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Mirrors > Home > NFE Home > Th. List > txpeq1 | GIF version |
Description: Equality theorem for tail cross product. (Contributed by Scott Fenton, 31-Jul-2019.) |
Ref | Expression |
---|---|
txpeq1 | ⊢ (A = B → (A ⊗ C) = (B ⊗ C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coeq2 4875 | . . 3 ⊢ (A = B → (◡1st ∘ A) = (◡1st ∘ B)) | |
2 | 1 | ineq1d 3456 | . 2 ⊢ (A = B → ((◡1st ∘ A) ∩ (◡2nd ∘ C)) = ((◡1st ∘ B) ∩ (◡2nd ∘ C))) |
3 | df-txp 5736 | . 2 ⊢ (A ⊗ C) = ((◡1st ∘ A) ∩ (◡2nd ∘ C)) | |
4 | df-txp 5736 | . 2 ⊢ (B ⊗ C) = ((◡1st ∘ B) ∩ (◡2nd ∘ C)) | |
5 | 2, 3, 4 | 3eqtr4g 2410 | 1 ⊢ (A = B → (A ⊗ C) = (B ⊗ C)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∩ cin 3208 1st c1st 4717 ∘ ccom 4721 ◡ccnv 4771 2nd c2nd 4783 ⊗ ctxp 5735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-opab 4623 df-br 4640 df-co 4726 df-txp 5736 |
This theorem is referenced by: pprodeq1 5834 |
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