grpasscan2d - Mathbox for Steven Nguyen

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Theorem grpasscan2d 43395

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 19126	. 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 2145 ‘cfv 6533 (class class class)co 7413 Basecbs 17301 +_gcplusg 17342 Grpcgrp 19057 inv_gcminusg 19058

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-riota 7370 df-ov 7416 df-0g 17526 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-grp 19060 df-minusg 19061

This theorem is used by: grpcominv1 43396

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