![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > 1t1e1ALT | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
1t1e1ALT | ⊢ (1 · 1) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 11259 | . 2 ⊢ 1 ∈ ℝ | |
2 | ax-1rid 11223 | . 2 ⊢ (1 ∈ ℝ → (1 · 1) = 1) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (1 · 1) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 (class class class)co 7431 ℝcr 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 |
Copyright terms: Public domain | W3C validator |