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Theorem metideq 33402
Description: Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
metideq ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) = (𝐡𝐷𝐹))
Proof of Theorem metideq
StepHypRef Expression
1 simpl 482 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐷 ∈ (PsMetβ€˜π‘‹))
2 metidss 33400 . . . . . . . . 9 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (~Metβ€˜π·) βŠ† (𝑋 Γ— 𝑋))
3 dmss 5895 . . . . . . . . 9 ((~Metβ€˜π·) βŠ† (𝑋 Γ— 𝑋) β†’ dom (~Metβ€˜π·) βŠ† dom (𝑋 Γ— 𝑋))
42, 3syl 17 . . . . . . . 8 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ dom (~Metβ€˜π·) βŠ† dom (𝑋 Γ— 𝑋))
5 dmxpid 5922 . . . . . . . 8 dom (𝑋 Γ— 𝑋) = 𝑋
64, 5sseqtrdi 4027 . . . . . . 7 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ dom (~Metβ€˜π·) βŠ† 𝑋)
71, 6syl 17 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ dom (~Metβ€˜π·) βŠ† 𝑋)
8 xpss 5685 . . . . . . . . . 10 (𝑋 Γ— 𝑋) βŠ† (V Γ— V)
92, 8sstrdi 3989 . . . . . . . . 9 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (~Metβ€˜π·) βŠ† (V Γ— V))
10 df-rel 5676 . . . . . . . . 9 (Rel (~Metβ€˜π·) ↔ (~Metβ€˜π·) βŠ† (V Γ— V))
119, 10sylibr 233 . . . . . . . 8 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ Rel (~Metβ€˜π·))
121, 11syl 17 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ Rel (~Metβ€˜π·))
13 simprl 768 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐴(~Metβ€˜π·)𝐡)
14 releldm 5936 . . . . . . 7 ((Rel (~Metβ€˜π·) ∧ 𝐴(~Metβ€˜π·)𝐡) β†’ 𝐴 ∈ dom (~Metβ€˜π·))
1512, 13, 14syl2anc 583 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐴 ∈ dom (~Metβ€˜π·))
167, 15sseldd 3978 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐴 ∈ 𝑋)
17 simprr 770 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐸(~Metβ€˜π·)𝐹)
18 releldm 5936 . . . . . . 7 ((Rel (~Metβ€˜π·) ∧ 𝐸(~Metβ€˜π·)𝐹) β†’ 𝐸 ∈ dom (~Metβ€˜π·))
1912, 17, 18syl2anc 583 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐸 ∈ dom (~Metβ€˜π·))
207, 19sseldd 3978 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐸 ∈ 𝑋)
21 psmetsym 24166 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋) β†’ (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
221, 16, 20, 21syl3anc 1368 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
23 psmetf 24162 . . . . . 6 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝐷:(𝑋 Γ— 𝑋)βŸΆβ„*)
2423fovcdmda 7574 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐸 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) β†’ (𝐸𝐷𝐴) ∈ ℝ*)
251, 20, 16, 24syl12anc 834 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐸𝐷𝐴) ∈ ℝ*)
2622, 25eqeltrd 2827 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) ∈ ℝ*)
27 rnss 5931 . . . . . . . 8 ((~Metβ€˜π·) βŠ† (𝑋 Γ— 𝑋) β†’ ran (~Metβ€˜π·) βŠ† ran (𝑋 Γ— 𝑋))
282, 27syl 17 . . . . . . 7 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ ran (~Metβ€˜π·) βŠ† ran (𝑋 Γ— 𝑋))
29 rnxpid 6165 . . . . . . 7 ran (𝑋 Γ— 𝑋) = 𝑋
3028, 29sseqtrdi 4027 . . . . . 6 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ ran (~Metβ€˜π·) βŠ† 𝑋)
311, 30syl 17 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ran (~Metβ€˜π·) βŠ† 𝑋)
32 relelrn 5937 . . . . . 6 ((Rel (~Metβ€˜π·) ∧ 𝐴(~Metβ€˜π·)𝐡) β†’ 𝐡 ∈ ran (~Metβ€˜π·))
3312, 13, 32syl2anc 583 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐡 ∈ ran (~Metβ€˜π·))
3431, 33sseldd 3978 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐡 ∈ 𝑋)
3523fovcdmda 7574 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐡 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) β†’ (𝐡𝐷𝐸) ∈ ℝ*)
361, 34, 20, 35syl12anc 834 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐸) ∈ ℝ*)
37 relelrn 5937 . . . . . . 7 ((Rel (~Metβ€˜π·) ∧ 𝐸(~Metβ€˜π·)𝐹) β†’ 𝐹 ∈ ran (~Metβ€˜π·))
3812, 17, 37syl2anc 583 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐹 ∈ ran (~Metβ€˜π·))
3931, 38sseldd 3978 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ 𝐹 ∈ 𝑋)
40 psmetsym 24166 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐹 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐹𝐷𝐡) = (𝐡𝐷𝐹))
411, 39, 34, 40syl3anc 1368 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐹𝐷𝐡) = (𝐡𝐷𝐹))
4223fovcdmda 7574 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐹 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐹𝐷𝐡) ∈ ℝ*)
431, 39, 34, 42syl12anc 834 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐹𝐷𝐡) ∈ ℝ*)
4441, 43eqeltrrd 2828 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐹) ∈ ℝ*)
45 psmettri2 24165 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐡 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) β†’ (𝐴𝐷𝐸) ≀ ((𝐡𝐷𝐴) +𝑒 (𝐡𝐷𝐸)))
461, 34, 16, 20, 45syl13anc 1369 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) ≀ ((𝐡𝐷𝐴) +𝑒 (𝐡𝐷𝐸)))
47 psmetsym 24166 . . . . . . . 8 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = (𝐡𝐷𝐴))
481, 16, 34, 47syl3anc 1368 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐡) = (𝐡𝐷𝐴))
4916, 34jca 511 . . . . . . . 8 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋))
50 metidv 33401 . . . . . . . . 9 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴(~Metβ€˜π·)𝐡 ↔ (𝐴𝐷𝐡) = 0))
5150biimpa 476 . . . . . . . 8 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) ∧ 𝐴(~Metβ€˜π·)𝐡) β†’ (𝐴𝐷𝐡) = 0)
521, 49, 13, 51syl21anc 835 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐡) = 0)
5348, 52eqtr3d 2768 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐴) = 0)
5453oveq1d 7419 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐡𝐷𝐴) +𝑒 (𝐡𝐷𝐸)) = (0 +𝑒 (𝐡𝐷𝐸)))
55 xaddlid 13224 . . . . . 6 ((𝐡𝐷𝐸) ∈ ℝ* β†’ (0 +𝑒 (𝐡𝐷𝐸)) = (𝐡𝐷𝐸))
5636, 55syl 17 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (0 +𝑒 (𝐡𝐷𝐸)) = (𝐡𝐷𝐸))
5754, 56eqtrd 2766 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐡𝐷𝐴) +𝑒 (𝐡𝐷𝐸)) = (𝐡𝐷𝐸))
5846, 57breqtrd 5167 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) ≀ (𝐡𝐷𝐸))
59 psmettri2 24165 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐹 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) β†’ (𝐡𝐷𝐸) ≀ ((𝐹𝐷𝐡) +𝑒 (𝐹𝐷𝐸)))
601, 39, 34, 20, 59syl13anc 1369 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐸) ≀ ((𝐹𝐷𝐡) +𝑒 (𝐹𝐷𝐸)))
61 psmetsym 24166 . . . . . . . 8 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐹 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋) β†’ (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
621, 39, 20, 61syl3anc 1368 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
6320, 39jca 511 . . . . . . . 8 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))
64 metidv 33401 . . . . . . . . 9 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) β†’ (𝐸(~Metβ€˜π·)𝐹 ↔ (𝐸𝐷𝐹) = 0))
6564biimpa 476 . . . . . . . 8 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) ∧ 𝐸(~Metβ€˜π·)𝐹) β†’ (𝐸𝐷𝐹) = 0)
661, 63, 17, 65syl21anc 835 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐸𝐷𝐹) = 0)
6762, 66eqtrd 2766 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐹𝐷𝐸) = 0)
6867oveq2d 7420 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐹𝐷𝐡) +𝑒 (𝐹𝐷𝐸)) = ((𝐹𝐷𝐡) +𝑒 0))
69 xaddrid 13223 . . . . . 6 ((𝐹𝐷𝐡) ∈ ℝ* β†’ ((𝐹𝐷𝐡) +𝑒 0) = (𝐹𝐷𝐡))
7043, 69syl 17 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐹𝐷𝐡) +𝑒 0) = (𝐹𝐷𝐡))
7168, 70, 413eqtrd 2770 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐹𝐷𝐡) +𝑒 (𝐹𝐷𝐸)) = (𝐡𝐷𝐹))
7260, 71breqtrd 5167 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐸) ≀ (𝐡𝐷𝐹))
7326, 36, 44, 58, 72xrletrd 13144 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) ≀ (𝐡𝐷𝐹))
7423fovcdmda 7574 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) β†’ (𝐴𝐷𝐹) ∈ ℝ*)
751, 16, 39, 74syl12anc 834 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐹) ∈ ℝ*)
76 psmettri2 24165 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) β†’ (𝐡𝐷𝐹) ≀ ((𝐴𝐷𝐡) +𝑒 (𝐴𝐷𝐹)))
771, 16, 34, 39, 76syl13anc 1369 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐹) ≀ ((𝐴𝐷𝐡) +𝑒 (𝐴𝐷𝐹)))
7852oveq1d 7419 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐴𝐷𝐡) +𝑒 (𝐴𝐷𝐹)) = (0 +𝑒 (𝐴𝐷𝐹)))
79 xaddlid 13224 . . . . . 6 ((𝐴𝐷𝐹) ∈ ℝ* β†’ (0 +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
8075, 79syl 17 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (0 +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
8178, 80eqtrd 2766 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐴𝐷𝐡) +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
8277, 81breqtrd 5167 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐹) ≀ (𝐴𝐷𝐹))
83 psmettri2 24165 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐸 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) β†’ (𝐴𝐷𝐹) ≀ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
841, 20, 16, 39, 83syl13anc 1369 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐹) ≀ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
85 xaddrid 13223 . . . . . 6 ((𝐸𝐷𝐴) ∈ ℝ* β†’ ((𝐸𝐷𝐴) +𝑒 0) = (𝐸𝐷𝐴))
8625, 85syl 17 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐸𝐷𝐴) +𝑒 0) = (𝐸𝐷𝐴))
8766oveq2d 7420 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = ((𝐸𝐷𝐴) +𝑒 0))
8886, 87, 223eqtr4d 2776 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = (𝐴𝐷𝐸))
8984, 88breqtrd 5167 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐹) ≀ (𝐴𝐷𝐸))
9044, 75, 26, 82, 89xrletrd 13144 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐡𝐷𝐹) ≀ (𝐴𝐷𝐸))
91 xrletri3 13136 . . 3 (((𝐴𝐷𝐸) ∈ ℝ* ∧ (𝐡𝐷𝐹) ∈ ℝ*) β†’ ((𝐴𝐷𝐸) = (𝐡𝐷𝐹) ↔ ((𝐴𝐷𝐸) ≀ (𝐡𝐷𝐹) ∧ (𝐡𝐷𝐹) ≀ (𝐴𝐷𝐸))))
9226, 44, 91syl2anc 583 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ ((𝐴𝐷𝐸) = (𝐡𝐷𝐹) ↔ ((𝐴𝐷𝐸) ≀ (𝐡𝐷𝐹) ∧ (𝐡𝐷𝐹) ≀ (𝐴𝐷𝐸))))
9373, 90, 92mpbir2and 710 1 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝐡 ∧ 𝐸(~Metβ€˜π·)𝐹)) β†’ (𝐴𝐷𝐸) = (𝐡𝐷𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   βŠ† wss 3943   class class class wbr 5141   Γ— cxp 5667  dom cdm 5669  ran crn 5670  Rel wrel 5674  β€˜cfv 6536  (class class class)co 7404  0cc0 11109  β„*cxr 11248   ≀ cle 11250   +𝑒 cxad 13093  PsMetcpsmet 21219  ~Metcmetid 33395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-xadd 13096  df-psmet 21227  df-metid 33397
This theorem is referenced by:  pstmfval  33405
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