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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fuco23a | Structured version Visualization version GIF version | ||
| Description: The morphism part of the functor composition bifunctor. An alternate definition of ∘F. See also fuco23 49582. (Contributed by Zhi Wang, 3-Oct-2025.) |
| Ref | Expression |
|---|---|
| fuco23a.a | ⊢ (𝜑 → 𝐴 ∈ (〈𝐹, 𝐺〉(𝐶 Nat 𝐷)〈𝑀, 𝑁〉)) |
| fuco23a.b | ⊢ (𝜑 → 𝐵 ∈ (〈𝐾, 𝐿〉(𝐷 Nat 𝐸)〈𝑅, 𝑆〉)) |
| fuco23a.x | ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
| fuco23a.p | ⊢ (𝜑 → (〈𝐶, 𝐷〉 ∘F 𝐸) = 〈𝑂, 𝑃〉) |
| fuco23a.u | ⊢ (𝜑 → 𝑈 = 〈〈𝐾, 𝐿〉, 〈𝐹, 𝐺〉〉) |
| fuco23a.v | ⊢ (𝜑 → 𝑉 = 〈〈𝑅, 𝑆〉, 〈𝑀, 𝑁〉〉) |
| fuco23a.o | ⊢ (𝜑 → ∗ = (〈(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝐹‘𝑋))〉(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))) |
| Ref | Expression |
|---|---|
| fuco23a | ⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) = ((((𝐹‘𝑋)𝑆(𝑀‘𝑋))‘(𝐴‘𝑋)) ∗ (𝐵‘(𝐹‘𝑋)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fuco23a.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ (〈𝐹, 𝐺〉(𝐶 Nat 𝐷)〈𝑀, 𝑁〉)) | |
| 2 | fuco23a.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ (〈𝐾, 𝐿〉(𝐷 Nat 𝐸)〈𝑅, 𝑆〉)) | |
| 3 | fuco23a.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) | |
| 4 | eqid 2736 | . . 3 ⊢ (comp‘𝐸) = (comp‘𝐸) | |
| 5 | 1, 2, 3, 4 | fuco23alem 49592 | . 2 ⊢ (𝜑 → ((𝐵‘(𝑀‘𝑋))(〈(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))〉(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))) = ((((𝐹‘𝑋)𝑆(𝑀‘𝑋))‘(𝐴‘𝑋))(〈(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝐹‘𝑋))〉(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))(𝐵‘(𝐹‘𝑋)))) |
| 6 | fuco23a.p | . . 3 ⊢ (𝜑 → (〈𝐶, 𝐷〉 ∘F 𝐸) = 〈𝑂, 𝑃〉) | |
| 7 | fuco23a.u | . . 3 ⊢ (𝜑 → 𝑈 = 〈〈𝐾, 𝐿〉, 〈𝐹, 𝐺〉〉) | |
| 8 | fuco23a.v | . . 3 ⊢ (𝜑 → 𝑉 = 〈〈𝑅, 𝑆〉, 〈𝑀, 𝑁〉〉) | |
| 9 | eqidd 2737 | . . 3 ⊢ (𝜑 → (〈(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))〉(comp‘𝐸)(𝑅‘(𝑀‘𝑋))) = (〈(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))〉(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))) | |
| 10 | 6, 7, 8, 1, 2, 3, 9 | fuco23 49582 | . 2 ⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) = ((𝐵‘(𝑀‘𝑋))(〈(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))〉(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)))) |
| 11 | fuco23a.o | . . 3 ⊢ (𝜑 → ∗ = (〈(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝐹‘𝑋))〉(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))) | |
| 12 | 11 | oveqd 7375 | . 2 ⊢ (𝜑 → ((((𝐹‘𝑋)𝑆(𝑀‘𝑋))‘(𝐴‘𝑋)) ∗ (𝐵‘(𝐹‘𝑋))) = ((((𝐹‘𝑋)𝑆(𝑀‘𝑋))‘(𝐴‘𝑋))(〈(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝐹‘𝑋))〉(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))(𝐵‘(𝐹‘𝑋)))) |
| 13 | 5, 10, 12 | 3eqtr4d 2781 | 1 ⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) = ((((𝐹‘𝑋)𝑆(𝑀‘𝑋))‘(𝐴‘𝑋)) ∗ (𝐵‘(𝐹‘𝑋)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 〈cop 4586 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 compcco 17189 Nat cnat 17868 ∘F cfuco 49557 |
| 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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-map 8765 df-ixp 8836 df-func 17782 df-cofu 17784 df-nat 17870 df-fuco 49558 |
| This theorem is referenced by: (None) |
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