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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 11214 | . . . . . . . 8 β’ 1 β V | |
2 | 2ex 12293 | . . . . . . . 8 β’ 2 β V | |
3 | 1, 2 | pm3.2i 470 | . . . . . . 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 12424 | . . . . . . 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 7149 | . . . . 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 4819 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β {π΄, π΅} β β) | |
12 | 10, 11 | fssd 6729 | . . 3 β’ ((π΄ β β β§ π΅ β β) β {β¨1, π΄β©, β¨2, π΅β©}:{1, 2}βΆβ) |
13 | reex 11203 | . . . . 5 β’ β β V | |
14 | prex 5425 | . . . . 5 β’ {1, 2} β V | |
15 | 13, 14 | pm3.2i 470 | . . . 4 β’ (β β V β§ {1, 2} β V) |
16 | elmapg 8835 | . . . 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 7416 | . . . 4 β’ (β βm πΌ) = (β βm {1, 2}) |
22 | 19, 21 | eqtri 2754 | . . 3 β’ π = (β βm {1, 2}) |
23 | 22 | eleq2i 2819 | . 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 Vcvv 3468 {cpr 4625 β¨cop 4629 βΆwf 6533 (class class class)co 7405 βm cmap 8822 βcr 11111 1c1 11113 2c2 12271 |
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 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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-reu 3371 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 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-2 12279 |
This theorem is referenced by: prelrrx2b 47675 rrx2xpref1o 47679 rrx2plordisom 47684 line2ylem 47712 line2xlem 47714 itscnhlinecirc02p 47746 inlinecirc02plem 47747 |
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