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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > prelrrx2 | Structured version Visualization version GIF version |
Description: An unordered pair of ordered pairs with first components 1 and 2 and real numbers as second components is a point in a real Euclidean space of dimension 2. (Contributed by AV, 4-Feb-2023.) |
Ref | Expression |
---|---|
prelrrx2.i | β’ πΌ = {1, 2} |
prelrrx2.b | β’ π = (β βm πΌ) |
Ref | Expression |
---|---|
prelrrx2 | β’ ((π΄ β β β§ π΅ β β) β {β¨1, π΄β©, β¨2, π΅β©} β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ex 11250 | . . . . . . . 8 β’ 1 β V | |
2 | 2ex 12329 | . . . . . . . 8 β’ 2 β V | |
3 | 1, 2 | pm3.2i 469 | . . . . . . 7 β’ (1 β V β§ 2 β V) |
4 | 3 | a1i 11 | . . . . . 6 β’ ((π΄ β β β§ π΅ β β) β (1 β V β§ 2 β V)) |
5 | id 22 | . . . . . 6 β’ ((π΄ β β β§ π΅ β β) β (π΄ β β β§ π΅ β β)) | |
6 | 1ne2 12460 | . . . . . . 7 β’ 1 β 2 | |
7 | 6 | a1i 11 | . . . . . 6 β’ ((π΄ β β β§ π΅ β β) β 1 β 2) |
8 | 4, 5, 7 | 3jca 1125 | . . . . 5 β’ ((π΄ β β β§ π΅ β β) β ((1 β V β§ 2 β V) β§ (π΄ β β β§ π΅ β β) β§ 1 β 2)) |
9 | fprg 7170 | . . . . 5 β’ (((1 β V β§ 2 β V) β§ (π΄ β β β§ π΅ β β) β§ 1 β 2) β {β¨1, π΄β©, β¨2, π΅β©}:{1, 2}βΆ{π΄, π΅}) | |
10 | 8, 9 | syl 17 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β {β¨1, π΄β©, β¨2, π΅β©}:{1, 2}βΆ{π΄, π΅}) |
11 | prssi 4829 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β {π΄, π΅} β β) | |
12 | 10, 11 | fssd 6745 | . . 3 β’ ((π΄ β β β§ π΅ β β) β {β¨1, π΄β©, β¨2, π΅β©}:{1, 2}βΆβ) |
13 | reex 11239 | . . . . 5 β’ β β V | |
14 | prex 5438 | . . . . 5 β’ {1, 2} β V | |
15 | 13, 14 | pm3.2i 469 | . . . 4 β’ (β β V β§ {1, 2} β V) |
16 | elmapg 8866 | . . . 4 β’ ((β β V β§ {1, 2} β V) β ({β¨1, π΄β©, β¨2, π΅β©} β (β βm {1, 2}) β {β¨1, π΄β©, β¨2, π΅β©}:{1, 2}βΆβ)) | |
17 | 15, 16 | ax-mp 5 | . . 3 β’ ({β¨1, π΄β©, β¨2, π΅β©} β (β βm {1, 2}) β {β¨1, π΄β©, β¨2, π΅β©}:{1, 2}βΆβ) |
18 | 12, 17 | sylibr 233 | . 2 β’ ((π΄ β β β§ π΅ β β) β {β¨1, π΄β©, β¨2, π΅β©} β (β βm {1, 2})) |
19 | prelrrx2.b | . . . 4 β’ π = (β βm πΌ) | |
20 | prelrrx2.i | . . . . 5 β’ πΌ = {1, 2} | |
21 | 20 | oveq2i 7437 | . . . 4 β’ (β βm πΌ) = (β βm {1, 2}) |
22 | 19, 21 | eqtri 2756 | . . 3 β’ π = (β βm {1, 2}) |
23 | 22 | eleq2i 2821 | . 2 β’ ({β¨1, π΄β©, β¨2, π΅β©} β π β {β¨1, π΄β©, β¨2, π΅β©} β (β βm {1, 2})) |
24 | 18, 23 | sylibr 233 | 1 β’ ((π΄ β β β§ π΅ β β) β {β¨1, π΄β©, β¨2, π΅β©} β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2937 Vcvv 3473 {cpr 4634 β¨cop 4638 βΆwf 6549 (class class class)co 7426 βm cmap 8853 βcr 11147 1c1 11149 2c2 12307 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-2 12315 |
This theorem is referenced by: prelrrx2b 47883 rrx2xpref1o 47887 rrx2plordisom 47892 line2ylem 47920 line2xlem 47922 itscnhlinecirc02p 47954 inlinecirc02plem 47955 |
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