cofuoppf - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cofuoppf		Structured version Visualization version GIF version

Theorem cofuoppf 50141

Description: Composition of opposite functors. (Contributed by Zhi Wang, 26-Nov-2025.)

**Hypotheses**
Ref	Expression
cofuoppf.k	⊢ (𝜑 → (𝐺 ∘_func 𝐹) = 𝐾)
cofuoppf.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
cofuoppf.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))

**Assertion**
Ref	Expression
cofuoppf	⊢ (𝜑 → (( oppFunc ‘𝐺) ∘_func ( oppFunc ‘𝐹)) = ( oppFunc ‘𝐾))

Proof of Theorem cofuoppf Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. . . 4 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
2		eqid 2760	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17831	. . 3 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝐶))
4		eqid 2760	. . . 4 ⊢ (oppCat‘𝐷) = (oppCat‘𝐷)
5		cofuoppf.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6	5	func1st2nd 50067	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
7	1, 4, 6	funcoppc 17989	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)((oppCat‘𝐶) Func (oppCat‘𝐷))tpos (2^nd ‘𝐹))
8		eqid 2760	. . . 4 ⊢ (oppCat‘𝐸) = (oppCat‘𝐸)
9		cofuoppf.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
10	9	func1st2nd 50067	. . . 4 ⊢ (𝜑 → (1^st ‘𝐺)(𝐷 Func 𝐸)(2^nd ‘𝐺))
11	4, 8, 10	funcoppc 17989	. . 3 ⊢ (𝜑 → (1^st ‘𝐺)((oppCat‘𝐷) Func (oppCat‘𝐸))tpos (2^nd ‘𝐺))
12	3, 7, 11	cofuval2 18001	. 2 ⊢ (𝜑 → (⟨(1^st ‘𝐺), tpos (2^nd ‘𝐺)⟩ ∘_func ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩) = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), (𝑦 ∈ (Base‘𝐶), 𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥)))⟩)
13		oppfval2 50128	. . . 4 ⊢ (𝐺 ∈ (𝐷 Func 𝐸) → ( oppFunc ‘𝐺) = ⟨(1^st ‘𝐺), tpos (2^nd ‘𝐺)⟩)
14	9, 13	syl 18	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐺) = ⟨(1^st ‘𝐺), tpos (2^nd ‘𝐺)⟩)
15		oppfval2 50128	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
16	5, 15	syl 18	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
17	14, 16	oveq12d 7434	. 2 ⊢ (𝜑 → (( oppFunc ‘𝐺) ∘_func ( oppFunc ‘𝐹)) = (⟨(1^st ‘𝐺), tpos (2^nd ‘𝐺)⟩ ∘_func ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩))
18		cofuoppf.k	. . . . 5 ⊢ (𝜑 → (𝐺 ∘_func 𝐹) = 𝐾)
19	2, 5, 9	cofuval 17996	. . . . 5 ⊢ (𝜑 → (𝐺 ∘_func 𝐹) = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)))⟩)
20	18, 19	eqtr3d 2797	. . . 4 ⊢ (𝜑 → 𝐾 = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)))⟩)
21	5, 9	cofucl 18002	. . . . 5 ⊢ (𝜑 → (𝐺 ∘_func 𝐹) ∈ (𝐶 Func 𝐸))
22	18, 21	eqeltrrd 2861	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝐶 Func 𝐸))
23	20, 22	oppfval3 50129	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐾) = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), tpos (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)))⟩)
24		ovtpos 8244	. . . . . . . . 9 ⊢ (((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) = (((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦))
25		ovtpos 8244	. . . . . . . . 9 ⊢ (𝑦tpos (2^nd ‘𝐹)𝑥) = (𝑥(2^nd ‘𝐹)𝑦)
26	24, 25	coeq12i 5845	. . . . . . . 8 ⊢ ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥)) = ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦))
27	26	eqcomi 2769	. . . . . . 7 ⊢ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)) = ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥))
28	27	a1i 11	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)) = ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥)))
29	28	mpoeq3ia 7494	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥)))
30	29	tposmpo 8266	. . . 4 ⊢ tpos (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦))) = (𝑦 ∈ (Base‘𝐶), 𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥)))
31	30	opeq2i 4837	. . 3 ⊢ ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), tpos (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)))⟩ = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), (𝑦 ∈ (Base‘𝐶), 𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥)))⟩
32	23, 31	eqtrdi 2811	. 2 ⊢ (𝜑 → ( oppFunc ‘𝐾) = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), (𝑦 ∈ (Base‘𝐶), 𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥)))⟩)
33	12, 17, 32	3eqtr4d 2805	1 ⊢ (𝜑 → (( oppFunc ‘𝐺) ∘_func ( oppFunc ‘𝐹)) = ( oppFunc ‘𝐾))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ⟨cop 4590 ∘ ccom 5659 ‘cfv 6535 (class class class)co 7416 ∈ cmpo 7418 1^st c1st 7990 2^nd c2nd 7991 tpos ctpos 8228 Basecbs 17326 oppCatcoppc 17824 Func cfunc 17968 ∘_func ccofu 17970 oppFunc coppf 50113

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-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 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-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 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-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7992 df-2nd 7993 df-tpos 8229 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-er 8703 df-map 8835 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-z 12641 df-dec 12762 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-hom 17391 df-cco 17392 df-cat 17781 df-cid 17782 df-oppc 17825 df-func 17972 df-cofu 17974 df-oppf 50114

This theorem is used by: lmdran 50662

W3C validator