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Theorem gtiso 28649
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
StepHypRef Expression
1 eqid 2514 . . . . 5 ((𝐴 × 𝐴) ∖ < ) = ((𝐴 × 𝐴) ∖ < )
2 eqid 2514 . . . . 5 ((𝐵 × 𝐵) ∖ < ) = ((𝐵 × 𝐵) ∖ < )
31, 2isocnv3 6359 . . . 4 (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵))
43a1i 11 . . 3 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵)))
5 df-le 9835 . . . . . . . . . 10 ≤ = ((ℝ* × ℝ*) ∖ < )
65cnveqi 5111 . . . . . . . . 9 ≤ = ((ℝ* × ℝ*) ∖ < )
7 cnvdif 5348 . . . . . . . . 9 ((ℝ* × ℝ*) ∖ < ) = ((ℝ* × ℝ*) ∖ < )
8 cnvxp 5360 . . . . . . . . . 10 (ℝ* × ℝ*) = (ℝ* × ℝ*)
9 ltrel 9850 . . . . . . . . . . 11 Rel <
10 dfrel2 5392 . . . . . . . . . . 11 (Rel < ↔ < = < )
119, 10mpbi 218 . . . . . . . . . 10 < = <
128, 11difeq12i 3592 . . . . . . . . 9 ((ℝ* × ℝ*) ∖ < ) = ((ℝ* × ℝ*) ∖ < )
136, 7, 123eqtri 2540 . . . . . . . 8 ≤ = ((ℝ* × ℝ*) ∖ < )
1413ineq1i 3675 . . . . . . 7 ( ≤ ∩ (𝐴 × 𝐴)) = (((ℝ* × ℝ*) ∖ < ) ∩ (𝐴 × 𝐴))
15 indif1 3733 . . . . . . 7 (((ℝ* × ℝ*) ∖ < ) ∩ (𝐴 × 𝐴)) = (((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) ∖ < )
1614, 15eqtri 2536 . . . . . 6 ( ≤ ∩ (𝐴 × 𝐴)) = (((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) ∖ < )
17 xpss12 5041 . . . . . . . . 9 ((𝐴 ⊆ ℝ*𝐴 ⊆ ℝ*) → (𝐴 × 𝐴) ⊆ (ℝ* × ℝ*))
1817anidms 674 . . . . . . . 8 (𝐴 ⊆ ℝ* → (𝐴 × 𝐴) ⊆ (ℝ* × ℝ*))
19 sseqin2 3682 . . . . . . . 8 ((𝐴 × 𝐴) ⊆ (ℝ* × ℝ*) ↔ ((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
2018, 19sylib 206 . . . . . . 7 (𝐴 ⊆ ℝ* → ((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
2120difeq1d 3593 . . . . . 6 (𝐴 ⊆ ℝ* → (((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) ∖ < ) = ((𝐴 × 𝐴) ∖ < ))
2216, 21syl5req 2561 . . . . 5 (𝐴 ⊆ ℝ* → ((𝐴 × 𝐴) ∖ < ) = ( ≤ ∩ (𝐴 × 𝐴)))
2322adantr 479 . . . 4 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → ((𝐴 × 𝐴) ∖ < ) = ( ≤ ∩ (𝐴 × 𝐴)))
24 isoeq2 6345 . . . 4 (((𝐴 × 𝐴) ∖ < ) = ( ≤ ∩ (𝐴 × 𝐴)) → (𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵)))
2523, 24syl 17 . . 3 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵)))
265ineq1i 3675 . . . . . . 7 ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ* × ℝ*) ∖ < ) ∩ (𝐵 × 𝐵))
27 indif1 3733 . . . . . . 7 (((ℝ* × ℝ*) ∖ < ) ∩ (𝐵 × 𝐵)) = (((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) ∖ < )
2826, 27eqtri 2536 . . . . . 6 ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) ∖ < )
29 xpss12 5041 . . . . . . . . 9 ((𝐵 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐵 × 𝐵) ⊆ (ℝ* × ℝ*))
3029anidms 674 . . . . . . . 8 (𝐵 ⊆ ℝ* → (𝐵 × 𝐵) ⊆ (ℝ* × ℝ*))
31 sseqin2 3682 . . . . . . . 8 ((𝐵 × 𝐵) ⊆ (ℝ* × ℝ*) ↔ ((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
3230, 31sylib 206 . . . . . . 7 (𝐵 ⊆ ℝ* → ((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
3332difeq1d 3593 . . . . . 6 (𝐵 ⊆ ℝ* → (((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) ∖ < ) = ((𝐵 × 𝐵) ∖ < ))
3428, 33syl5req 2561 . . . . 5 (𝐵 ⊆ ℝ* → ((𝐵 × 𝐵) ∖ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
3534adantl 480 . . . 4 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → ((𝐵 × 𝐵) ∖ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
36 isoeq3 6346 . . . 4 (((𝐵 × 𝐵) ∖ < ) = ( ≤ ∩ (𝐵 × 𝐵)) → (𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
3735, 36syl 17 . . 3 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
384, 25, 373bitrd 292 . 2 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
39 isocnv2 6358 . . 3 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵))
40 isores2 6360 . . . 4 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
41 isores1 6361 . . . 4 (𝐹 Isom ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
4240, 41bitri 262 . . 3 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
43 lerel 9852 . . . . 5 Rel ≤
44 dfrel2 5392 . . . . 5 (Rel ≤ ↔ ≤ = ≤ )
4543, 44mpbi 218 . . . 4 ≤ = ≤
46 isoeq2 6345 . . . 4 ( ≤ = ≤ → (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
4745, 46ax-mp 5 . . 3 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵))
4839, 42, 473bitr3ri 289 . 2 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
4938, 48syl6bbr 276 1 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  cdif 3441  cin 3443  wss 3444   × cxp 4930  ccnv 4931  Rel wrel 4937   Isom wiso 5690  *cxr 9828   < clt 9829  cle 9830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-isom 5698  df-xr 9833  df-ltxr 9834  df-le 9835
This theorem is referenced by:  xrge0iifhmeo  29106
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