Theorem mirtrcgr 28667

Description: Point inversion of one point of a triangle around another point preserves triangle congruence. (Contributed by Thierry Arnoux, 4-Oct-2020.)

Hypotheses

Ref	Expression
mirval.p	⊢ 𝑃 = (Base‘𝐺)
mirval.d	⊢ − = (dist‘𝐺)
mirval.i	⊢ 𝐼 = (Itv‘𝐺)
mirval.l	⊢ 𝐿 = (LineG‘𝐺)
mirval.s	⊢ 𝑆 = (pInvG‘𝐺)
mirval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
mirtrcgr.e	⊢ ∼ = (cgrG‘𝐺)
mirtrcgr.m	⊢ 𝑀 = (𝑆‘𝐵)
mirtrcgr.n	⊢ 𝑁 = (𝑆‘𝑌)
mirtrcgr.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mirtrcgr.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
mirtrcgr.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
mirtrcgr.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
mirtrcgr.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
mirtrcgr.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
mirtrcgr.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)
mirtrcgr.2	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝑋𝑌𝑍”⟩)

Assertion

Ref	Expression
mirtrcgr	⊢ (𝜑 → ⟨“(𝑀‘𝐴)𝐵𝐶”⟩ ∼ ⟨“(𝑁‘𝑋)𝑌𝑍”⟩)

Proof of Theorem mirtrcgr

Step	Hyp	Ref	Expression
1		mirval.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		mirval.d	. 2 ⊢ − = (dist‘𝐺)
3		mirtrcgr.e	. 2 ⊢ ∼ = (cgrG‘𝐺)
4		mirval.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		mirval.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
6		mirval.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
7		mirval.s	. . 3 ⊢ 𝑆 = (pInvG‘𝐺)
8		mirtrcgr.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9		mirtrcgr.m	. . 3 ⊢ 𝑀 = (𝑆‘𝐵)
10		mirtrcgr.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11	1, 2, 5, 6, 7, 4, 8, 9, 10	mircl 28645	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
12		mirtrcgr.c	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
13		mirtrcgr.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
14		mirtrcgr.n	. . 3 ⊢ 𝑁 = (𝑆‘𝑌)
15		mirtrcgr.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
16	1, 2, 5, 6, 7, 4, 13, 14, 15	mircl 28645	. 2 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝑃)
17		mirtrcgr.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
18		mirtrcgr.2	. . . . . 6 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝑋𝑌𝑍”⟩)
19	1, 2, 5, 3, 4, 10, 8, 12, 15, 13, 17, 18	cgr3simp1 28504	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌))
20	1, 2, 5, 4, 10, 8, 15, 13, 19	tgcgrcomlr 28464	. . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝑌 − 𝑋))
21	1, 2, 5, 6, 7, 4, 8, 9, 10	mircgr 28641	. . . 4 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝐵 − 𝐴))
22	1, 2, 5, 6, 7, 4, 13, 14, 15	mircgr 28641	. . . 4 ⊢ (𝜑 → (𝑌 − (𝑁‘𝑋)) = (𝑌 − 𝑋))
23	20, 21, 22	3eqtr4d 2776	. . 3 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝑌 − (𝑁‘𝑋)))
24	1, 2, 5, 4, 8, 11, 13, 16, 23	tgcgrcomlr 28464	. 2 ⊢ (𝜑 → ((𝑀‘𝐴) − 𝐵) = ((𝑁‘𝑋) − 𝑌))
25	1, 2, 5, 3, 4, 10, 8, 12, 15, 13, 17, 18	cgr3simp2 28505	. 2 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝑌 − 𝑍))
26	1, 2, 5, 6, 7, 4, 8, 9, 10	mirbtwn 28642	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((𝑀‘𝐴)𝐼𝐴))
27	1, 6, 5, 4, 11, 10, 8, 26	btwncolg1 28539	. . . . 5 ⊢ (𝜑 → (𝐵 ∈ ((𝑀‘𝐴)𝐿𝐴) ∨ (𝑀‘𝐴) = 𝐴))
28	1, 6, 5, 4, 11, 10, 8, 27	colcom 28542	. . . 4 ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐿(𝑀‘𝐴)) ∨ 𝐴 = (𝑀‘𝐴)))
29	1, 2, 5, 6, 7, 4, 3, 9, 14, 10, 8, 15, 13, 19	mircgrextend 28666	. . . . . 6 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋)))
30	1, 2, 5, 4, 10, 11, 15, 16, 29	tgcgrcomlr 28464	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝐴) − 𝐴) = ((𝑁‘𝑋) − 𝑋))
31	1, 2, 3, 4, 10, 8, 11, 15, 13, 16, 19, 23, 30	trgcgr 28500	. . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵(𝑀‘𝐴)”⟩ ∼ ⟨“𝑋𝑌(𝑁‘𝑋)”⟩)
32	1, 2, 5, 3, 4, 10, 8, 12, 15, 13, 17, 18	cgr3simp3 28506	. . . . 5 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝑍 − 𝑋))
33	1, 2, 5, 4, 12, 10, 17, 15, 32	tgcgrcomlr 28464	. . . 4 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝑋 − 𝑍))
34		mirtrcgr.1	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
35	1, 6, 5, 4, 10, 8, 11, 3, 15, 13, 2, 12, 16, 17, 28, 31, 33, 25, 34	tgfscgr 28552	. . 3 ⊢ (𝜑 → ((𝑀‘𝐴) − 𝐶) = ((𝑁‘𝑋) − 𝑍))
36	1, 2, 5, 4, 11, 12, 16, 17, 35	tgcgrcomlr 28464	. 2 ⊢ (𝜑 → (𝐶 − (𝑀‘𝐴)) = (𝑍 − (𝑁‘𝑋)))
37	1, 2, 3, 4, 11, 8, 12, 16, 13, 17, 24, 25, 36	trgcgr 28500	1 ⊢ (𝜑 → ⟨“(𝑀‘𝐴)𝐵𝐶”⟩ ∼ ⟨“(𝑁‘𝑋)𝑌𝑍”⟩)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111   ≠ wne 2928   class class class wbr 5093  ‘cfv 6487  (class class class)co 7352  ⟨“cs3 14755  Basecbs 17126  distcds 17176  TarskiGcstrkg 28411  Itvcitv 28417  LineGclng 28418  cgrGccgrg 28494  pInvGcmir 28636

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11068  ax-resscn 11069  ax-1cn 11070  ax-icn 11071  ax-addcl 11072  ax-addrcl 11073  ax-mulcl 11074  ax-mulrcl 11075  ax-mulcom 11076  ax-addass 11077  ax-mulass 11078  ax-distr 11079  ax-i2m1 11080  ax-1ne0 11081  ax-1rid 11082  ax-rnegex 11083  ax-rrecex 11084  ax-cnre 11085  ax-pre-lttri 11086  ax-pre-lttrn 11087  ax-pre-ltadd 11088  ax-pre-mulgt0 11089

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-oadd 8395  df-er 8628  df-pm 8759  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-dju 9800  df-card 9838  df-pnf 11154  df-mnf 11155  df-xr 11156  df-ltxr 11157  df-le 11158  df-sub 11352  df-neg 11353  df-nn 12132  df-2 12194  df-3 12195  df-n0 12388  df-xnn0 12461  df-z 12475  df-uz 12739  df-fz 13414  df-fzo 13561  df-hash 14244  df-word 14427  df-concat 14484  df-s1 14510  df-s2 14761  df-s3 14762  df-trkgc 28432  df-trkgb 28433  df-trkgcb 28434  df-trkg 28437  df-cgrg 28495  df-mir 28637

This theorem is referenced by:  sacgr  28815