![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 26632 | . . . . . 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 235 | . . . . 5 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)))) |
12 | 11 | simpld 498 | . . . 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 499 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))) |
18 | eqid 2798 | . . . . . . 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 26282 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃 ∧ 𝑥 ∈ (𝐴𝐼𝐶)) → 𝑥 ∈ (𝐶𝐼𝐴)) |
25 | 24 | 3expia 1118 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑥 ∈ (𝐴𝐼𝐶) → 𝑥 ∈ (𝐶𝐼𝐴))) |
26 | 25 | anim1d 613 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)) → (𝑥 ∈ (𝐶𝐼𝐴) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)))) |
27 | 26 | reximdva 3233 | . . 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 26632 | . 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 712 | 1 ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐶𝐵𝐴”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 〈“cs3 14195 Basecbs 16475 distcds 16566 TarskiGcstrkg 26224 Itvcitv 26230 hlGchlg 26394 inAcinag 26629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-s3 14202 df-trkgc 26242 df-trkgb 26243 df-trkgcb 26244 df-trkg 26247 df-inag 26631 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |