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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 11156 | . . . . . . . 8 β’ 1 β V | |
2 | 2ex 12235 | . . . . . . . 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 12366 | . . . . . . 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 7102 | . . . . 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 4782 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β {π΄, π΅} β β) | |
12 | 10, 11 | fssd 6687 | . . 3 β’ ((π΄ β β β§ π΅ β β) β {β¨1, π΄β©, β¨2, π΅β©}:{1, 2}βΆβ) |
13 | reex 11147 | . . . . 5 β’ β β V | |
14 | prex 5390 | . . . . 5 β’ {1, 2} β V | |
15 | 13, 14 | pm3.2i 472 | . . . 4 β’ (β β V β§ {1, 2} β V) |
16 | elmapg 8781 | . . . 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 7369 | . . . 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 2940 Vcvv 3444 {cpr 4589 β¨cop 4593 βΆwf 6493 (class class class)co 7358 βm cmap 8768 βcr 11055 1c1 11057 2c2 12213 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-2 12221 |
This theorem is referenced by: prelrrx2b 46886 rrx2xpref1o 46890 rrx2plordisom 46895 line2ylem 46923 line2xlem 46925 itscnhlinecirc02p 46957 inlinecirc02plem 46958 |
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