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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 11210 | . . . . . . . 8 β’ 1 β V | |
2 | 2ex 12289 | . . . . . . . 8 β’ 2 β V | |
3 | 1, 2 | pm3.2i 472 | . . . . . . 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 12420 | . . . . . . 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 1129 | . . . . 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 7153 | . . . . 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 4825 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β {π΄, π΅} β β) | |
12 | 10, 11 | fssd 6736 | . . 3 β’ ((π΄ β β β§ π΅ β β) β {β¨1, π΄β©, β¨2, π΅β©}:{1, 2}βΆβ) |
13 | reex 11201 | . . . . 5 β’ β β V | |
14 | prex 5433 | . . . . 5 β’ {1, 2} β V | |
15 | 13, 14 | pm3.2i 472 | . . . 4 β’ (β β V β§ {1, 2} β V) |
16 | elmapg 8833 | . . . 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 7420 | . . . 4 β’ (β βm πΌ) = (β βm {1, 2}) |
22 | 19, 21 | eqtri 2761 | . . 3 β’ π = (β βm {1, 2}) |
23 | 22 | eleq2i 2826 | . 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2941 Vcvv 3475 {cpr 4631 β¨cop 4635 βΆwf 6540 (class class class)co 7409 βm cmap 8820 βcr 11109 1c1 11111 2c2 12267 |
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-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
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-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-2 12275 |
This theorem is referenced by: prelrrx2b 47400 rrx2xpref1o 47404 rrx2plordisom 47409 line2ylem 47437 line2xlem 47439 itscnhlinecirc02p 47471 inlinecirc02plem 47472 |
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