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Theorem ismgmOLD 35651
Description: Obsolete version of ismgm 17969 as of 3-Feb-2020. The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
ismgmOLD.1 𝑋 = dom dom 𝐺
Assertion
Ref Expression
ismgmOLD (𝐺𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))

Proof of Theorem ismgmOLD
Dummy variables 𝑔 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 feq1 6485 . . . . 5 (𝑔 = 𝐺 → (𝑔:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑡 × 𝑡)⟶𝑡))
21exbidv 1928 . . . 4 (𝑔 = 𝐺 → (∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡 ↔ ∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡))
3 df-mgmOLD 35650 . . . 4 Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}
42, 3elab2g 3575 . . 3 (𝐺𝐴 → (𝐺 ∈ Magma ↔ ∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡))
5 f00 6560 . . . . . . . 8 (𝐺:(∅ × ∅)⟶∅ ↔ (𝐺 = ∅ ∧ (∅ × ∅) = ∅))
6 dmeq 5746 . . . . . . . . . 10 (𝐺 = ∅ → dom 𝐺 = dom ∅)
7 dmeq 5746 . . . . . . . . . . 11 (dom 𝐺 = dom ∅ → dom dom 𝐺 = dom dom ∅)
8 dm0 5763 . . . . . . . . . . . . 13 dom ∅ = ∅
98dmeqi 5747 . . . . . . . . . . . 12 dom dom ∅ = dom ∅
109, 8eqtri 2761 . . . . . . . . . . 11 dom dom ∅ = ∅
117, 10eqtr2di 2790 . . . . . . . . . 10 (dom 𝐺 = dom ∅ → ∅ = dom dom 𝐺)
126, 11syl 17 . . . . . . . . 9 (𝐺 = ∅ → ∅ = dom dom 𝐺)
1312adantr 484 . . . . . . . 8 ((𝐺 = ∅ ∧ (∅ × ∅) = ∅) → ∅ = dom dom 𝐺)
145, 13sylbi 220 . . . . . . 7 (𝐺:(∅ × ∅)⟶∅ → ∅ = dom dom 𝐺)
15 xpeq12 5550 . . . . . . . . . 10 ((𝑡 = ∅ ∧ 𝑡 = ∅) → (𝑡 × 𝑡) = (∅ × ∅))
1615anidms 570 . . . . . . . . 9 (𝑡 = ∅ → (𝑡 × 𝑡) = (∅ × ∅))
17 feq23 6488 . . . . . . . . 9 (((𝑡 × 𝑡) = (∅ × ∅) ∧ 𝑡 = ∅) → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(∅ × ∅)⟶∅))
1816, 17mpancom 688 . . . . . . . 8 (𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(∅ × ∅)⟶∅))
19 eqeq1 2742 . . . . . . . 8 (𝑡 = ∅ → (𝑡 = dom dom 𝐺 ↔ ∅ = dom dom 𝐺))
2018, 19imbi12d 348 . . . . . . 7 (𝑡 = ∅ → ((𝐺:(𝑡 × 𝑡)⟶𝑡𝑡 = dom dom 𝐺) ↔ (𝐺:(∅ × ∅)⟶∅ → ∅ = dom dom 𝐺)))
2114, 20mpbiri 261 . . . . . 6 (𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡𝑡 = dom dom 𝐺))
22 fdm 6513 . . . . . . . 8 (𝐺:(𝑡 × 𝑡)⟶𝑡 → dom 𝐺 = (𝑡 × 𝑡))
23 dmeq 5746 . . . . . . . 8 (dom 𝐺 = (𝑡 × 𝑡) → dom dom 𝐺 = dom (𝑡 × 𝑡))
24 df-ne 2935 . . . . . . . . . . . 12 (𝑡 ≠ ∅ ↔ ¬ 𝑡 = ∅)
25 dmxp 5772 . . . . . . . . . . . 12 (𝑡 ≠ ∅ → dom (𝑡 × 𝑡) = 𝑡)
2624, 25sylbir 238 . . . . . . . . . . 11 𝑡 = ∅ → dom (𝑡 × 𝑡) = 𝑡)
2726eqeq1d 2740 . . . . . . . . . 10 𝑡 = ∅ → (dom (𝑡 × 𝑡) = dom dom 𝐺𝑡 = dom dom 𝐺))
2827biimpcd 252 . . . . . . . . 9 (dom (𝑡 × 𝑡) = dom dom 𝐺 → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
2928eqcoms 2746 . . . . . . . 8 (dom dom 𝐺 = dom (𝑡 × 𝑡) → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
3022, 23, 293syl 18 . . . . . . 7 (𝐺:(𝑡 × 𝑡)⟶𝑡 → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
3130com12 32 . . . . . 6 𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡𝑡 = dom dom 𝐺))
3221, 31pm2.61i 185 . . . . 5 (𝐺:(𝑡 × 𝑡)⟶𝑡𝑡 = dom dom 𝐺)
3332pm4.71ri 564 . . . 4 (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ (𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡))
3433exbii 1854 . . 3 (∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ ∃𝑡(𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡))
354, 34bitrdi 290 . 2 (𝐺𝐴 → (𝐺 ∈ Magma ↔ ∃𝑡(𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡)))
36 dmexg 7634 . . 3 (𝐺𝐴 → dom 𝐺 ∈ V)
37 dmexg 7634 . . 3 (dom 𝐺 ∈ V → dom dom 𝐺 ∈ V)
38 xpeq12 5550 . . . . . . 7 ((𝑡 = dom dom 𝐺𝑡 = dom dom 𝐺) → (𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺))
3938anidms 570 . . . . . 6 (𝑡 = dom dom 𝐺 → (𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺))
40 feq23 6488 . . . . . 6 (((𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺) ∧ 𝑡 = dom dom 𝐺) → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺))
4139, 40mpancom 688 . . . . 5 (𝑡 = dom dom 𝐺 → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺))
42 ismgmOLD.1 . . . . . . . 8 𝑋 = dom dom 𝐺
4342eqcomi 2747 . . . . . . 7 dom dom 𝐺 = 𝑋
4443, 43xpeq12i 5553 . . . . . 6 (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋)
4544, 43feq23i 6498 . . . . 5 (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺𝐺:(𝑋 × 𝑋)⟶𝑋)
4641, 45bitrdi 290 . . . 4 (𝑡 = dom dom 𝐺 → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑋 × 𝑋)⟶𝑋))
4746ceqsexgv 3550 . . 3 (dom dom 𝐺 ∈ V → (∃𝑡(𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡) ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
4836, 37, 473syl 18 . 2 (𝐺𝐴 → (∃𝑡(𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡) ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
4935, 48bitrd 282 1 (𝐺𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1542  wex 1786  wcel 2114  wne 2934  Vcvv 3398  c0 4211   × cxp 5523  dom cdm 5525  wf 6335  Magmacmagm 35649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-fun 6341  df-fn 6342  df-f 6343  df-mgmOLD 35650
This theorem is referenced by:  clmgmOLD  35652  opidonOLD  35653  issmgrpOLD  35664
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