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Theorem infxp 10252
Description: Absorption law for multiplication with an infinite cardinal. Equivalent to Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
infxp (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))

Proof of Theorem infxp
StepHypRef Expression
1 sdomdom 9019 . . 3 (𝐵𝐴𝐵𝐴)
2 infxpabs 10249 . . . . . 6 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
3 infunabs 10244 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
433expa 1117 . . . . . . . 8 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
54adantrl 716 . . . . . . 7 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴𝐵) ≈ 𝐴)
65ensymd 9044 . . . . . 6 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → 𝐴 ≈ (𝐴𝐵))
7 entr 9045 . . . . . 6 (((𝐴 × 𝐵) ≈ 𝐴𝐴 ≈ (𝐴𝐵)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
82, 6, 7syl2anc 584 . . . . 5 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
98expr 456 . . . 4 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝐵 ≠ ∅) → (𝐵𝐴 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
109adantrl 716 . . 3 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐵𝐴 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
111, 10syl5 34 . 2 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐵𝐴 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
12 domtri2 10027 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
1312ad2ant2r 747 . . 3 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
14 xpcomeng 9103 . . . . . 6 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
1514ad2ant2r 747 . . . . 5 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
16 simplrl 777 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐵 ∈ dom card)
17 domtr 9046 . . . . . . . 8 ((ω ≼ 𝐴𝐴𝐵) → ω ≼ 𝐵)
1817ad4ant24 754 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → ω ≼ 𝐵)
19 infn0 9338 . . . . . . . 8 (ω ≼ 𝐴𝐴 ≠ ∅)
2019ad3antlr 731 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐴 ≠ ∅)
21 simpr 484 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐴𝐵)
22 infxpabs 10249 . . . . . . 7 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (𝐴 ≠ ∅ ∧ 𝐴𝐵)) → (𝐵 × 𝐴) ≈ 𝐵)
2316, 18, 20, 21, 22syl22anc 839 . . . . . 6 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐵 × 𝐴) ≈ 𝐵)
24 uncom 4168 . . . . . . . 8 (𝐴𝐵) = (𝐵𝐴)
25 infunabs 10244 . . . . . . . . 9 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵𝐴𝐵) → (𝐵𝐴) ≈ 𝐵)
2616, 18, 21, 25syl3anc 1370 . . . . . . . 8 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐵𝐴) ≈ 𝐵)
2724, 26eqbrtrid 5183 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐴𝐵) ≈ 𝐵)
2827ensymd 9044 . . . . . 6 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐵 ≈ (𝐴𝐵))
29 entr 9045 . . . . . 6 (((𝐵 × 𝐴) ≈ 𝐵𝐵 ≈ (𝐴𝐵)) → (𝐵 × 𝐴) ≈ (𝐴𝐵))
3023, 28, 29syl2anc 584 . . . . 5 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐵 × 𝐴) ≈ (𝐴𝐵))
31 entr 9045 . . . . 5 (((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ≈ (𝐴𝐵)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
3215, 30, 31syl2an2r 685 . . . 4 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
3332ex 412 . . 3 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴𝐵 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
3413, 33sylbird 260 . 2 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (¬ 𝐵𝐴 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
3511, 34pm2.61d 179 1 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wcel 2106  wne 2938  cun 3961  c0 4339   class class class wbr 5148   × cxp 5687  dom cdm 5689  ωcom 7887  cen 8981  cdom 8982  csdm 8983  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-dju 9939  df-card 9977
This theorem is referenced by:  alephmul  10616
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