Theorem lrold

Description: The union of the left and right options of a surreal make its old set. (Contributed by Scott Fenton, 9-Oct-2024)

	Ref	Expression
Assertion	lrold	Could not format assertion : No typesetting found for |- ( ( _Left ` A ) u. ( _Right ` A ) ) = ( _Old ` ( bday ` A ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		leftval	Could not format ( _Left ` A ) = { x e. ( _Old ` ( bday ` A ) ) | x
2		rightval	Could not format ( _Right ` A ) = { x e. ( _Old ` ( bday ` A ) ) | A
3	1 2	uneq12i	Could not format ( ( _Left ` A ) u. ( _Right ` A ) ) = ( { x e. ( _Old ` ( bday ` A ) ) | x
4		unrab	$$⊢\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|x{<}_{s}A\right\}\cup \left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|A{<}_{s}x\right\}=\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|\left(x{<}_{s}A\vee A{<}_{s}x\right)\right\}$$
5	3 4	eqtri	Could not format ( ( _Left ` A ) u. ( _Right ` A ) ) = { x e. ( _Old ` ( bday ` A ) ) | ( x
6		oldirr	$$⊢\neg A\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)$$
7		eleq1	$$⊢x=A\to \left(x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)\leftrightarrow A\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)\right)$$
8	6 7	mtbiri	$$⊢x=A\to \neg x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)$$
9	8	necon2ai	$$⊢x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)\to x\ne A$$
10	9	adantl	$$⊢\left(A\in \mathrm{No}\wedge x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)\right)\to x\ne A$$
11		oldssno	$$⊢\mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)\subseteq \mathrm{No}$$
12	11	sseli	$$⊢x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)\to x\in \mathrm{No}$$
13		slttrine	$$⊢\left(x\in \mathrm{No}\wedge A\in \mathrm{No}\right)\to \left(x\ne A\leftrightarrow \left(x{<}_{s}A\vee A{<}_{s}x\right)\right)$$
14	13	ancoms	$$⊢\left(A\in \mathrm{No}\wedge x\in \mathrm{No}\right)\to \left(x\ne A\leftrightarrow \left(x{<}_{s}A\vee A{<}_{s}x\right)\right)$$
15	12 14	sylan2	$$⊢\left(A\in \mathrm{No}\wedge x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)\right)\to \left(x\ne A\leftrightarrow \left(x{<}_{s}A\vee A{<}_{s}x\right)\right)$$
16	10 15	mpbid	$$⊢\left(A\in \mathrm{No}\wedge x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)\right)\to \left(x{<}_{s}A\vee A{<}_{s}x\right)$$
17	16	rabeqcda	$$⊢A\in \mathrm{No}\to \left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|\left(x{<}_{s}A\vee A{<}_{s}x\right)\right\}=\mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)$$
18	5 17	eqtrid	Could not format ( A e. No -> ( ( _Left ` A ) u. ( _Right ` A ) ) = ( _Old ` ( bday ` A ) ) ) : No typesetting found for |- ( A e. No -> ( ( _Left ` A ) u. ( _Right ` A ) ) = ( _Old ` ( bday ` A ) ) ) with typecode |-
19		un0	$$⊢\varnothing \cup \varnothing =\varnothing$$
20		leftf	Could not format _Left : No --> ~P No : No typesetting found for |- _Left : No --> ~P No with typecode |-
21	20	fdmi	Could not format dom _Left = No : No typesetting found for |- dom _Left = No with typecode |-
22	21	eleq2i	Could not format ( A e. dom _Left <-> A e. No ) : No typesetting found for |- ( A e. dom _Left <-> A e. No ) with typecode |-
23		ndmfv	Could not format ( -. A e. dom _Left -> ( _Left ` A ) = (/) ) : No typesetting found for |- ( -. A e. dom _Left -> ( _Left ` A ) = (/) ) with typecode |-
24	22 23	sylnbir	Could not format ( -. A e. No -> ( _Left ` A ) = (/) ) : No typesetting found for |- ( -. A e. No -> ( _Left ` A ) = (/) ) with typecode |-
25		rightf	Could not format _Right : No --> ~P No : No typesetting found for |- _Right : No --> ~P No with typecode |-
26	25	fdmi	Could not format dom _Right = No : No typesetting found for |- dom _Right = No with typecode |-
27	26	eleq2i	Could not format ( A e. dom _Right <-> A e. No ) : No typesetting found for |- ( A e. dom _Right <-> A e. No ) with typecode |-
28		ndmfv	Could not format ( -. A e. dom _Right -> ( _Right ` A ) = (/) ) : No typesetting found for |- ( -. A e. dom _Right -> ( _Right ` A ) = (/) ) with typecode |-
29	27 28	sylnbir	Could not format ( -. A e. No -> ( _Right ` A ) = (/) ) : No typesetting found for |- ( -. A e. No -> ( _Right ` A ) = (/) ) with typecode |-
30	24 29	uneq12d	Could not format ( -. A e. No -> ( ( _Left ` A ) u. ( _Right ` A ) ) = ( (/) u. (/) ) ) : No typesetting found for |- ( -. A e. No -> ( ( _Left ` A ) u. ( _Right ` A ) ) = ( (/) u. (/) ) ) with typecode |-
31		bdaydm	$$⊢\mathrm{dom}\mathrm{bday}=\mathrm{No}$$
32	31	eleq2i	$$⊢A\in \mathrm{dom}\mathrm{bday}\leftrightarrow A\in \mathrm{No}$$
33		ndmfv	$$⊢\neg A\in \mathrm{dom}\mathrm{bday}\to \mathrm{bday}\left(A\right)=\varnothing$$
34	32 33	sylnbir	$$⊢\neg A\in \mathrm{No}\to \mathrm{bday}\left(A\right)=\varnothing$$
35	34	fveq2d	$$⊢\neg A\in \mathrm{No}\to \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)=\mathrm{Old}\left(\varnothing \right)$$
36		old0	$$⊢\mathrm{Old}\left(\varnothing \right)=\varnothing$$
37	35 36	eqtrdi	$$⊢\neg A\in \mathrm{No}\to \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)=\varnothing$$
38	19 30 37	3eqtr4a	Could not format ( -. A e. No -> ( ( _Left ` A ) u. ( _Right ` A ) ) = ( _Old ` ( bday ` A ) ) ) : No typesetting found for |- ( -. A e. No -> ( ( _Left ` A ) u. ( _Right ` A ) ) = ( _Old ` ( bday ` A ) ) ) with typecode |-
39	18 38	pm2.61i	Could not format ( ( _Left ` A ) u. ( _Right ` A ) ) = ( _Old ` ( bday ` A ) ) : No typesetting found for |- ( ( _Left ` A ) u. ( _Right ` A ) ) = ( _Old ` ( bday ` A ) ) with typecode |-