gtiso - Mathbox for Thierry Arnoux

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Theorem gtiso 31033

Description: Two ways to write a strictly decreasing function on the reals. (Contributed by Thierry Arnoux, 6-Apr-2017.)

**Assertion**
Ref	Expression
gtiso	⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ^◡ ≤ (𝐴, 𝐵)))

**Proof of Theorem gtiso**
Step	Hyp	Ref	Expression
1		eqid 2738	. . . . 5 ⊢ ((𝐴 × 𝐴) ∖ < ) = ((𝐴 × 𝐴) ∖ < )
2		eqid 2738	. . . . 5 ⊢ ((𝐵 × 𝐵) ∖ ^◡ < ) = ((𝐵 × 𝐵) ∖ ^◡ < )
3	1, 2	isocnv3 7203	. . . 4 ⊢ (𝐹 Isom < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵))
4	3	a1i 11	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵)))
5		df-le 11015	. . . . . . . . . 10 ⊢ ≤ = ((ℝ^* × ℝ^*) ∖ ^◡ < )
6	5	cnveqi 5783	. . . . . . . . 9 ⊢ ^◡ ≤ = ^◡((ℝ^* × ℝ^*) ∖ ^◡ < )
7		cnvdif 6047	. . . . . . . . 9 ⊢ ^◡((ℝ^* × ℝ^) ∖ ^◡ < ) = (^◡(ℝ^ × ℝ^*) ∖ ^◡^◡ < )
8		cnvxp 6060	. . . . . . . . . 10 ⊢ ^◡(ℝ^* × ℝ^) = (ℝ^ × ℝ^*)
9		ltrel 11037	. . . . . . . . . . 11 ⊢ Rel <
10		dfrel2 6092	. . . . . . . . . . 11 ⊢ (Rel < ↔ ^◡^◡ < = < )
11	9, 10	mpbi 229	. . . . . . . . . 10 ⊢ ^◡^◡ < = <
12	8, 11	difeq12i 4055	. . . . . . . . 9 ⊢ (^◡(ℝ^* × ℝ^) ∖ ^◡^◡ < ) = ((ℝ^ × ℝ^*) ∖ < )
13	6, 7, 12	3eqtri 2770	. . . . . . . 8 ⊢ ^◡ ≤ = ((ℝ^* × ℝ^*) ∖ < )
14	13	ineq1i 4142	. . . . . . 7 ⊢ (^◡ ≤ ∩ (𝐴 × 𝐴)) = (((ℝ^* × ℝ^*) ∖ < ) ∩ (𝐴 × 𝐴))
15		indif1 4205	. . . . . . 7 ⊢ (((ℝ^* × ℝ^) ∖ < ) ∩ (𝐴 × 𝐴)) = (((ℝ^ × ℝ^*) ∩ (𝐴 × 𝐴)) ∖ < )
16	14, 15	eqtri 2766	. . . . . 6 ⊢ (^◡ ≤ ∩ (𝐴 × 𝐴)) = (((ℝ^* × ℝ^*) ∩ (𝐴 × 𝐴)) ∖ < )
17		xpss12 5604	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐴 ⊆ ℝ^) → (𝐴 × 𝐴) ⊆ (ℝ^ × ℝ^*))
18	17	anidms 567	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → (𝐴 × 𝐴) ⊆ (ℝ^* × ℝ^*))
19		sseqin2 4149	. . . . . . . 8 ⊢ ((𝐴 × 𝐴) ⊆ (ℝ^* × ℝ^) ↔ ((ℝ^ × ℝ^*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
20	18, 19	sylib 217	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → ((ℝ^* × ℝ^*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
21	20	difeq1d 4056	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → (((ℝ^* × ℝ^*) ∩ (𝐴 × 𝐴)) ∖ < ) = ((𝐴 × 𝐴) ∖ < ))
22	16, 21	eqtr2id 2791	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → ((𝐴 × 𝐴) ∖ < ) = (^◡ ≤ ∩ (𝐴 × 𝐴)))
23	22	adantr 481	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → ((𝐴 × 𝐴) ∖ < ) = (^◡ ≤ ∩ (𝐴 × 𝐴)))
24		isoeq2 7189	. . . 4 ⊢ (((𝐴 × 𝐴) ∖ < ) = (^◡ ≤ ∩ (𝐴 × 𝐴)) → (𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵)))
25	23, 24	syl 17	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵)))
26	5	ineq1i 4142	. . . . . . 7 ⊢ ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ^* × ℝ^*) ∖ ^◡ < ) ∩ (𝐵 × 𝐵))
27		indif1 4205	. . . . . . 7 ⊢ (((ℝ^* × ℝ^) ∖ ^◡ < ) ∩ (𝐵 × 𝐵)) = (((ℝ^ × ℝ^*) ∩ (𝐵 × 𝐵)) ∖ ^◡ < )
28	26, 27	eqtri 2766	. . . . . 6 ⊢ ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ^* × ℝ^*) ∩ (𝐵 × 𝐵)) ∖ ^◡ < )
29		xpss12 5604	. . . . . . . . 9 ⊢ ((𝐵 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) → (𝐵 × 𝐵) ⊆ (ℝ^ × ℝ^*))
30	29	anidms 567	. . . . . . . 8 ⊢ (𝐵 ⊆ ℝ^* → (𝐵 × 𝐵) ⊆ (ℝ^* × ℝ^*))
31		sseqin2 4149	. . . . . . . 8 ⊢ ((𝐵 × 𝐵) ⊆ (ℝ^* × ℝ^) ↔ ((ℝ^ × ℝ^*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
32	30, 31	sylib 217	. . . . . . 7 ⊢ (𝐵 ⊆ ℝ^* → ((ℝ^* × ℝ^*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
33	32	difeq1d 4056	. . . . . 6 ⊢ (𝐵 ⊆ ℝ^* → (((ℝ^* × ℝ^*) ∩ (𝐵 × 𝐵)) ∖ ^◡ < ) = ((𝐵 × 𝐵) ∖ ^◡ < ))
34	28, 33	eqtr2id 2791	. . . . 5 ⊢ (𝐵 ⊆ ℝ^* → ((𝐵 × 𝐵) ∖ ^◡ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
35	34	adantl 482	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → ((𝐵 × 𝐵) ∖ ^◡ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
36		isoeq3 7190	. . . 4 ⊢ (((𝐵 × 𝐵) ∖ ^◡ < ) = ( ≤ ∩ (𝐵 × 𝐵)) → (𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
37	35, 36	syl 17	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
38	4, 25, 37	3bitrd 305	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
39		isocnv2 7202	. . 3 ⊢ (𝐹 Isom ^◡ ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ^◡^◡ ≤ , ^◡ ≤ (𝐴, 𝐵))
40		isores2 7204	. . . 4 ⊢ (𝐹 Isom ^◡ ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ^◡ ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
41		isores1 7205	. . . 4 ⊢ (𝐹 Isom ^◡ ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
42	40, 41	bitri 274	. . 3 ⊢ (𝐹 Isom ^◡ ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
43		lerel 11039	. . . . 5 ⊢ Rel ≤
44		dfrel2 6092	. . . . 5 ⊢ (Rel ≤ ↔ ^◡^◡ ≤ = ≤ )
45	43, 44	mpbi 229	. . . 4 ⊢ ^◡^◡ ≤ = ≤
46		isoeq2 7189	. . . 4 ⊢ (^◡^◡ ≤ = ≤ → (𝐹 Isom ^◡^◡ ≤ , ^◡ ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ^◡ ≤ (𝐴, 𝐵)))
47	45, 46	ax-mp 5	. . 3 ⊢ (𝐹 Isom ^◡^◡ ≤ , ^◡ ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ^◡ ≤ (𝐴, 𝐵))
48	39, 42, 47	3bitr3ri 302	. 2 ⊢ (𝐹 Isom ≤ , ^◡ ≤ (𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
49	38, 48	bitr4di 289	1 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ^◡ ≤ (𝐴, 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∖ cdif 3884 ∩ cin 3886 ⊆ wss 3887 × cxp 5587 ^◡ccnv 5588 Rel wrel 5594 Isom wiso 6434 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-xr 11013 df-ltxr 11014 df-le 11015

This theorem is referenced by: xrge0iifhmeo 31886

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