grpasscan2d - Mathbox for Steven Nguyen

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Theorem grpasscan2d 43322

Description: An associative cancellation law for groups. (Contributed by SN, 29-Jan-2025.)

**Hypotheses**
Ref	Expression
grpasscan2d.b	⊢ 𝐵 = (Base‘𝐺)
grpasscan2d.p	⊢ + = (+_g‘𝐺)
grpasscan2d.n	⊢ 𝑁 = (inv_g‘𝐺)
grpasscan2d.g	⊢ (𝜑 → 𝐺 ∈ Grp)
grpasscan2d.1	⊢ (𝜑 → 𝑋 ∈ 𝐵)
grpasscan2d.2	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
grpasscan2d	⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)

**Proof of Theorem grpasscan2d**
Step	Hyp	Ref	Expression
1		grpasscan2d.g	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
2		grpasscan2d.1	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		grpasscan2d.2	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4		grpasscan2d.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
5		grpasscan2d.p	. . 3 ⊢ + = (+_g‘𝐺)
6		grpasscan2d.n	. . 3 ⊢ 𝑁 = (inv_g‘𝐺)
7	4, 5, 6	grpasscan2 19100	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)
8	1, 2, 3, 7	syl3anc 1398	1 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2146 ‘cfv 6543 (class class class)co 7423 Basecbs 17294 +_gcplusg 17335 Grpcgrp 19031 inv_gcminusg 19032

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-riota 7380 df-ov 7426 df-0g 17519 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-grp 19034 df-minusg 19035

This theorem is used by: grpcominv1 43323

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