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Theorem 1t1e1ALT 42275
Description: Alternate proof of 1t1e1 12426 using a different set of axioms (add ax-mulrcl 11216, ax-i2m1 11221, ax-1ne0 11222, ax-rrecex 11225 and remove ax-resscn 11210, ax-mulcom 11217, ax-mulass 11219, ax-distr 11220). (Contributed by Steven Nguyen, 20-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
1t1e1ALT (1 · 1) = 1

Proof of Theorem 1t1e1ALT
StepHypRef Expression
1 1re 11259 . 2 1 ∈ ℝ
2 ax-1rid 11223 . 2 (1 ∈ ℝ → (1 · 1) = 1)
31, 2ax-mp 5 1 (1 · 1) = 1
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  (class class class)co 7431  cr 11152  1c1 11154   · cmul 11158
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-ext 2706  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-mulcl 11215  ax-mulrcl 11216  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rrecex 11225  ax-cnre 11226
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  nnmul1com  42285  remulinvcom  42439  sn-0tie0  42446
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