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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 28765 | . . . . . 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 231 | . . . . 5 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)))) |
12 | 11 | simpld 493 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵)) |
13 | 12 | simp2d 1140 | . . 3 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
14 | 12 | simp1d 1139 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
15 | 12 | simp3d 1141 | . . 3 ⊢ (𝜑 → 𝑋 ≠ 𝐵) |
16 | 13, 14, 15 | 3jca 1125 | . 2 ⊢ (𝜑 → (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵)) |
17 | 11 | simprd 494 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))) |
18 | eqid 2726 | . . . . . . 7 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
19 | 9 | 3ad2ant1 1130 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃 ∧ 𝑥 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG) |
20 | 6 | 3ad2ant1 1130 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃 ∧ 𝑥 ∈ (𝐴𝐼𝐶)) → 𝐴 ∈ 𝑃) |
21 | simp2 1134 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃 ∧ 𝑥 ∈ (𝐴𝐼𝐶)) → 𝑥 ∈ 𝑃) | |
22 | 8 | 3ad2ant1 1130 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃 ∧ 𝑥 ∈ (𝐴𝐼𝐶)) → 𝐶 ∈ 𝑃) |
23 | simp3 1135 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃 ∧ 𝑥 ∈ (𝐴𝐼𝐶)) → 𝑥 ∈ (𝐴𝐼𝐶)) | |
24 | 2, 18, 3, 19, 20, 21, 22, 23 | tgbtwncom 28415 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃 ∧ 𝑥 ∈ (𝐴𝐼𝐶)) → 𝑥 ∈ (𝐶𝐼𝐴)) |
25 | 24 | 3expia 1118 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑥 ∈ (𝐴𝐼𝐶) → 𝑥 ∈ (𝐶𝐼𝐴))) |
26 | 25 | anim1d 609 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)) → (𝑥 ∈ (𝐶𝐼𝐴) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)))) |
27 | 26 | reximdva 3158 | . . 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 28765 | . 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 394 ∨ wo 845 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 〈“cs3 14851 Basecbs 17213 distcds 17275 TarskiGcstrkg 28354 Itvcitv 28360 hlGchlg 28527 inAcinag 28762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-fzo 13682 df-hash 14348 df-word 14523 df-concat 14579 df-s1 14604 df-s2 14857 df-s3 14858 df-trkgc 28375 df-trkgb 28376 df-trkgcb 28377 df-trkg 28380 df-inag 28764 |
This theorem is referenced by: (None) |
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