdf 29930 . . 3 (π‘ˆ ∈ NrmCVec β†’ 𝐷:(𝑋 Γ— 𝑋)βŸΆβ„)
63, 5ax-mp 5 . 2 𝐷:(𝑋 Γ— 𝑋)βŸΆβ„
7 imsmetlem.2 . . . . . 6 𝐺 = ( +𝑣 β€˜π‘ˆ)
8 imsmetlem.4 . . . . . 6 𝑆 = ( ·𝑠OLD β€˜π‘ˆ)
9 imsmetlem.6 . . . . . 6 𝑁 = (normCVβ€˜π‘ˆ)
101, 7, 8, 9, 4imsdval2 29928 . . . . 5 ((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝐷𝑦) = (π‘β€˜(π‘₯𝐺(-1𝑆𝑦))))
113, 10mp3an1 1449 . . . 4 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝐷𝑦) = (π‘β€˜(π‘₯𝐺(-1𝑆𝑦))))
1211eqeq1d 2735 . . 3 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ ((π‘₯𝐷𝑦) = 0 ↔ (π‘β€˜(π‘₯𝐺(-1𝑆𝑦))) = 0))
13 neg1cn 12323 . . . . . 6 -1 ∈ β„‚
141, 8nvscl 29867 . . . . . 6 ((π‘ˆ ∈ NrmCVec ∧ -1 ∈ β„‚ ∧ 𝑦 ∈ 𝑋) β†’ (-1𝑆𝑦) ∈ 𝑋)
153, 13, 14mp3an12 1452 . . . . 5 (𝑦 ∈ 𝑋 β†’ (-1𝑆𝑦) ∈ 𝑋)
161, 7nvgcl 29861 . . . . . 6 ((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) β†’ (π‘₯𝐺(-1𝑆𝑦)) ∈ 𝑋)
173, 16mp3an1 1449 . . . . 5 ((π‘₯ ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) β†’ (π‘₯𝐺(-1𝑆𝑦)) ∈ 𝑋)
1815, 17sylan2 594 . . . 4 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝐺(-1𝑆𝑦)) ∈ 𝑋)
19 imsmetlem.5 . . . . 5 𝑍 = (0vecβ€˜π‘ˆ)
201, 19, 9nvz 29910 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ (π‘₯𝐺(-1𝑆𝑦)) ∈ 𝑋) β†’ ((π‘β€˜(π‘₯𝐺(-1𝑆𝑦))) = 0 ↔ (π‘₯𝐺(-1𝑆𝑦)) = 𝑍))
213, 18, 20sylancr 588 . . 3 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ ((π‘β€˜(π‘₯𝐺(-1𝑆𝑦))) = 0 ↔ (π‘₯𝐺(-1𝑆𝑦)) = 𝑍))
221, 19nvzcl 29875 . . . . . . 7 (π‘ˆ ∈ NrmCVec β†’ 𝑍 ∈ 𝑋)
233, 22ax-mp 5 . . . . . 6 𝑍 ∈ 𝑋
241, 7nvrcan 29865 . . . . . . 7 ((π‘ˆ ∈ NrmCVec ∧ ((π‘₯𝐺(-1𝑆𝑦)) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ (((π‘₯𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (π‘₯𝐺(-1𝑆𝑦)) = 𝑍))
253, 24mpan 689 . . . . . 6 (((π‘₯𝐺(-1𝑆𝑦)) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (((π‘₯𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (π‘₯𝐺(-1𝑆𝑦)) = 𝑍))
2623, 25mp3an2 1450 . . . . 5 (((π‘₯𝐺(-1𝑆𝑦)) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (((π‘₯𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (π‘₯𝐺(-1𝑆𝑦)) = 𝑍))
2718, 26sylancom 589 . . . 4 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (((π‘₯𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (π‘₯𝐺(-1𝑆𝑦)) = 𝑍))
28 simpl 484 . . . . . . 7 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
2915adantl 483 . . . . . . 7 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (-1𝑆𝑦) ∈ 𝑋)
30 simpr 486 . . . . . . 7 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ 𝑦 ∈ 𝑋)
311, 7nvass 29863 . . . . . . . 8 ((π‘ˆ ∈ NrmCVec ∧ (π‘₯ ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ ((π‘₯𝐺(-1𝑆𝑦))𝐺𝑦) = (π‘₯𝐺((-1𝑆𝑦)𝐺𝑦)))
323, 31mpan 689 . . . . . . 7 ((π‘₯ ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ ((π‘₯𝐺(-1𝑆𝑦))𝐺𝑦) = (π‘₯𝐺((-1𝑆𝑦)𝐺𝑦)))
3328, 29, 30, 32syl3anc 1372 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ ((π‘₯𝐺(-1𝑆𝑦))𝐺𝑦) = (π‘₯𝐺((-1𝑆𝑦)𝐺𝑦)))
341, 7, 8, 19nvlinv 29893 . . . . . . . . 9 ((π‘ˆ ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) β†’ ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
353, 34mpan 689 . . . . . . . 8 (𝑦 ∈ 𝑋 β†’ ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
3635adantl 483 . . . . . . 7 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
3736oveq2d 7422 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝐺((-1𝑆𝑦)𝐺𝑦)) = (π‘₯𝐺𝑍))
381, 7, 19nv0rid 29876 . . . . . . . 8 ((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯𝐺𝑍) = π‘₯)
393, 38mpan 689 . . . . . . 7 (π‘₯ ∈ 𝑋 β†’ (π‘₯𝐺𝑍) = π‘₯)
4039adantr 482 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝐺𝑍) = π‘₯)
4133, 37, 403eqtrd 2777 . . . . 5 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ ((π‘₯𝐺(-1𝑆𝑦))𝐺𝑦) = π‘₯)
421, 7, 19nv0lid 29877 . . . . . . 7 ((π‘ˆ ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) β†’ (𝑍𝐺𝑦) = 𝑦)
433, 42mpan 689 . . . . . 6 (𝑦 ∈ 𝑋 β†’ (𝑍𝐺𝑦) = 𝑦)
4443adantl 483 . . . . 5 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝑍𝐺𝑦) = 𝑦)
4541, 44eqeq12d 2749 . . . 4 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (((π‘₯𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ π‘₯ = 𝑦))
4627, 45bitr3d 281 . . 3 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ ((π‘₯𝐺(-1𝑆𝑦)) = 𝑍 ↔ π‘₯ = 𝑦))
4712, 21, 463bitrd 305 . 2 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ ((π‘₯𝐷𝑦) = 0 ↔ π‘₯ = 𝑦))
48 simpr 486 . . . . . . 7 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
491, 8nvscl 29867 . . . . . . . . 9 ((π‘ˆ ∈ NrmCVec ∧ -1 ∈ β„‚ ∧ 𝑧 ∈ 𝑋) β†’ (-1𝑆𝑧) ∈ 𝑋)
503, 13, 49mp3an12 1452 . . . . . . . 8 (𝑧 ∈ 𝑋 β†’ (-1𝑆𝑧) ∈ 𝑋)
5150adantr 482 . . . . . . 7 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (-1𝑆𝑧) ∈ 𝑋)
521, 7nvgcl 29861 . . . . . . . 8 ((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋) β†’ (π‘₯𝐺(-1𝑆𝑧)) ∈ 𝑋)
533, 52mp3an1 1449 . . . . . . 7 ((π‘₯ ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋) β†’ (π‘₯𝐺(-1𝑆𝑧)) ∈ 𝑋)
5448, 51, 53syl2anc 585 . . . . . 6 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯𝐺(-1𝑆𝑧)) ∈ 𝑋)
55543adant3 1133 . . . . 5 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝐺(-1𝑆𝑧)) ∈ 𝑋)
561, 7nvgcl 29861 . . . . . . . 8 ((π‘ˆ ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) β†’ (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
573, 56mp3an1 1449 . . . . . . 7 ((𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) β†’ (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
5815, 57sylan2 594 . . . . . 6 ((𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
59583adant2 1132 . . . . 5 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
601, 7, 9nvtri 29911 . . . . . 6 ((π‘ˆ ∈ NrmCVec ∧ (π‘₯𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋) β†’ (π‘β€˜((π‘₯𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) ≀ ((π‘β€˜(π‘₯𝐺(-1𝑆𝑧))) + (π‘β€˜(𝑧𝐺(-1𝑆𝑦)))))
613, 60mp3an1 1449 . . . . 5 (((π‘₯𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋) β†’ (π‘β€˜((π‘₯𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) ≀ ((π‘β€˜(π‘₯𝐺(-1𝑆𝑧))) + (π‘β€˜(𝑧𝐺(-1𝑆𝑦)))))
6255, 59, 61syl2anc 585 . . . 4 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘β€˜((π‘₯𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) ≀ ((π‘β€˜(π‘₯𝐺(-1𝑆𝑧))) + (π‘β€˜(𝑧𝐺(-1𝑆𝑦)))))
63113adant1 1131 . . . . 5 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝐷𝑦) = (π‘β€˜(π‘₯𝐺(-1𝑆𝑦))))
64 simp1 1137 . . . . . . . 8 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ 𝑧 ∈ 𝑋)
65153ad2ant3 1136 . . . . . . . 8 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (-1𝑆𝑦) ∈ 𝑋)
661, 7nvass 29863 . . . . . . . . 9 ((π‘ˆ ∈ NrmCVec ∧ ((π‘₯𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋)) β†’ (((π‘₯𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((π‘₯𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
673, 66mpan 689 . . . . . . . 8 (((π‘₯𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) β†’ (((π‘₯𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((π‘₯𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
6855, 64, 65, 67syl3anc 1372 . . . . . . 7 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (((π‘₯𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((π‘₯𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
69 simpl 484 . . . . . . . . . . 11 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ 𝑧 ∈ 𝑋)
701, 7nvass 29863 . . . . . . . . . . . 12 ((π‘ˆ ∈ NrmCVec ∧ (π‘₯ ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) β†’ ((π‘₯𝐺(-1𝑆𝑧))𝐺𝑧) = (π‘₯𝐺((-1𝑆𝑧)𝐺𝑧)))
713, 70mpan 689 . . . . . . . . . . 11 ((π‘₯ ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) β†’ ((π‘₯𝐺(-1𝑆𝑧))𝐺𝑧) = (π‘₯𝐺((-1𝑆𝑧)𝐺𝑧)))
7248, 51, 69, 71syl3anc 1372 . . . . . . . . . 10 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯𝐺(-1𝑆𝑧))𝐺𝑧) = (π‘₯𝐺((-1𝑆𝑧)𝐺𝑧)))
731, 7, 8, 19nvlinv 29893 . . . . . . . . . . . . 13 ((π‘ˆ ∈ NrmCVec ∧ 𝑧 ∈ 𝑋) β†’ ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
743, 73mpan 689 . . . . . . . . . . . 12 (𝑧 ∈ 𝑋 β†’ ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
7574adantr 482 . . . . . . . . . . 11 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
7675oveq2d 7422 . . . . . . . . . 10 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯𝐺((-1𝑆𝑧)𝐺𝑧)) = (π‘₯𝐺𝑍))
7739adantl 483 . . . . . . . . . 10 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯𝐺𝑍) = π‘₯)
7872, 76, 773eqtrd 2777 . . . . . . . . 9 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯𝐺(-1𝑆𝑧))𝐺𝑧) = π‘₯)
79783adant3 1133 . . . . . . . 8 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ ((π‘₯𝐺(-1𝑆𝑧))𝐺𝑧) = π‘₯)
8079oveq1d 7421 . . . . . . 7 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (((π‘₯𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = (π‘₯𝐺(-1𝑆𝑦)))
8168, 80eqtr3d 2775 . . . . . 6 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ ((π‘₯𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))) = (π‘₯𝐺(-1𝑆𝑦)))
8281fveq2d 6893 . . . . 5 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘β€˜((π‘₯𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) = (π‘β€˜(π‘₯𝐺(-1𝑆𝑦))))
8363, 82eqtr4d 2776 . . . 4 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝐷𝑦) = (π‘β€˜((π‘₯𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))))
841, 7, 8, 9, 4imsdval2 29928 . . . . . . . 8 ((π‘ˆ ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (𝑧𝐷π‘₯) = (π‘β€˜(𝑧𝐺(-1𝑆π‘₯))))
853, 84mp3an1 1449 . . . . . . 7 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (𝑧𝐷π‘₯) = (π‘β€˜(𝑧𝐺(-1𝑆π‘₯))))
861, 7, 8, 9nvdif 29907 . . . . . . . 8 ((π‘ˆ ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (π‘β€˜(𝑧𝐺(-1𝑆π‘₯))) = (π‘β€˜(π‘₯𝐺(-1𝑆𝑧))))
873, 86mp3an1 1449 . . . . . . 7 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (π‘β€˜(𝑧𝐺(-1𝑆π‘₯))) = (π‘β€˜(π‘₯𝐺(-1𝑆𝑧))))
8885, 87eqtrd 2773 . . . . . 6 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (𝑧𝐷π‘₯) = (π‘β€˜(π‘₯𝐺(-1𝑆𝑧))))
89883adant3 1133 . . . . 5 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝑧𝐷π‘₯) = (π‘β€˜(π‘₯𝐺(-1𝑆𝑧))))
901, 7, 8, 9, 4imsdval2 29928 . . . . . . 7 ((π‘ˆ ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝑧𝐷𝑦) = (π‘β€˜(𝑧𝐺(-1𝑆𝑦))))
913, 90mp3an1 1449 . . . . . 6 ((𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝑧𝐷𝑦) = (π‘β€˜(𝑧𝐺(-1𝑆𝑦))))
92913adant2 1132 . . . . 5 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝑧𝐷𝑦) = (π‘β€˜(𝑧𝐺(-1𝑆𝑦))))
9389, 92oveq12d 7424 . . . 4 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ ((𝑧𝐷π‘₯) + (𝑧𝐷𝑦)) = ((π‘β€˜(π‘₯𝐺(-1𝑆𝑧))) + (π‘β€˜(𝑧𝐺(-1𝑆𝑦)))))
9462, 83, 933brtr4d 5180 . . 3 ((𝑧 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝐷𝑦) ≀ ((𝑧𝐷π‘₯) + (𝑧𝐷𝑦)))
95943coml 1128 . 2 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) β†’ (π‘₯𝐷𝑦) ≀ ((𝑧𝐷π‘₯) + (𝑧𝐷𝑦)))
962, 6, 47, 95ismeti 23823 1 𝐷 ∈ (Metβ€˜π‘‹)
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5148   Γ— cxp 5674  βŸΆwf 6537  β€˜cfv 6541  (class class class)co 7406  β„‚cc 11105  β„cr 11106  0cc0 11107  1c1 11108   + caddc 11110   ≀ cle 11246  -cneg 11442  Metcmet 20923  invcgn 29732  NrmCVeccnv 29825   +𝑣 cpv 29826  BaseSetcba 29827   ·𝑠OLD cns 29828  0veccn0v 29829  normCVcnmcv 29831  IndMetcims 29832
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 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-sup 9434  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-rp 12972  df-seq 13964  df-exp 14025  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-met 20931  df-grpo 29734  df-gid 29735  df-ginv 29736  df-gdiv 29737  df-ablo 29786  df-vc 29800  df-nv 29833  df-va 29836  df-ba 29837  df-sm 29838  df-0v 29839  df-vs 29840  df-nmcv 29841  df-ims 29842
This theorem is referenced by:  imsmet  29932
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