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Mirrors > Home > MPE Home > Th. List > inagswap | Structured version Visualization version GIF version |
Description: Swap the order of the half lines delimiting the angle. Theorem 11.24 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 15-Aug-2020.) |
Ref | Expression |
---|---|
isinag.p | ⊢ 𝑃 = (Base‘𝐺) |
isinag.i | ⊢ 𝐼 = (Itv‘𝐺) |
isinag.k | ⊢ 𝐾 = (hlG‘𝐺) |
isinag.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
isinag.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
isinag.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
isinag.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
inagflat.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
inagswap.1 | ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐴𝐵𝐶”〉) |
Ref | Expression |
---|---|
inagswap | ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐶𝐵𝐴”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inagswap.1 | . . . . . 6 ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐴𝐵𝐶”〉) | |
2 | isinag.p | . . . . . . 7 ⊢ 𝑃 = (Base‘𝐺) | |
3 | isinag.i | . . . . . . 7 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | isinag.k | . . . . . . 7 ⊢ 𝐾 = (hlG‘𝐺) | |
5 | isinag.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
6 | isinag.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | isinag.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
8 | isinag.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
9 | inagflat.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | isinag 26610 | . . . . . 6 ⊢ (𝜑 → (𝑋(inA‘𝐺)〈“𝐴𝐵𝐶”〉 ↔ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))))) |
11 | 1, 10 | mpbid 234 | . . . . 5 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)))) |
12 | 11 | simpld 497 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵)) |
13 | 12 | simp2d 1139 | . . 3 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
14 | 12 | simp1d 1138 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
15 | 12 | simp3d 1140 | . . 3 ⊢ (𝜑 → 𝑋 ≠ 𝐵) |
16 | 13, 14, 15 | 3jca 1124 | . 2 ⊢ (𝜑 → (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵)) |
17 | 11 | simprd 498 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))) |
18 | eqid 2821 | . . . . . . 7 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
19 | 9 | 3ad2ant1 1129 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃 ∧ 𝑥 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG) |
20 | 6 | 3ad2ant1 1129 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃 ∧ 𝑥 ∈ (𝐴𝐼𝐶)) → 𝐴 ∈ 𝑃) |
21 | simp2 1133 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃 ∧ 𝑥 ∈ (𝐴𝐼𝐶)) → 𝑥 ∈ 𝑃) | |
22 | 8 | 3ad2ant1 1129 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃 ∧ 𝑥 ∈ (𝐴𝐼𝐶)) → 𝐶 ∈ 𝑃) |
23 | simp3 1134 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃 ∧ 𝑥 ∈ (𝐴𝐼𝐶)) → 𝑥 ∈ (𝐴𝐼𝐶)) | |
24 | 2, 18, 3, 19, 20, 21, 22, 23 | tgbtwncom 26260 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃 ∧ 𝑥 ∈ (𝐴𝐼𝐶)) → 𝑥 ∈ (𝐶𝐼𝐴)) |
25 | 24 | 3expia 1117 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑥 ∈ (𝐴𝐼𝐶) → 𝑥 ∈ (𝐶𝐼𝐴))) |
26 | 25 | anim1d 612 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)) → (𝑥 ∈ (𝐶𝐼𝐴) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)))) |
27 | 26 | reximdva 3274 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐶𝐼𝐴) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)))) |
28 | 17, 27 | mpd 15 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐶𝐼𝐴) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))) |
29 | 2, 3, 4, 5, 8, 7, 6, 9 | isinag 26610 | . 2 ⊢ (𝜑 → (𝑋(inA‘𝐺)〈“𝐶𝐵𝐴”〉 ↔ ((𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐶𝐼𝐴) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))))) |
30 | 16, 28, 29 | mpbir2and 711 | 1 ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐶𝐵𝐴”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 class class class wbr 5052 ‘cfv 6341 (class class class)co 7142 〈“cs3 14189 Basecbs 16466 distcds 16557 TarskiGcstrkg 26202 Itvcitv 26208 hlGchlg 26372 inAcinag 26607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-map 8394 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-n0 11885 df-z 11969 df-uz 12231 df-fz 12883 df-fzo 13024 df-hash 13681 df-word 13852 df-concat 13908 df-s1 13935 df-s2 14195 df-s3 14196 df-trkgc 26220 df-trkgb 26221 df-trkgcb 26222 df-trkg 26225 df-inag 26609 |
This theorem is referenced by: (None) |
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