![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iscgrad | Structured version Visualization version GIF version |
Description: Sufficient conditions for angle congruence, deduction version. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
Ref | Expression |
---|---|
iscgra.p | β’ π = (BaseβπΊ) |
iscgra.i | β’ πΌ = (ItvβπΊ) |
iscgra.k | β’ πΎ = (hlGβπΊ) |
iscgra.g | β’ (π β πΊ β TarskiG) |
iscgra.a | β’ (π β π΄ β π) |
iscgra.b | β’ (π β π΅ β π) |
iscgra.c | β’ (π β πΆ β π) |
iscgra.d | β’ (π β π· β π) |
iscgra.e | β’ (π β πΈ β π) |
iscgra.f | β’ (π β πΉ β π) |
iscgrad.x | β’ (π β π β π) |
iscgrad.y | β’ (π β π β π) |
iscgrad.1 | β’ (π β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βππΈπββ©) |
iscgrad.2 | β’ (π β π(πΎβπΈ)π·) |
iscgrad.3 | β’ (π β π(πΎβπΈ)πΉ) |
Ref | Expression |
---|---|
iscgrad | β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscgrad.x | . . 3 β’ (π β π β π) | |
2 | iscgrad.y | . . 3 β’ (π β π β π) | |
3 | iscgrad.1 | . . 3 β’ (π β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βππΈπββ©) | |
4 | iscgrad.2 | . . 3 β’ (π β π(πΎβπΈ)π·) | |
5 | iscgrad.3 | . . 3 β’ (π β π(πΎβπΈ)πΉ) | |
6 | id 22 | . . . . . . 7 β’ (π₯ = π β π₯ = π) | |
7 | eqidd 2727 | . . . . . . 7 β’ (π₯ = π β πΈ = πΈ) | |
8 | eqidd 2727 | . . . . . . 7 β’ (π₯ = π β π¦ = π¦) | |
9 | 6, 7, 8 | s3eqd 14818 | . . . . . 6 β’ (π₯ = π β β¨βπ₯πΈπ¦ββ© = β¨βππΈπ¦ββ©) |
10 | 9 | breq2d 5153 | . . . . 5 β’ (π₯ = π β (β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ₯πΈπ¦ββ© β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βππΈπ¦ββ©)) |
11 | breq1 5144 | . . . . 5 β’ (π₯ = π β (π₯(πΎβπΈ)π· β π(πΎβπΈ)π·)) | |
12 | 10, 11 | 3anbi12d 1433 | . . . 4 β’ (π₯ = π β ((β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ₯πΈπ¦ββ© β§ π₯(πΎβπΈ)π· β§ π¦(πΎβπΈ)πΉ) β (β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βππΈπ¦ββ© β§ π(πΎβπΈ)π· β§ π¦(πΎβπΈ)πΉ))) |
13 | eqidd 2727 | . . . . . . 7 β’ (π¦ = π β π = π) | |
14 | eqidd 2727 | . . . . . . 7 β’ (π¦ = π β πΈ = πΈ) | |
15 | id 22 | . . . . . . 7 β’ (π¦ = π β π¦ = π) | |
16 | 13, 14, 15 | s3eqd 14818 | . . . . . 6 β’ (π¦ = π β β¨βππΈπ¦ββ© = β¨βππΈπββ©) |
17 | 16 | breq2d 5153 | . . . . 5 β’ (π¦ = π β (β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βππΈπ¦ββ© β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βππΈπββ©)) |
18 | breq1 5144 | . . . . 5 β’ (π¦ = π β (π¦(πΎβπΈ)πΉ β π(πΎβπΈ)πΉ)) | |
19 | 17, 18 | 3anbi13d 1434 | . . . 4 β’ (π¦ = π β ((β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βππΈπ¦ββ© β§ π(πΎβπΈ)π· β§ π¦(πΎβπΈ)πΉ) β (β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βππΈπββ© β§ π(πΎβπΈ)π· β§ π(πΎβπΈ)πΉ))) |
20 | 12, 19 | rspc2ev 3619 | . . 3 β’ ((π β π β§ π β π β§ (β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βππΈπββ© β§ π(πΎβπΈ)π· β§ π(πΎβπΈ)πΉ)) β βπ₯ β π βπ¦ β π (β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ₯πΈπ¦ββ© β§ π₯(πΎβπΈ)π· β§ π¦(πΎβπΈ)πΉ)) |
21 | 1, 2, 3, 4, 5, 20 | syl113anc 1379 | . 2 β’ (π β βπ₯ β π βπ¦ β π (β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ₯πΈπ¦ββ© β§ π₯(πΎβπΈ)π· β§ π¦(πΎβπΈ)πΉ)) |
22 | iscgra.p | . . 3 β’ π = (BaseβπΊ) | |
23 | iscgra.i | . . 3 β’ πΌ = (ItvβπΊ) | |
24 | iscgra.k | . . 3 β’ πΎ = (hlGβπΊ) | |
25 | iscgra.g | . . 3 β’ (π β πΊ β TarskiG) | |
26 | iscgra.a | . . 3 β’ (π β π΄ β π) | |
27 | iscgra.b | . . 3 β’ (π β π΅ β π) | |
28 | iscgra.c | . . 3 β’ (π β πΆ β π) | |
29 | iscgra.d | . . 3 β’ (π β π· β π) | |
30 | iscgra.e | . . 3 β’ (π β πΈ β π) | |
31 | iscgra.f | . . 3 β’ (π β πΉ β π) | |
32 | 22, 23, 24, 25, 26, 27, 28, 29, 30, 31 | iscgra 28563 | . 2 β’ (π β (β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ© β βπ₯ β π βπ¦ β π (β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ₯πΈπ¦ββ© β§ π₯(πΎβπΈ)π· β§ π¦(πΎβπΈ)πΉ))) |
33 | 21, 32 | mpbird 257 | 1 β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βwrex 3064 class class class wbr 5141 βcfv 6536 β¨βcs3 14796 Basecbs 17150 TarskiGcstrkg 28181 Itvcitv 28187 cgrGccgrg 28264 hlGchlg 28354 cgrAccgra 28561 |
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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
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-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 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-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-hash 14293 df-word 14468 df-concat 14524 df-s1 14549 df-s2 14802 df-s3 14803 df-cgra 28562 |
This theorem is referenced by: cgrahl1 28570 cgrahl2 28571 cgraid 28573 cgrcgra 28575 dfcgra2 28584 sacgr 28585 tgsas2 28610 tgsas3 28611 tgasa1 28612 |
Copyright terms: Public domain | W3C validator |