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Theorem angmgm 29398

Description: The angle addition magma is a magma. (Contributed by Thierry Arnoux, 31-Aug-2026.)

**Hypotheses**
Ref	Expression
angmgmval.p	⊢ 𝑃 = (Base‘𝐺)
angmgmval.a	⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}
angmgmval.i	⊢ 𝐼 = (Itv‘𝐺)
angmgmval.d	⊢ − = (dist‘𝐺)
angmgmval.c	⊢ ∼ = (cgrA‘𝐺)
angmgmval.l	⊢ 𝐿 = (LineG‘𝐺)
angmgmval.o	⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))
angmgmval.j	⊢ 𝐽 = (AngMgm‘𝐺)
angmgm.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
angmgm.1	⊢ (𝜑 → 2 ≤ (♯‘𝑃))

**Assertion**
Ref	Expression
angmgm	⊢ (𝜑 → 𝐽 ∈ Mgm)

Distinct variable groups: 𝐴,𝑒,𝑓,𝑠 𝑒,𝐺,𝑓,𝑠 𝑃,𝑑,𝑒,𝑓,𝑠 + ,𝑑,𝑒,𝑓,𝑠 − ,𝑑,𝑒,𝑓,𝑠 ∼ ,𝑑,𝑒,𝑓,𝑠 𝐴,𝑑 𝐺,𝑑 𝐼,𝑑,𝑒,𝑓,𝑠 𝑒,𝐽,𝑓 𝐿,𝑑,𝑒,𝑓,𝑠 𝜑,𝑑,𝑒,𝑓,𝑠 Allowed substitution hints: 𝐽(𝑠, 𝑑)

Proof of Theorem angmgm Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		angmgmval.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		angmgmval.a	. . . 4 ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}
3		angmgmval.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		angmgmval.d	. . . 4 ⊢ − = (dist‘𝐺)
5		angmgmval.c	. . . 4 ⊢ ∼ = (cgrA‘𝐺)
6		angmgmval.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
7		angmgmval.o	. . . 4 ⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))
8		angmgmval.j	. . . 4 ⊢ 𝐽 = (AngMgm‘𝐺)
9		angmgm.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
10	9	ad3antrrr 743	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝐺 ∈ TarskiG)
11		simpllr 788	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ 𝑃)
12		simplr 781	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ 𝑃)
13		simpr 490	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝑥 ≠ 𝑦)
14	13	necomd 3010	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝑦 ≠ 𝑥)
15	12, 14	eldifsnd 4749	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ (𝑃 ∖ {𝑥}))
16	1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 15	angmgmlem 29396	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → (𝐽 ∈ Mgm ∧ [⟨“𝑥𝑦𝑥”⟩] ∼ = (0_g‘𝐽)))
17	16	simpld 500	. 2 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝐽 ∈ Mgm)
18		angmgm.1	. . 3 ⊢ (𝜑 → 2 ≤ (♯‘𝑃))
19	1, 4, 3, 9, 18	tglowdim1 28964	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑥 ≠ 𝑦)
20	17, 19	r19.29vva 3222	1 ⊢ (𝜑 → 𝐽 ∈ Mgm)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 {crab 3412 ∩ cin 4089 ∅c0 4278 ifcif 4481 class class class wbr 5102 ‘cfv 6538 ℩crio 7376 (class class class)co 7420 ∈ cmpo 7422 [cec 8716 ↑_m cmap 8848 0cc0 11201 1c1 11202 ≤ cle 11345 2c2 12398 3c3 12399 ..^cfzo 13789 ♯chash 14475 ⟨“cs3 14994 Basecbs 17388 distcds 17438 0_gc0g 17611 Mgmcmgm 18815 TarskiGcstrkg 28890 Itvcitv 28896 LineGclng 28897 cgrAccgra 29315 AngMgmcangmgm 29374

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-oadd 8481 df-er 8718 df-ec 8720 df-qs 8724 df-map 8850 df-pm 8851 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9435 df-inf 9436 df-dju 9983 df-card 10021 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-nn 12337 df-2 12406 df-3 12407 df-4 12408 df-5 12409 df-6 12410 df-7 12411 df-8 12412 df-9 12413 df-n0 12608 df-xnn0 12681 df-z 12695 df-dec 12816 df-uz 12967 df-fz 13641 df-fzo 13790 df-hash 14476 df-word 14660 df-concat 14717 df-s1 14744 df-s2 15000 df-s3 15001 df-struct 17326 df-slot 17361 df-ndx 17373 df-base 17389 df-plusg 17442 df-mulr 17443 df-sca 17445 df-vsca 17446 df-ip 17447 df-tset 17448 df-ple 17449 df-ds 17451 df-0g 17613 df-imas 17681 df-qus 17682 df-mgm 18817 df-trkgc 28911 df-trkgb 28912 df-trkgcb 28913 df-trkgld 28915 df-trkg 28916 df-cgrg 28975 df-ismt 28997 df-leg 29047 df-hlg 29065 df-mir 29126 df-rag 29170 df-perpg 29172 df-hpg 29237 df-mid 29280 df-lmi 29281 df-cgra 29316 df-angmgm 29375

This theorem is used by: (None)

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