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Theorem imsmetlem 26929
 Description: Lemma for imsmet 26930. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
imsmetlem.1 𝑋 = (BaseSet‘𝑈)
imsmetlem.2 𝐺 = ( +𝑣𝑈)
imsmetlem.7 𝑀 = (inv‘𝐺)
imsmetlem.4 𝑆 = ( ·𝑠OLD𝑈)
imsmetlem.5 𝑍 = (0vec𝑈)
imsmetlem.6 𝑁 = (normCV𝑈)
imsmetlem.8 𝐷 = (IndMet‘𝑈)
imsmetlem.9 𝑈 ∈ NrmCVec
Assertion
Ref Expression
imsmetlem 𝐷 ∈ (Met‘𝑋)

Proof of Theorem imsmetlem
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imsmetlem.1 . . 3 𝑋 = (BaseSet‘𝑈)
2 fvex 6113 . . 3 (BaseSet‘𝑈) ∈ V
31, 2eqeltri 2684 . 2 𝑋 ∈ V
4 imsmetlem.9 . . 3 𝑈 ∈ NrmCVec
5 imsmetlem.8 . . . 4 𝐷 = (IndMet‘𝑈)
61, 5imsdf 26928 . . 3 (𝑈 ∈ NrmCVec → 𝐷:(𝑋 × 𝑋)⟶ℝ)
74, 6ax-mp 5 . 2 𝐷:(𝑋 × 𝑋)⟶ℝ
8 imsmetlem.2 . . . . . 6 𝐺 = ( +𝑣𝑈)
9 imsmetlem.4 . . . . . 6 𝑆 = ( ·𝑠OLD𝑈)
10 imsmetlem.6 . . . . . 6 𝑁 = (normCV𝑈)
111, 8, 9, 10, 5imsdval2 26926 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐷𝑦) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
124, 11mp3an1 1403 . . . 4 ((𝑥𝑋𝑦𝑋) → (𝑥𝐷𝑦) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
1312eqeq1d 2612 . . 3 ((𝑥𝑋𝑦𝑋) → ((𝑥𝐷𝑦) = 0 ↔ (𝑁‘(𝑥𝐺(-1𝑆𝑦))) = 0))
14 neg1cn 11001 . . . . . 6 -1 ∈ ℂ
151, 9nvscl 26865 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑦𝑋) → (-1𝑆𝑦) ∈ 𝑋)
164, 14, 15mp3an12 1406 . . . . 5 (𝑦𝑋 → (-1𝑆𝑦) ∈ 𝑋)
171, 8nvgcl 26859 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑥𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋)
184, 17mp3an1 1403 . . . . 5 ((𝑥𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋)
1916, 18sylan2 490 . . . 4 ((𝑥𝑋𝑦𝑋) → (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋)
20 imsmetlem.5 . . . . 5 𝑍 = (0vec𝑈)
211, 20, 10nvz 26908 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋) → ((𝑁‘(𝑥𝐺(-1𝑆𝑦))) = 0 ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
224, 19, 21sylancr 694 . . 3 ((𝑥𝑋𝑦𝑋) → ((𝑁‘(𝑥𝐺(-1𝑆𝑦))) = 0 ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
231, 20nvzcl 26873 . . . . . . 7 (𝑈 ∈ NrmCVec → 𝑍𝑋)
244, 23ax-mp 5 . . . . . 6 𝑍𝑋
251, 8nvrcan 26863 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ ((𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋𝑍𝑋𝑦𝑋)) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
264, 25mpan 702 . . . . . 6 (((𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋𝑍𝑋𝑦𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
2724, 26mp3an2 1404 . . . . 5 (((𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋𝑦𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
2819, 27sylancom 698 . . . 4 ((𝑥𝑋𝑦𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
29 simpl 472 . . . . . . 7 ((𝑥𝑋𝑦𝑋) → 𝑥𝑋)
3016adantl 481 . . . . . . 7 ((𝑥𝑋𝑦𝑋) → (-1𝑆𝑦) ∈ 𝑋)
31 simpr 476 . . . . . . 7 ((𝑥𝑋𝑦𝑋) → 𝑦𝑋)
321, 8nvass 26861 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ (𝑥𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋𝑦𝑋)) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)))
334, 32mpan 702 . . . . . . 7 ((𝑥𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋𝑦𝑋) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)))
3429, 30, 31, 33syl3anc 1318 . . . . . 6 ((𝑥𝑋𝑦𝑋) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)))
351, 8, 9, 20nvlinv 26891 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑦𝑋) → ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
364, 35mpan 702 . . . . . . . 8 (𝑦𝑋 → ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
3736adantl 481 . . . . . . 7 ((𝑥𝑋𝑦𝑋) → ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
3837oveq2d 6565 . . . . . 6 ((𝑥𝑋𝑦𝑋) → (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)) = (𝑥𝐺𝑍))
391, 8, 20nv0rid 26874 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑥𝑋) → (𝑥𝐺𝑍) = 𝑥)
404, 39mpan 702 . . . . . . 7 (𝑥𝑋 → (𝑥𝐺𝑍) = 𝑥)
4140adantr 480 . . . . . 6 ((𝑥𝑋𝑦𝑋) → (𝑥𝐺𝑍) = 𝑥)
4234, 38, 413eqtrd 2648 . . . . 5 ((𝑥𝑋𝑦𝑋) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = 𝑥)
431, 8, 20nv0lid 26875 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑦𝑋) → (𝑍𝐺𝑦) = 𝑦)
444, 43mpan 702 . . . . . 6 (𝑦𝑋 → (𝑍𝐺𝑦) = 𝑦)
4544adantl 481 . . . . 5 ((𝑥𝑋𝑦𝑋) → (𝑍𝐺𝑦) = 𝑦)
4642, 45eqeq12d 2625 . . . 4 ((𝑥𝑋𝑦𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ 𝑥 = 𝑦))
4728, 46bitr3d 269 . . 3 ((𝑥𝑋𝑦𝑋) → ((𝑥𝐺(-1𝑆𝑦)) = 𝑍𝑥 = 𝑦))
4813, 22, 473bitrd 293 . 2 ((𝑥𝑋𝑦𝑋) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
49 simpr 476 . . . . . . 7 ((𝑧𝑋𝑥𝑋) → 𝑥𝑋)
501, 9nvscl 26865 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑧𝑋) → (-1𝑆𝑧) ∈ 𝑋)
514, 14, 50mp3an12 1406 . . . . . . . 8 (𝑧𝑋 → (-1𝑆𝑧) ∈ 𝑋)
5251adantr 480 . . . . . . 7 ((𝑧𝑋𝑥𝑋) → (-1𝑆𝑧) ∈ 𝑋)
531, 8nvgcl 26859 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑥𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
544, 53mp3an1 1403 . . . . . . 7 ((𝑥𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
5549, 52, 54syl2anc 691 . . . . . 6 ((𝑧𝑋𝑥𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
56553adant3 1074 . . . . 5 ((𝑧𝑋𝑥𝑋𝑦𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
571, 8nvgcl 26859 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑧𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
584, 57mp3an1 1403 . . . . . . 7 ((𝑧𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
5916, 58sylan2 490 . . . . . 6 ((𝑧𝑋𝑦𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
60593adant2 1073 . . . . 5 ((𝑧𝑋𝑥𝑋𝑦𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
611, 8, 10nvtri 26909 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋) → (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) ≤ ((𝑁‘(𝑥𝐺(-1𝑆𝑧))) + (𝑁‘(𝑧𝐺(-1𝑆𝑦)))))
624, 61mp3an1 1403 . . . . 5 (((𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋) → (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) ≤ ((𝑁‘(𝑥𝐺(-1𝑆𝑧))) + (𝑁‘(𝑧𝐺(-1𝑆𝑦)))))
6356, 60, 62syl2anc 691 . . . 4 ((𝑧𝑋𝑥𝑋𝑦𝑋) → (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) ≤ ((𝑁‘(𝑥𝐺(-1𝑆𝑧))) + (𝑁‘(𝑧𝐺(-1𝑆𝑦)))))
64123adant1 1072 . . . . 5 ((𝑧𝑋𝑥𝑋𝑦𝑋) → (𝑥𝐷𝑦) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
65 simp1 1054 . . . . . . . 8 ((𝑧𝑋𝑥𝑋𝑦𝑋) → 𝑧𝑋)
66163ad2ant3 1077 . . . . . . . 8 ((𝑧𝑋𝑥𝑋𝑦𝑋) → (-1𝑆𝑦) ∈ 𝑋)
671, 8nvass 26861 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ ((𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋𝑧𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋)) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
684, 67mpan 702 . . . . . . . 8 (((𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋𝑧𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
6956, 65, 66, 68syl3anc 1318 . . . . . . 7 ((𝑧𝑋𝑥𝑋𝑦𝑋) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
70 simpl 472 . . . . . . . . . . 11 ((𝑧𝑋𝑥𝑋) → 𝑧𝑋)
711, 8nvass 26861 . . . . . . . . . . . 12 ((𝑈 ∈ NrmCVec ∧ (𝑥𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋𝑧𝑋)) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)))
724, 71mpan 702 . . . . . . . . . . 11 ((𝑥𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋𝑧𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)))
7349, 52, 70, 72syl3anc 1318 . . . . . . . . . 10 ((𝑧𝑋𝑥𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)))
741, 8, 9, 20nvlinv 26891 . . . . . . . . . . . . 13 ((𝑈 ∈ NrmCVec ∧ 𝑧𝑋) → ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
754, 74mpan 702 . . . . . . . . . . . 12 (𝑧𝑋 → ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
7675adantr 480 . . . . . . . . . . 11 ((𝑧𝑋𝑥𝑋) → ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
7776oveq2d 6565 . . . . . . . . . 10 ((𝑧𝑋𝑥𝑋) → (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)) = (𝑥𝐺𝑍))
7840adantl 481 . . . . . . . . . 10 ((𝑧𝑋𝑥𝑋) → (𝑥𝐺𝑍) = 𝑥)
7973, 77, 783eqtrd 2648 . . . . . . . . 9 ((𝑧𝑋𝑥𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = 𝑥)
80793adant3 1074 . . . . . . . 8 ((𝑧𝑋𝑥𝑋𝑦𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = 𝑥)
8180oveq1d 6564 . . . . . . 7 ((𝑧𝑋𝑥𝑋𝑦𝑋) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = (𝑥𝐺(-1𝑆𝑦)))
8269, 81eqtr3d 2646 . . . . . 6 ((𝑧𝑋𝑥𝑋𝑦𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))) = (𝑥𝐺(-1𝑆𝑦)))
8382fveq2d 6107 . . . . 5 ((𝑧𝑋𝑥𝑋𝑦𝑋) → (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
8464, 83eqtr4d 2647 . . . 4 ((𝑧𝑋𝑥𝑋𝑦𝑋) → (𝑥𝐷𝑦) = (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))))
851, 8, 9, 10, 5imsdval2 26926 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑧𝑋𝑥𝑋) → (𝑧𝐷𝑥) = (𝑁‘(𝑧𝐺(-1𝑆𝑥))))
864, 85mp3an1 1403 . . . . . . 7 ((𝑧𝑋𝑥𝑋) → (𝑧𝐷𝑥) = (𝑁‘(𝑧𝐺(-1𝑆𝑥))))
871, 8, 9, 10nvdif 26905 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑧𝑋𝑥𝑋) → (𝑁‘(𝑧𝐺(-1𝑆𝑥))) = (𝑁‘(𝑥𝐺(-1𝑆𝑧))))
884, 87mp3an1 1403 . . . . . . 7 ((𝑧𝑋𝑥𝑋) → (𝑁‘(𝑧𝐺(-1𝑆𝑥))) = (𝑁‘(𝑥𝐺(-1𝑆𝑧))))
8986, 88eqtrd 2644 . . . . . 6 ((𝑧𝑋𝑥𝑋) → (𝑧𝐷𝑥) = (𝑁‘(𝑥𝐺(-1𝑆𝑧))))
90893adant3 1074 . . . . 5 ((𝑧𝑋𝑥𝑋𝑦𝑋) → (𝑧𝐷𝑥) = (𝑁‘(𝑥𝐺(-1𝑆𝑧))))
911, 8, 9, 10, 5imsdval2 26926 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑧𝑋𝑦𝑋) → (𝑧𝐷𝑦) = (𝑁‘(𝑧𝐺(-1𝑆𝑦))))
924, 91mp3an1 1403 . . . . . 6 ((𝑧𝑋𝑦𝑋) → (𝑧𝐷𝑦) = (𝑁‘(𝑧𝐺(-1𝑆𝑦))))
93923adant2 1073 . . . . 5 ((𝑧𝑋𝑥𝑋𝑦𝑋) → (𝑧𝐷𝑦) = (𝑁‘(𝑧𝐺(-1𝑆𝑦))))
9490, 93oveq12d 6567 . . . 4 ((𝑧𝑋𝑥𝑋𝑦𝑋) → ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)) = ((𝑁‘(𝑥𝐺(-1𝑆𝑧))) + (𝑁‘(𝑧𝐺(-1𝑆𝑦)))))
9563, 84, 943brtr4d 4615 . . 3 ((𝑧𝑋𝑥𝑋𝑦𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
96953coml 1264 . 2 ((𝑥𝑋𝑦𝑋𝑧𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
973, 7, 48, 96ismeti 21940 1 𝐷 ∈ (Met‘𝑋)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   ≤ cle 9954  -cneg 10146  Metcme 19553  invcgn 26729  NrmCVeccnv 26823   +𝑣 cpv 26824  BaseSetcba 26825   ·𝑠OLD cns 26826  0veccn0v 26827  normCVcnmcv 26829  IndMetcims 26830 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-met 19561  df-grpo 26731  df-gid 26732  df-ginv 26733  df-gdiv 26734  df-ablo 26783  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-vs 26838  df-nmcv 26839  df-ims 26840 This theorem is referenced by:  imsmet  26930
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