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Theorem gtiso 32715
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 2734 . . . . 5 ((𝐴 × 𝐴) ∖ < ) = ((𝐴 × 𝐴) ∖ < )
2 eqid 2734 . . . . 5 ((𝐵 × 𝐵) ∖ < ) = ((𝐵 × 𝐵) ∖ < )
31, 2isocnv3 7351 . . . 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 11298 . . . . . . . . . 10 ≤ = ((ℝ* × ℝ*) ∖ < )
65cnveqi 5887 . . . . . . . . 9 ≤ = ((ℝ* × ℝ*) ∖ < )
7 cnvdif 6165 . . . . . . . . 9 ((ℝ* × ℝ*) ∖ < ) = ((ℝ* × ℝ*) ∖ < )
8 cnvxp 6178 . . . . . . . . . 10 (ℝ* × ℝ*) = (ℝ* × ℝ*)
9 ltrel 11320 . . . . . . . . . . 11 Rel <
10 dfrel2 6210 . . . . . . . . . . 11 (Rel < ↔ < = < )
119, 10mpbi 230 . . . . . . . . . 10 < = <
128, 11difeq12i 4133 . . . . . . . . 9 ((ℝ* × ℝ*) ∖ < ) = ((ℝ* × ℝ*) ∖ < )
136, 7, 123eqtri 2766 . . . . . . . 8 ≤ = ((ℝ* × ℝ*) ∖ < )
1413ineq1i 4223 . . . . . . 7 ( ≤ ∩ (𝐴 × 𝐴)) = (((ℝ* × ℝ*) ∖ < ) ∩ (𝐴 × 𝐴))
15 indif1 4287 . . . . . . 7 (((ℝ* × ℝ*) ∖ < ) ∩ (𝐴 × 𝐴)) = (((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) ∖ < )
1614, 15eqtri 2762 . . . . . 6 ( ≤ ∩ (𝐴 × 𝐴)) = (((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) ∖ < )
17 xpss12 5703 . . . . . . . . 9 ((𝐴 ⊆ ℝ*𝐴 ⊆ ℝ*) → (𝐴 × 𝐴) ⊆ (ℝ* × ℝ*))
1817anidms 566 . . . . . . . 8 (𝐴 ⊆ ℝ* → (𝐴 × 𝐴) ⊆ (ℝ* × ℝ*))
19 sseqin2 4230 . . . . . . . 8 ((𝐴 × 𝐴) ⊆ (ℝ* × ℝ*) ↔ ((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
2018, 19sylib 218 . . . . . . 7 (𝐴 ⊆ ℝ* → ((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
2120difeq1d 4134 . . . . . 6 (𝐴 ⊆ ℝ* → (((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) ∖ < ) = ((𝐴 × 𝐴) ∖ < ))
2216, 21eqtr2id 2787 . . . . 5 (𝐴 ⊆ ℝ* → ((𝐴 × 𝐴) ∖ < ) = ( ≤ ∩ (𝐴 × 𝐴)))
2322adantr 480 . . . 4 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → ((𝐴 × 𝐴) ∖ < ) = ( ≤ ∩ (𝐴 × 𝐴)))
24 isoeq2 7337 . . . 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 4223 . . . . . . 7 ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ* × ℝ*) ∖ < ) ∩ (𝐵 × 𝐵))
27 indif1 4287 . . . . . . 7 (((ℝ* × ℝ*) ∖ < ) ∩ (𝐵 × 𝐵)) = (((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) ∖ < )
2826, 27eqtri 2762 . . . . . 6 ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) ∖ < )
29 xpss12 5703 . . . . . . . . 9 ((𝐵 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐵 × 𝐵) ⊆ (ℝ* × ℝ*))
3029anidms 566 . . . . . . . 8 (𝐵 ⊆ ℝ* → (𝐵 × 𝐵) ⊆ (ℝ* × ℝ*))
31 sseqin2 4230 . . . . . . . 8 ((𝐵 × 𝐵) ⊆ (ℝ* × ℝ*) ↔ ((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
3230, 31sylib 218 . . . . . . 7 (𝐵 ⊆ ℝ* → ((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
3332difeq1d 4134 . . . . . 6 (𝐵 ⊆ ℝ* → (((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) ∖ < ) = ((𝐵 × 𝐵) ∖ < ))
3428, 33eqtr2id 2787 . . . . 5 (𝐵 ⊆ ℝ* → ((𝐵 × 𝐵) ∖ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
3534adantl 481 . . . 4 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → ((𝐵 × 𝐵) ∖ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
36 isoeq3 7338 . . . 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 305 . 2 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
39 isocnv2 7350 . . 3 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵))
40 isores2 7352 . . . 4 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
41 isores1 7353 . . . 4 (𝐹 Isom ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
4240, 41bitri 275 . . 3 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
43 lerel 11322 . . . . 5 Rel ≤
44 dfrel2 6210 . . . . 5 (Rel ≤ ↔ ≤ = ≤ )
4543, 44mpbi 230 . . . 4 ≤ = ≤
46 isoeq2 7337 . . . 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 302 . 2 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
4938, 48bitr4di 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 206  wa 395   = wceq 1536  cdif 3959  cin 3961  wss 3962   × cxp 5686  ccnv 5687  Rel wrel 5693   Isom wiso 6563  *cxr 11291   < clt 11292  cle 11293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-xr 11296  df-ltxr 11297  df-le 11298
This theorem is referenced by:  xrge0iifhmeo  33896
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