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Mirrors > Home > MPE Home > Th. List > imasdsf1o | Structured version Visualization version GIF version |
Description: The distance function is transferred across an image structure under a bijection. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
imasdsf1o.u | ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
imasdsf1o.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
imasdsf1o.f | ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) |
imasdsf1o.r | ⊢ (𝜑 → 𝑅 ∈ 𝑍) |
imasdsf1o.e | ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) |
imasdsf1o.d | ⊢ 𝐷 = (dist‘𝑈) |
imasdsf1o.m | ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉)) |
imasdsf1o.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
imasdsf1o.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
imasdsf1o | ⊢ (𝜑 → ((𝐹‘𝑋)𝐷(𝐹‘𝑌)) = (𝑋𝐸𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasdsf1o.u | . 2 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | |
2 | imasdsf1o.v | . 2 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
3 | imasdsf1o.f | . 2 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) | |
4 | imasdsf1o.r | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
5 | imasdsf1o.e | . 2 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) | |
6 | imasdsf1o.d | . 2 ⊢ 𝐷 = (dist‘𝑈) | |
7 | imasdsf1o.m | . 2 ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉)) | |
8 | imasdsf1o.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
9 | imasdsf1o.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
10 | eqid 2824 | . 2 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
11 | eqid 2824 | . 2 ⊢ {ℎ ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = (𝐹‘𝑋) ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = (𝐹‘𝑌) ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} = {ℎ ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = (𝐹‘𝑋) ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = (𝐹‘𝑌) ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} | |
12 | eqid 2824 | . 2 ⊢ ∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = (𝐹‘𝑋) ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = (𝐹‘𝑌) ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))) = ∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = (𝐹‘𝑋) ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = (𝐹‘𝑌) ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | imasdsf1olem 22547 | 1 ⊢ (𝜑 → ((𝐹‘𝑋)𝐷(𝐹‘𝑌)) = (𝑋𝐸𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∀wral 3116 {crab 3120 ∖ cdif 3794 {csn 4396 ∪ ciun 4739 ↦ cmpt 4951 × cxp 5339 ran crn 5342 ↾ cres 5343 ∘ ccom 5345 –1-1-onto→wf1o 6121 ‘cfv 6122 (class class class)co 6904 1st c1st 7425 2nd c2nd 7426 ↑𝑚 cmap 8121 1c1 10252 + caddc 10254 -∞cmnf 10388 ℝ*cxr 10389 − cmin 10584 ℕcn 11349 ...cfz 12618 Basecbs 16221 ↾s cress 16222 distcds 16313 Σg cgsu 16453 ℝ*𝑠cxrs 16512 “s cimas 16516 ∞Metcxmet 20090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-iin 4742 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-of 7156 df-om 7326 df-1st 7427 df-2nd 7428 df-supp 7559 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-er 8008 df-map 8123 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-fsupp 8544 df-sup 8616 df-inf 8617 df-oi 8683 df-card 9077 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-5 11416 df-6 11417 df-7 11418 df-8 11419 df-9 11420 df-n0 11618 df-z 11704 df-dec 11821 df-uz 11968 df-rp 12112 df-xneg 12231 df-xadd 12232 df-xmul 12233 df-fz 12619 df-fzo 12760 df-seq 13095 df-hash 13410 df-struct 16223 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-mulr 16318 df-sca 16320 df-vsca 16321 df-ip 16322 df-tset 16323 df-ple 16324 df-ds 16326 df-0g 16454 df-gsum 16455 df-xrs 16514 df-imas 16520 df-mre 16598 df-mrc 16599 df-acs 16601 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-submnd 17688 df-mulg 17894 df-cntz 18099 df-cmn 18547 df-xmet 20098 |
This theorem is referenced by: imasf1oxmet 22549 imasf1omet 22550 xpsdsval 22555 imasf1obl 22662 |
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