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Mirrors > Home > MPE Home > Th. List > grpoidinvlem1 | Structured version Visualization version GIF version |
Description: Lemma for grpoidinv 28285. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpfo.1 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
grpoidinvlem1 | ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑈𝐺𝐴) = 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . 5 ⊢ ((𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) | |
2 | 1 | 3anidm23 1417 | . . . 4 ⊢ ((𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) |
3 | grpfo.1 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
4 | 3 | grpoass 28280 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑌𝐺𝐴)𝐺𝐴) = (𝑌𝐺(𝐴𝐺𝐴))) |
5 | 2, 4 | sylan2 594 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑌𝐺𝐴)𝐺𝐴) = (𝑌𝐺(𝐴𝐺𝐴))) |
6 | 5 | adantr 483 | . 2 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → ((𝑌𝐺𝐴)𝐺𝐴) = (𝑌𝐺(𝐴𝐺𝐴))) |
7 | oveq1 7163 | . . 3 ⊢ ((𝑌𝐺𝐴) = 𝑈 → ((𝑌𝐺𝐴)𝐺𝐴) = (𝑈𝐺𝐴)) | |
8 | 7 | ad2antrl 726 | . 2 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → ((𝑌𝐺𝐴)𝐺𝐴) = (𝑈𝐺𝐴)) |
9 | oveq2 7164 | . . . 4 ⊢ ((𝐴𝐺𝐴) = 𝐴 → (𝑌𝐺(𝐴𝐺𝐴)) = (𝑌𝐺𝐴)) | |
10 | 9 | ad2antll 727 | . . 3 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑌𝐺(𝐴𝐺𝐴)) = (𝑌𝐺𝐴)) |
11 | simprl 769 | . . 3 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑌𝐺𝐴) = 𝑈) | |
12 | 10, 11 | eqtrd 2856 | . 2 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑌𝐺(𝐴𝐺𝐴)) = 𝑈) |
13 | 6, 8, 12 | 3eqtr3d 2864 | 1 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑈𝐺𝐴) = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ran crn 5556 (class class class)co 7156 GrpOpcgr 28266 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fo 6361 df-fv 6363 df-ov 7159 df-grpo 28270 |
This theorem is referenced by: grpoidinvlem3 28283 |
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