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Mirrors > Home > MPE Home > Th. List > grpomuldivass | Structured version Visualization version GIF version |
Description: Associative-type law for multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpdivf.1 | ⊢ 𝑋 = ran 𝐺 |
grpdivf.3 | ⊢ 𝐷 = ( /𝑔 ‘𝐺) |
Ref | Expression |
---|---|
grpomuldivass | ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = (𝐴𝐺(𝐵𝐷𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr1 1195 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐴 ∈ 𝑋) | |
2 | simpr2 1196 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐵 ∈ 𝑋) | |
3 | grpdivf.1 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
4 | eqid 2739 | . . . . . 6 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
5 | 3, 4 | grpoinvcl 28471 | . . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐶 ∈ 𝑋) → ((inv‘𝐺)‘𝐶) ∈ 𝑋) |
6 | 5 | 3ad2antr3 1191 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋) |
7 | 1, 2, 6 | 3jca 1129 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋)) |
8 | 3 | grpoass 28450 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺((inv‘𝐺)‘𝐶)) = (𝐴𝐺(𝐵𝐺((inv‘𝐺)‘𝐶)))) |
9 | 7, 8 | syldan 594 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺((inv‘𝐺)‘𝐶)) = (𝐴𝐺(𝐵𝐺((inv‘𝐺)‘𝐶)))) |
10 | simpl 486 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐺 ∈ GrpOp) | |
11 | 3 | grpocl 28447 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) |
12 | 11 | 3adant3r3 1185 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋) |
13 | simpr3 1197 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐶 ∈ 𝑋) | |
14 | grpdivf.3 | . . . 4 ⊢ 𝐷 = ( /𝑔 ‘𝐺) | |
15 | 3, 4, 14 | grpodivval 28482 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐺𝐵)𝐺((inv‘𝐺)‘𝐶))) |
16 | 10, 12, 13, 15 | syl3anc 1372 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐺𝐵)𝐺((inv‘𝐺)‘𝐶))) |
17 | 3, 4, 14 | grpodivval 28482 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐷𝐶) = (𝐵𝐺((inv‘𝐺)‘𝐶))) |
18 | 17 | 3adant3r1 1183 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐷𝐶) = (𝐵𝐺((inv‘𝐺)‘𝐶))) |
19 | 18 | oveq2d 7198 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐺(𝐵𝐷𝐶)) = (𝐴𝐺(𝐵𝐺((inv‘𝐺)‘𝐶)))) |
20 | 9, 16, 19 | 3eqtr4d 2784 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = (𝐴𝐺(𝐵𝐷𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ran crn 5536 ‘cfv 6349 (class class class)co 7182 GrpOpcgr 28436 invcgn 28438 /𝑔 cgs 28439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-1st 7726 df-2nd 7727 df-grpo 28440 df-gid 28441 df-ginv 28442 df-gdiv 28443 |
This theorem is referenced by: ablomuldiv 28499 ablodivdiv 28500 ablo4pnp 35693 |
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